Tosio Kato’s work on nonrelativistic quantum mechanics: part 1
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Abstract
We review the work of Tosio Kato on the mathematics of nonrelativistic quantum mechanics and some of the research that was motivated by this. Topics in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, selfadjointness results, and quadratic forms including monotone convergence theorems.
Keywords
Kato Schrödinger operators Quantum mechanicsMathematics Subject Classification
Primary 81Q10 81U05 47A55 Secondary 35Q40 46N50 81Q151 Introduction
Note: There are four pictures in this part and one picture in Part 2.
In 2017, we are celebrating the 100th anniversary of the birth of Tosio Kato (August 25, 1917–October 2, 1999). While there can be arguments as to which of his work is the deepest or most beautiful, there is no question that the most significant is his discovery, published in 1951, of the selfadjointness of the quantum mechanical Hamiltonian for atoms and molecules [314]. This is the founding document and Kato is the founding father of what has come to be called the theory of Schrödinger operators. So it seems appropriate to commemorate Kato with a comprehensive review of his work on nonrelativistic quantum mechanics (NRQM) that includes the context and later impact of this work.
One might wonder why I date this field only from Kato’s 1951 paper. After all, quantum theory was invented in 1925–1926 as matrix mechanics in Göttingen (by Heisenberg, Born and Jordan) and as wave mechanics in Zürich (by Schrödinger) and within a few years, books appeared on the mathematical foundations of quantum mechanics by two of the greatest mathematicians of their generation: Hermann Weyl [681] (not coincidentally, in Zürich; indeed the connection between Weyl and Schrödinger was more than professional—Weyl had a passionate love affair with Schrödinger’s wife) and John von Neumann [664] (von Neumann, whose thesis had been in logic, went to Göttingen to work with Hilbert on that subject, but was swept up in the local enthusiasm for quantum theory, in response to which, he developed the spectral theory of unbounded selfadjoint operators and his foundational work). One should also mention the work of Bargmann and Wigner (prior to Kato, summarized in [579] with references) on quantum dynamics. I think of this earlier work as first level foundations and the theory of Schrödinger operators as second level. Another way of explaining the distinction is that the Weyl–von Neumann work is an analog of setting up a formalism for classical mechanics like the Hamiltonian or Lagrangian while the theory initiated by Kato is the analog of celestial mechanics—the application of the general framework to concrete systems.
When I began this project I decided to write about all of Kato’s major contributions to the field in a larger context and this turned into a much larger article than I originally planned. As such, it is a review of a significant fraction of the work of the last 65 years on the mathematics of NRQM. Two important areas only touched on or totally missing are Nbody systems and the large N limit. Of course, Kato’s selfadjointness work includes Nbody systems, and there are papers on bound states in Helium and on properties of many body eigenfunctions. As we’ll see, his theory of smooth perturbations applies to give a complete spectral analysis of certain Nbody systems with only one scattering channel and is one tool in the study of general Nbody systems. But there is much more to the Nbody theory—for reviews, see [101, 116, 197, 212, 264]. Except for the 1972 work of Lieb–Simon on Thomas Fermi almost all the large N limit work is after 1980 when Kato mostly left the field; for recent reviews of different aspects of this subfield, see [51, 424, 425, 428, 429, 529, 551].
While this review will cover a huge array of work, it is important to realize it is only a fraction, albeit a substantial fraction, of Kato’s opus. I’d classify his work into four broad areas, NRQM, nonlinear PDE’s, linear semigroup theory and miscellaneous contributions to functional analysis. We will not give references to all this work. The reader can get an (almost) complete bibliography from MathSciNet or, for papers up to 1987, the dedication of the special issue of JMAA on the occasion of Kato’s 70th birthday [122] has a bibliography.
Around 1980, one can detect a clear shift in Kato’s interest. Before 1980, the bulk of his papers are on NRQM with a sprinkling in the other three areas while after 1980, the bulk are on nonlinear equations with a sprinkling in the other areas including NRQM. Kato’s nonlinear work includes looking at the Euler, Navier–Stokes, KdV and nonlinear Schrödinger equations. He was a pioneer in existence results—we note that his famous 1951 paper can be viewed as a result on existence of solutions for the time dependent linear Schrödinger equation! It is almost that when NRQM became too crowded with workers drawn by his work, he moved to a new area which took some time to become popular. Terry Tao said of this work: the Kato smoothing effect for Schrödinger equations is fundamental to the modern theory of nonlinear Schrödinger equations, perhaps second only to the Strichartz estimates in importance...Kato developed a beautiful abstract (functional analytic) theory for local well posedness for evolution equations; it is not used directly too much these days because it often requires quite a bit more regularity than we would like, but I think it was influential in inspiring more modern approaches to local existence based on more sophisticated function space estimates.
And here is what Carlos Kenig told me: T. Kato played a pioneering role in the study of nonlinear evolution equations. He not only developed an abstract framework for their study, but also introduced the tools to study many fundamental nonlinear evolutions coming from mathematical physics. Some remarkable examples of this are: Kato’s introduction of the “local smoothing effect” in his pioneering study of the Korteweg–de Vries equation, which has played a key role in the development of the theory of nonlinear dispersive equations.
Kato’s unified proof of the global wellposedness of the Euler and Navier–Stokes equations in 2d, which led to the development of the Beale–Kato–Majda blowup criterion for these equations. Kato’s works with Ponce on strong solutions of the Euler and Navier–Stokes equations, which developed the tools for the systematic application of fractional derivatives in the study of evolutions, which now completely permeates the subject. These contributions and many others, have left an indelible and enduring impact for the work of Kato on nonlinear evolutions.
The basic results on generators of semigroups on Banach spaces date back to the early 1950s going under the name Feller–Miyadera–Phillips and Hille–Yosida theorems (with a later 1961 paper of Lumer–Phillips). A basic book with references to this work is Pazy [475]. This is a subject that Kato returned to often, especially in the 1960s. Pazy [475] lists 19 papers by Kato on the subject. There is overlap with the NRQM work and the semigroup work. Perhaps the most important of these results are the Trotter–Kato theorems (discussed below briefly after Theorem 3.7) and the definition of fractional powers for generators of (not necessarily selfadjoint) semigroups. There are also connections between quantum statistical mechanics and contraction semigroup on operator algebras. To keep this review within bounds, we will not discuss this work.
Returning to the timing of Kato’s fundamental 1951 paper [314], I note that he was 34 when it was published (it was submitted a few years earlier as we’ll discuss in Sect. 7). Before it, his most important work was his thesis, awarded in 1951 and published in 1949–1951. One might be surprised at his age when this work was published but not if one understands the impact of the war. Kato got his BS from the University of Tokyo in 1941, a year in which he published two (not mathematical) papers in theoretical physics. But during the war, he was evacuated to the countryside. We were at a conference together one evening and Kato described rather harrowing experiences in the camp he was assigned to, especially an evacuation of the camp down a steep wet hill. He contracted TB in the camp. In his acceptance for the Wiener Prize [1], Kato says that his work on essential selfadjointness and on perturbation theory were essentially complete “by the end of the war.” Recently, several of Kato’s notebook were discovered dated 1945 that contain most of results published in Kato [314, 316] sometimes with different proofs from the later publications (these notes have recently been edited for publication in [358]).
In 1946, Kato returned to the University of Tokyo as an Assistant (a position common for students progressing towards their degrees) in physics, was appointed Assistant Professor of Physics in 1951 and full professor in 1958. I’ve sometimes wondered what his colleagues in physics made of him. He was perhaps influenced by the distinguished Japanese algebraic geometer, Kunihiko Kodaira (1915–1997) 2 years his senior and a 1954 Fields medalist. Kodaira got a BS in physics after his BA in mathematics and was given a joint appointment in 1944, so there was clearly some sympathy towards pure mathematics in the physics department. In 1948, Kato and Kodaira wrote a 2 page note [360] to a physics journal whose point was that every \(L^2\) wave function was acceptable for quantum mechanics, something about which there was confusion in the physics literature.
Beginning in 1954, Kato started visiting the United States. This bland statement masks some drama. In 1954, Kato was invited to visit Berkeley for a year, I presume arranged by F. Wolf. Of course, Kato needed a visa and it is likely it would have been denied due to his history of TB. Fortunately, just at the time (and only for a period of about a year), the scientific attaché at the US embassy in Tokyo was Otto Laporte (1902–1971) on leave from a professorship in Physics at the University of Michigan. Charles Dolph (1919–1994), a mathematician at Michigan, learned of the problem and contacted Laporte who intervened to get Kato a visa. Dolph once told me that he thought his most important contribution to American mathematics was his helping to allow Kato to come to the US. In 1987, in honor of Kato’s 70th birthday, there was a special issue of the Journal of Mathematical Analysis and Applications and the issue was jointly dedicated [122] to Laporte (he passed away in 1971) and Kato and edited by Dolph and Kato’s student Jim Howland.
During the mid 1950s, Kato spent close to 3 years visiting US institutions, mainly Berkeley, but also the Courant Institute, American University, National Bureau of Standards and Caltech. In 1962, he accepted a professorship in Mathematics from Berkeley where he spent the rest of his career and remained after his retirement. One should not underestimate the courage it takes for a 45 year old to move to a very different culture because of a scientific opportunity. That said, I’m told that when he retired and some of his students urged him to live in Japan, he said he liked the weather in Northern California too much to consider it. The reader can consult the Mathematics Genealogy Project (http://www.genealogy.ams.org/id.php?id=32842) for a list of Kato’s students (24 listed there, 3 from Tokyo and 21 from Berkeley; the best known are Ikebe and Kuroda from Tokyo and Balslev and Howland from Berkeley) and [98] for a memorial article with lots of reminisces of Kato.
One can get a feel for Kato’s impact by considering the number of theorems, theories and inequalities with his name on them. Here are some: Kato’s theorem (which usually refers to his result on selfadjointness of atomic Hamiltonians), the Kato–Rellich theorem (which Rellich had first), the Kato–Rosenblum theorem and the Kato–Birman theory (where Kato had the most significant results although, as we’ll see, Rosenblum should get more credit than he does), the Kato projection lemma and Kato dynamics (used in the adiabatic theorem), the Putnam–Kato theorem, the Trotter–Kato theorem (which is used for three results; see Sect. 3), the Kato cusp condition (see Sect. 19 in Part 2), Kato smoothness theory, the Kato class of potentials and Kato–Kuroda eigenfunction expansions. To me Kato’s inequality refers to the selfadjointness technique discussed in Sect. 9, but the term has also been used for the Hardy like inequality with best constant for \(r^{1}\) in three dimensions (which we discuss in Sect. 10), for a result on hyponormal operators that follows from Kato smoothness theory (the book [441] has a section called “Kato’s inequality” on it) and for the above mentioned variant of the Heinz–Loewner inequality for maximal accretive operators. There are also Heinz–Kato, Ponce–Kato and Kato–Temple inequalities. In [550], Erhard Seiler and I proved that if \(f,g \in L^p({\mathbb {R}}^\nu ),\, p\ge 2\), then \(f(X)g(i\nabla )\) is in the trace ideal \({\mathcal {I}}_p\). At the time, Kato and I had correspondence about the issue and about some results for \(p<2\). In [496], Reed and I mentioned that Kato had this result independently. Although Kato never published anything on the subject, in recent times, it has come to be called the Kato–Seiler–Simon inequality.
Of course, when discussing the impact of Kato’s work, one must emphasize the importance of his book Perturbation Theory for Linear Operators [345] which has been a bible for several generations of mathematicians. One of its virtues is its comprehensive nature. Percy Deift told me that Peter Lax told him that Friedrichs remarked on the book: “Oh, its easy to write a book when you put everything in it!”
We will not discuss every piece of work that Kato did in NRQM—for example, he wrote several papers on variational bounds on scattering phase shifts whose lasting impact was limited. And we will discuss Kato’s work on the definition of a selfadjoint Dirac Hamiltonian which of course isn’t nonrelativistic. It is closely related to the Schrödinger work and so belongs here. Perhaps I should have dropped “nonrelativistic” from the title but since almost all of Kato’s work on quantum theory is nonrelativistic and even the Dirac stuff is not quantum field theory, I decided to leave it.
Roughly speaking, this article is in five parts. Sections 2–6 discuss eigenvalue perturbation theory in both the analytic (where many of his results were rediscoveries of results of Rellich and SzNagy) and asymptotic (where he was the pioneer). There is a section on situations where either an eigenvalue is initially embedded in continuous spectrum or where as soon the perturbation is turned on the location of the spectrum is swamped by continuous spectrum (i.e. on the theory of QM resonances). There are a pair of sections on two issues that Kato studied in connection with eigenvalue perturbation theory: pairs of projections and on the Temple–Kato inequalities.
Next come four sections on selfadjointness. One focuses on the Kato–Rellich theorem and its applications to atomic physics, one on his work with Ikebe and one on what has come to be called Kato’s inequality. Finally his work on quadratic forms is discussed including his work on monotone convergence for forms. That will end Part 1.
Part 2 begins with two pioneering works on aspects of bound states—his result on nonexistence of positive energy bound states in certain two body systems and his paper on the infinity of bound states for Helium, at least for infinite nuclear mass.
Next four sections on scattering and spectral theory which discuss the Kato–Birman theory (trace class scattering), Kato smoothness, Kato–Kuroda eigenfunction expansions and the Jensen–Kato paper on threshold behavior.
Last is a set of three miscellaneous gems: his work on the adiabatic theorem, on the Trotter product formula and his pioneering look at eigenfunction regularity.
With apologies to those inadvertently left out, I’d like to thank a number of people for useful information Yosi Avron, Jan Dereziński, Pavel Exner, Rupert Frank, Fritz Gesztesy, Gian Michele Graf, Sandro Graffi, Vincenzo Grecchi, Evans Harrell, Ira Herbst, Bernard Helffer, Arne Jensen, Carlos Kenig, Toshi Kuroda, Peter Lax, Hiroshi Oguri, Sasha Pushnitski, Derek Robinson, Robert Seiringer, Heinz Siedentop, Israel Michael Sigal, Erik Skibsted, Terry Tao, Dimitri Yafaev and Kenji Yajima. The pictures here are all from the estate of Mizue Kato, Tosio’s wife who passed away in 2011. Her will gave control of the pictures to H. Fujita, M. Ishiguro and S. T. Kuroda. I thank them for permission to use the pictures and H. Okamoto for providing digital versions.
2 Eigenvalue perturbation theory, I: regular perturbations
This is the first of five sections on eigenvalue perturbation theory; this section deals with the analytic case. Section 3 begins with examples that delimit some of the possibilities when the analytic theory doesn’t apply and that section and the next discuss two sets of those examples after which there are two sections on related mathematical issues which are connected to the subject and where Kato made important contributions.
Eigenvalue perturbation theory in the case where the eigenvalues are analytic (aka regular perturbation theory or analytic perturbation theory) is central to Kato’s opus—it is both a main topic of his famous book on Perturbation Theory and the main subject of his thesis. We’ll begin this section by sketching the modern theory as presented in Kato’s book [345] or as sketched in Simon [616, Sections 1.4 and 2.3] (other book presentations include Baumgärtel [44], Friedrichs [174], Reed–Simon [497] and Rellich [511]). Then we’ll give a Kato–centric discussion of the history.
As a preliminary, we want to recall the theory of spectral projections for general bounded operators, A, on a Banach space, X. If the spectrum of A, \(\sigma (A)=\sigma _1\cup \sigma _2\) is a decomposition into disjoint closed sets, one can find a chain (finite sum and/or difference of contours), \(\Gamma \), so that if \(w(z,\Gamma )\) is the winding number about \(z \notin \Gamma \), (i.e. \(w(z,\Gamma ) = (2\pi i)^{1} \oint _{\zeta \in \Gamma } (\zeta z)^{1} d\zeta \)), then \(\Gamma \cap \sigma (A) = \emptyset \), \(w(z,\Gamma )=0\) or 1 for all \(z \in {\mathbb {C}}{\setminus }\Gamma \), \(w(z,\Gamma )=1\) for \(z \in \sigma _1\), and \(w(z,\Gamma )=0\) for \(z \in \sigma _2\) (see [613, Section 4.4]).
If \(\sigma _1 \cup \sigma _2\) is a decomposition, f can be taken to be 1 in a neighborhood of \(\sigma _1\) and 0 in a neighborhood of \(\sigma _2\). \(P_\lambda ^2 = P_\lambda \) is then a special case of his functional calculus result \((fg)(x)=f(x)g(x)\). In 1942–1943, this functional calculus was further developed in the United States by Dunford [125, 126], Lorch [437] and Taylor [636]. In his book, Kato calls (2.4) a Dunford–Taylor integral.
With this formalism out of the way, we can turn to sketch the theory of regular perturbations. For details see the book presentations of Kato [345, Chaps. II and VII], Reed–Simon [497, Chap XII] and Simon [616, Sections 1.4 and 2.3].
The set of early significant results include two theorems of Rellich [504, 505, 506, 507, 508, Part I]. If \(A(\beta )\) is selfadjoint (i.e. \(\Omega \) is invariant under complex conjugations and \(A(\bar{\beta })=A(\beta )^*)\), then \(\lambda (\beta )\) and \(P(\beta )\) are real analytic on \(\Omega \cap {\mathbb {R}}\), i.e. no fractional powers in \(\lambda (\beta )\) at points of \(S \cap {\mathbb {R}}\) and no polar singularities of \(P(\beta )\) there. The first comes from the fact that if a Puiseux series based at \(\beta _0 \in {\mathbb {R}}\) has a nontrivial fractional power term, then some branch must have nonreal values for some real values of \(\beta \) near \(\beta _0\) (interestingly enough, in his book, Kato [345] appeals to Butler’s theorem instead of using this simple argument of Rellich). The second relies on the fact that if \(P(\beta )\) has polar terms at \(\beta _0\), since there are only finitely many negative index terms, one has that \(\lim _{\beta \beta _0 \downarrow 0} P(\beta ) = \infty \) which is inconsistent with the fact that spectral projections for selfadjoint matrices are selfadjoint, so with norm 1.
Step 5 Regular Families of Closed Operators. For \(\beta \in \Omega \), a domain, we consider a family, \(A(\beta )\) of closed, densely defined (but not necessarily bounded) operators on a Banach space, X. We say that A is a regular family if, for every \(\beta _0 \in \Omega \), there is a \(z_0 \in {\mathbb {C}}\) and \(\epsilon >0\) so that for \(\beta \beta _0 < \epsilon \), we have that \(z_0 \notin \sigma (A(\beta ))\) and \(\beta \mapsto (A(\beta )z_0)^{1}\) is a bounded analytic function near \(\beta _0\). Kato [345, Section VII.1.2] has a more general definition that applies even to closed operators between two Banach spaces X and Y but he proves that it is equivalent to the above definition so long as \(X=Y\) and every \(A(\beta )\) has a nonempty resolvent set (which is no restriction if you want to consider isolated eigenvalues).
With this definition, all the eigenvalue perturbation theory for the bounded case carries over since \(\lambda _0\) is a discrete eigenvalue of \(A(\beta _0)\) if and only if \((\lambda _0z_0)^{1}\) is a discrete eigenvalue of \((A(\beta _0)z_0)^{1}\).
Step 6 Criteria for Regular Families. A type (A) family is a function, \(A(\beta )\), for \(\beta \in \Omega \), a region in \({\mathbb {C}}\), so that \(A(\beta )\) is a closed, densely defined operator on a Banach space, X, with domain \(D(A(\beta )) = {\mathcal {D}}\) independent of \(\beta \) and so that for all \(\varphi \in {\mathcal {D}}\) we have that \(\beta \mapsto A(\beta )\varphi \) is an analytic vector valued function. If \(A(\beta _0)\) has nonempty resolvent set, it is easy to see that \(A(\beta )\) is a regular family for \(\beta \) near \(\beta _0\). In particular, if the resolvent set is nonempty for all \(\beta \in \Omega \), then \(A(\beta )\) is a regular family on \(\Omega \).
Example 2.1
Kato was concerned with rigorous estimates on the radius of convergence, \(\rho \), of the power series for \(E_0(1/Z)\). He discussed this in his thesis and, in his book [345, Section VII.4.9], was able to show that \(\rho > 0.24\) and he noted that this didn’t cover the physically important cases \(1/Z = 1/2\), i.e, Helium (\(Z=2\)). In fact the case \(1/Z=1\) is also important because it describes the \(H^\) ion which is known to exist.
This completes our discussion of the theory of eigenvalue perturbation theory so we turn to some remarks on its history. Eigenvalue perturbation theory goes back to fundamental work of Lord Rayleigh on sound waves in 1897 [492, pp. 115–118] and [493] and by Schrödinger at the dawn of (new) quantum mechanics [544] and is often called Rayleigh–Schrödinger perturbation theory.
SzNagy followed up Rellich’s work in two papers published in 1947 and 1951 [454, 455] in which he treated the selfadjoint Hilbert space case and general closed operators on Banach spaces respectively. The first paper had a 1942 Hungarian language version [453]. He defined type (A) perturbations via (2.15). His main advance is to exploit the definition of spectral projections via (2.1). As a student of F. Riesz, this is not surprising. This was also the first place that it was proven (in the Hilbert space case) that two orthogonal projections, P and Q with \(PQ < 1\) are related via \(Q=UPU^{1}\) for a unitary which is analytic function of Q, i.e. he implemented Step 4 above.
Wolf [689] also extended the Nagy approach to the Banach space case is 1952. Perhaps the most significant aspect of this work is that it served eventually to introduce Kato to Wolf for Wolf was a Professor at Berkeley who was essential to recruiting Kato to come to Berkeley both in 1954 and 1962.
František Wolf (1904–1989) was a Czech mathematician who had a junior position at Charles University in Prague. Wolf had spent time in Cambridge and did some significant work on trigonometric series under the influence of Littlewood. When the Germans invaded Czechoslovakia in March 1938, he was able to get an invitation to Mittag–Leffler. He got permission from the Germans for a 3 weeks visa but stayed in Sweden! He was then able to get an instructorship at Macalester College in Minnesota. He made what turned out to be a fateful decision in terms of later developments. Because travel across the Atlantic was difficult, he took the transSiberian railroad across the Soviet Union and then through Japan and across the Pacific to the US. This was mid1941 before the US entered the war and made travel across the Pacific difficult.
Wolf stopped in Berkeley to talk with G. C. Evans (known for his work on potential theory) who was then department chair. Evans knew of Wolf’s work and offered him a position on the spot!! After the year he promised to Macalester, Wolf returned to Berkeley and worked his way up the ranks. In 1952, Wolf extended SzNagy’s work to the Banach space case. At about the same time Nagy himself did similar work and in so did Kato. While Wolf and Kato didn’t know of each other’s work, Wolf learned of Kato’s work and that led to his invitation for Kato to visit Berkeley.
Kato’s thesis dealt with both analytic and asymptotic perturbation theory (we’ll discuss the later in the next section). It appears that Kato found much of this in about 1944 without knowing about the work of Rellich or Nagy although he did know about Rellich by the time his thesis was written and he learned about the work of Nagy before the publication of the last of his early papers on perturbation theory [320, 322].
Interestingly enough, Kato’s first published work on the perturbation theory of eigenvalues [306] was a brief 1948 note with examples where the theory didn’t apply  these will be discussed in the next section (Examples 3.5, 3.6). His thesis was published in a university journal in full [316] in 1951 with parts published a year early in broader journals in both English [308, 309] and Japanese [310]. Two final early papers [320, 322] dealt with the Banach space case and with further results on asymptotic perturbation theory (discussed further in Sect. 6).
Many of the most significant results in Kato’s work on regular eigenvalue perturbation theory had been found (independently but) earlier by Rellich and Nagy. Kato’s work, especially if you include his book [345], was more systematic. His main contribution beyond theirs concerns the use of reduced resolvents. And, as we’ll see, he was the pioneer in the theory of asymptotic perturbation theory.
3 Eigenvalue perturbation theory, II: asymptotic perturbation theory
In this section and the next, we discuss situations where the Kato–Nagy–Rellich theory of regular perturbations does not apply. Lest the reader think this is a strange pathology, we begin with six (!) simple examples, four from the standard physics literature and then two that appeared in Kato’s first paper—a brief note—on perturbation theory [306].
Example 3.1
Example 3.2
(Autoionizing States of Two Electron Atoms) We further consider the Hamiltonian A(1 / Z) of Example 2.1; see (2.17). For \(1/Z = 0\), A(0) is the Hamiltonian of two uncoupled Hydrogen atoms so its eigenvalues are \(E_{n,m} = \tfrac{1}{4n^2}\tfrac{1}{4m^2}, \, m,n=1,2,\ldots \). The continuous spectrum starts at \(\tfrac{1}{4}\) (for \(n=1, m\rightarrow \infty \)), so, for example, \(E_{2,2}\) at energy \(\tfrac{1}{8}\) is an eigenvalue but not isolated, rather it is embedded in the continuous spectrum on \([\tfrac{1}{4},\infty )\). According to the physicist’s expectation, this eigenvalue becomes a decaying state, where in a finite time, one electron drops to the ground state and the other gets kicked out of the atom with the left over energy (i.e. \(\tfrac{1}{8}(\tfrac{1}{4})=\tfrac{1}{8}\)). For obvious reasons, these are called autoionizing states. These states are actually seen as electron scattering resonances (under \(e + He^+ \rightarrow e+He^+\)) or as photo ionization resonances (\(\gamma +He \rightarrow He^+ + e\)) called Auger resonances.
The situation has a complication we’ll ignore. The eigenvalue at energy \(\tfrac{1}{8}\) has multiplicity 16 which one can reduce by using exchange, rotation and parity symmetry. For our purposes, it is useful to look at states with angular momentum 2 and azimuthal angular momentum 2 which are simple. In fact, there are states of unnatural parity (with angular momentum 1 but parity +); the continuous spectrum below \(\tfrac{1}{16}\) is only of natural parity states so these unnatural parity eigenvalues are not embedded in continuous spectrum and so they don’t disappear. There are actually 15 subspaces with definite symmetry. In one, there is a doubly degenerate embedded eigenvalue, in 3 an isolated eigenvalue and in 11 a simple embedded eigenvalue.
So this example causes lots of problems we’ll look at in Sect. 4: What is a resonance? What does the perturbation series have to do with the resonance energy? Can one mathematically justify the Fermi golden rule? What are the higher terms? Is there a convergent series?
In 1948, Friedrichs [172] considered a model (related to some earlier work of his [170]) with operators acting on \(L^2([a,b],dx)\oplus {\mathbb {C}}\) with \(A_0(f(x),\zeta ) = (xf(x),\zeta )\) where \(a< 1 < b\) so that \(A_0\) has an embedded eigenvalue at \(E_0=1\). \(A(\beta )=A_0+\beta B\) where B is the rank two operator \(B(f(x),\zeta ) = (\zeta h(x), \langle h,f \rangle )\) for some \(h \in L^2([a,b],dx)\). For suitable h and small \(\beta > 0\), Friedrichs proved that \(A(\beta )\) has no eigenvalues in spite of the fact of a first order perturbation term so the eigenvalue indeed dissolves. He did not discuss resonances but this was an early attempt to study a model which in his words “is clearly related to the Auger effect.”
Example 3.3
In fact, one can prove that for any \(F \ne 0\), and any Z including \(Z=0\), A(F, Z) has spectrum \((\infty ,\infty )\) with infinite multiplicity, purely absolutely continuous spectrum. Titchmarsh [654] proved there are no embedded eigenvalues using the separability in parabolic coordinates we’ll use again below, Avron–Herbst [24] proved the existence of wave operators from \(A(F,Z=0)\) to A(F, Z) (wave operators are discussed in Sect. 13 in Part 2) and Herbst [240] proved that those wave operators were unitaries, U, with \(UA(F,Z=0)U^{1} = A(F,Z)\).
In this regard, I should mention what I’ve called [588] Howland’s Razor after [258, 259] and Occam’s Razor: “Resonances cannot be intrinsic to an abstract operator on a Hilbert space but must involve additional structure.” For \(\{A(F,1)\}_{F \ne 0}\) are all unitarily equivalent but we believe they have Fdependent resonance energies. We’ll discuss the possible extra structures in the next section.
Many of the same questions occur as for Example 3.2 which we’ll study in Sect. 4: What is a resonance? What is the meaning of the divergent perturbation series? What is the difference between (3.9) where \(\Gamma (\beta ) = \text {O}(\beta ^2)\) and (3.12) where \(\Gamma (\beta ) = \text {O}(\beta ^k)\) for all k.
Example 3.4
So far as I know, Kato never discussed anything like double wells in print, but we’ll see shortly that it illuminates the meaning of stability, a subject that Kato was the first to emphasize.
Example 3.5
In his first example, he takes B to be multiplication by x. This model is the poor man’s Stark effect. He doesn’t mention this connection in the paper but does in the thesis. He states without proof in the Note (but does have a proof in the thesis) that for \(\beta \ne 0\), \(A(\beta )\) has no eigenvalues but has a purely continuous spectrum. He remarked that this example shows that the formal perturbation series may be quite meaningless even if no “divergence” occurs. In his later work, as we’ll see in Sect. 4, he did discuss a possible significance of such series.
Example 3.6
The function \(g(\beta ) = 10^6 \exp (1/10^6 \beta )\) has a zero asymptotic series. \(f(\beta )\) and \(f(\beta )+g(\beta )\) thus have the same asymptotic series so an asymptotic series tells us nothing about the value, \(f(\beta _0)\), for a fixed \(\beta _0\). Typically however, for \(\beta _0\) small, a few terms approximate \(f(\beta _0)\) well but too many terms diverge. A good example is given [614, Table after (15.1.18)] for the error function \(\text {Erfc}(x) = \tfrac{2}{\sqrt{\pi }} \int _{x}^{\infty } \exp (y^2) dy\) for which \(h(x) \equiv \pi x \exp (x^2) \text {Erfc}(x)\) has an asymptotic series in 1 / x about \(x = \infty \). At \(x=10\), \(h(x) = .99507\ldots \). The order \(N=2\) asymptotic series is good to 5 decimal places and for \(N=108\) to more than 22 decimal places. But for \(N=1000\), the series is about \(10^{565}\). So it is interesting and important to know that a series is asymptotic but if one knows the series and wants to know f, it is disappointing not to know more.
One often considers \(A(\beta )\) defined in a truncated sector \(\{\beta \in {\mathbb {C}}\,\, 0<\beta < B, \arg \beta  <A\}\) and demands (3.25) (with \(\beta ^N\) in the error replaced by \(\beta ^N\)) in the whole sector.
In his thesis, Kato [316] only considered \(A(\beta ) = A_0+\beta B\) with \(A \ge 0, \,B \ge 0\) where \(A(\beta )\) is selfadjoint (with a suitable interpretation of the sum). He used what are now called Temple–Kato inequalities to obtain asymptotic series to all orders in [316, 322]. We discuss this approach in Sect. 6 below.
About the same time, Titchmarsh started a series of papers [647, 648, 649, 650, 651, 654] on eigenvalues of second order differential equations including asymptotic perturbation results for \(A(\beta ) = \tfrac{d^2}{dx^2}+V(x)+\beta W(x)\) on \(L^2({\mathbb {R}},dx)\) (or \(L^2((0,\infty ),dx)\) with a boundary condition at \(x=0\)). Typically both V(x) and W(x) go to infinity as \(x \rightarrow \infty \) (so the spectra are discrete) and W goes to \(\infty \) faster (so analytic perturbation theory fails; think \(V(x) = x^2, W(x) = x^4\)). His work relied heavily on ODE techniques. They have overlap of applicability with Kato’s operator theoretic approach, but Kato’s method is more broadly applicable.
In his book, Kato totally changed his approach to be able to say something about the Banach space (and also nonselfadjoint operators in Hilbert space) so he couldn’t use the Temple–Kato inequality which relies on the spectral theorem. There is some overlap of this work from his book and work of Huet [260], Kramer [388, 389], Krieger [392] and Simon [568].
Central to Kato’s approach is the notion of strong resolvent convergence and of stability. Kato often discusses this for sequences \(A_n\) converging to A in some sense as \(n \rightarrow \infty \); for our purposes here, it is more natural to consider \(A(\beta )\) depending on a positive real parameter as \(\beta \downarrow 0\). To avoid various technicalities, we’ll also focus initially on the selfadjoint case were there are a priori bounds on \((Bz)^{1}\) for \(z \in {\mathbb {C}}{\setminus }{\mathbb {R}}\), although we’ll consider some nonselfadjoint operators later.
For (possibly unbounded) selfadjoint \(\{A(\beta )\}_{0< \beta < B}\) and selfadjoint \(A_0\), we say that \(A(\beta )\) converges in strong resolvent sense (srs) if and only if for all \(z \in {\mathbb {C}}{\setminus }{\mathbb {R}}\), we have that \((A(\beta )z)^{1} \rightarrow (A_0z)^{1}\) in the strong (bounded) operator topology. Here is a theorem, going back to Rellich [504, 505, 506, 507, 508, Part 2] describing some results critical for asymptotic perturbation theory:
Theorem 3.7
 (a)
If \({\mathcal {D}}\subset {\mathcal {H}}\) is a dense subspace with \({\mathcal {D}}\subset D(A_0)\) and for all \(\beta \in (0,B), \, {\mathcal {D}}\subset D(A(\beta ))\), and if \({\mathcal {D}}\) is a core for \(A_0\) and for all \(\varphi \in {\mathcal {D}},\) we have that \(A(\beta )\varphi \rightarrow A_0\varphi \) as \(\beta \downarrow 0\), then \(A(\beta ) \rightarrow A_0\) in srs.
 (b)If \(a,b \in {\mathbb {R}}\) are not eigenvalues of \(A_0\) and \(A(\beta ) \rightarrow A_0\) in srs, thenwhere \(P_\Omega (B)\) is the spectral projection for B associated to the set \(\Omega \subset {\mathbb {R}}\) [616, Chapter 5 and Section 7.2]$$\begin{aligned} P_{(a,b)}(A(\beta )) \overset{s}{\rightarrow } P_{(a,b)}(A_0) \end{aligned}$$(3.26)
Proof
(a) follows from a simple use of the second resolvent formula; see [616, Theorem 7.2.11]. For (b), one first proves (3.26) when \(P_{(a,b)}\) is replaced by a continuous function [616, Theorem 7.2.10] and then approximates \(P_{(a,b)}\) with continuous functions [616, Problem 7.2.5]. \(\square \)
Remark
Before leaving the subject of abstract srs results, we should mention two results known as the Trotter–Kato theorem (Kato’s ultimate Trotter product formula, the subject of Sect. 18, is also sometimes called the Trotter–Kato theorem). One version says that if \(A_n\) and A are generators of contraction semigroups on a Banach space, X, then \(e^{tA_n} \overset{s}{\rightarrow } e^{tA}\) for all \(t>0\) if and only if for one (or for all) \(\lambda \) with \(\text {Re}\,(\lambda ) > 0\), one has \((A_n+\lambda )^{1} \overset{s}{\rightarrow } (A+\lambda )^{1}\). Related, sometimes part of the statement of the theorem, is that one doesn’t require A to exist a priori but only that for some \(\lambda \) in the open half plane that \((A_n+\lambda )^{1}\) have a strong limit whose range is dense. The basic theorem is then due to Trotter [655] in his thesis (written under the direction of Feller, whose interest in semigroups was motivated by Markov processes). Kato’s name is often on the theorem because he clarified an obscure point in this second version [331]. This theorem has also been called the Trotter–Kato–Neveu or Trotter–Kato–Neveu–Kurtz–Sova theorem after related contributions by these authors [402, 403, 466, 622]. There is another related result of this genre sometimes called the Trotter–Kato theorem. It says that if \(A_n\) is a family of selfadjoint operators, they have a srs limit for some A if and only if \((A_nz)^{1}\) has a strong limit with dense range for one z in \({\mathbb {C}}_+\) and one z in \({\mathbb {C}}_\).
Returning to perturbation theory, Kato introduced and developed the key notion of stability. Let \(\{A(\beta )\}_{0< \beta < B}\) (or \(\beta \) in a sector) be a family of closed operators in a Banach space, X. Let \(A_0\) be a closed operator so that as \(\beta \downarrow 0\), \(A(\beta )\) converges to \(A_0\) in some sense. Let \(E_0\) be an isolated, discrete, eigenvalue of \(A_0\). We say that \(E_0\) is stable if there exists \(\epsilon > 0\) so that \(\sigma (A_0) \cap \{z \,\, zE_0 \le \epsilon \} = \{E_0\}\) and so that
The second way one can have stability is illustrated by
Example 3.8
A similar argument works for \(p^2+\gamma x^2+\beta x^4\) for any \(\gamma \in \partial {\mathbb {D}}{\setminus } \{1\}\) so using scaling and the ideas below, one proves that for each n, the nth eigenvalue, \(E_n(\beta )\), of \(p^2+x^2+\beta x^4\) has an asymptotic series in each sector \(\{\beta \,\, 0<\beta <B_A; \arg \beta  < A\}\) so long as \(A \in (0,\tfrac{3\pi }{2})\) [568].
The above argument doesn’t work for \(\beta x^{2m}; \, m > 2\) but by using that \(\beta x^{2m}(p^2+x^2+\beta x^{2m}+1)^{1}\) is bounded, one sees that the norm of the difference of the resolvents is O\((\beta ^{1/m})\) which also goes to zero.
To state results on asymptotic series, we focus on getting series for all orders. Kato [345] is interested mainly in first and second order, so he needs much weaker hypotheses. Let \(C \ge 1\) be a selfadjoint operator on a Hilbert space, \({\mathcal {H}}\). Then \(D^\infty (C) \equiv \cap _{n \ge 0}D(C^n)\) is a countably normed Fréchet space with the norms \(\varphi _n \equiv C^n\varphi _{\mathcal {H}}\) (see [612, Section 6.1]). A densely defined operator, X, on \(D^\infty (C)\) is continuous in the Fréchet topology if and only if for all m, there is k(m) and \(c_m\) so that \(D^{k(m)}(C) \subset D(X), \, X\left[ D^{k(m)}(C)\right] \subset D^m(C)\) and \(X\varphi _m \le c_m \varphi _{k(m)}\). Typically, for some \(\ell \), k(m) can be chosen to be \(m+\ell \).
Theorem 3.9
Remarks
2. The set of algebraic terms obtained by the above proof are the same for asymptotic and analytic perturbation theory so the \(a_n\) are given by Rayleigh–Schrödinger perturbation theory.
3. Two useful choices for C are \(C=A_0+1\) and \(C=x^2+1\). For \(A_0=\tfrac{d^2}{dx^2}+x^2\), there are very good estimates on \((A_0+1)^m\varphi _0_2\) (see [612, Section 6.4]). If \(A_0 = \Delta +W+1\), for extremely general W’s, it is known that for \(z \notin \sigma (A_0)\), \((A_0z)^{1}\) has an integral kernel with exponential decay [600, Theorem B.7.1], which implies that \({(1+x^2)^m(A_0z)^{1}(1+x^2)^{m}}\) is bounded on \(L^2({\mathbb {R}})\), so \((A_0z)^{1}\) is bounded on \(D^\infty (1+x^2)\).
Asymptotic series have the virtue of uniquely determining the perturbation coefficients from the eigenvalues as functions and they often give good numeric results if \(\beta \) is small and one takes only a few terms. But mathematically, the situation is unsatisfactory—one would like the coefficients to uniquely determine \(E(\beta )\) (as they do in the regular case) or even better, one would like to have an algorithm to compute \(E(\beta )\) from \(\{a_n\}_{n=0}^\infty \). This is not an issue that Kato seems to have written about but it is an important part of the picture, so we will say a little about it.
It is a theorem of Carleman [82] that if \(\epsilon > 0\) and g is analytic in \(R_{\epsilon ,B} = \{z \,\, \arg z< \tfrac{\pi }{2}+ \epsilon ,\, 0< z < B\}\), if \(g(z) \le b_nz^n\) there and \(\sum _{n=1}^{\infty } b_n^{1/n} = \infty \) (e.g. \(b_n = n!\)), then \(g \equiv 0\) on \(R_{\epsilon ,B}\). This leads to a notion of strong asymptotic condition and an associated result of there being at most one function obeying that condition (and so a strong asymptotic series determines E)—see Simon [571, 572] or Reed–Simon [497, Section XII.4].
Algorithms for recovering a function from a possibly divergent series are called summability methods. Hardy [219] has a famous book on the subject. Many methods, such as Abel summability (i.e. \(\lim _{t \uparrow 1} \sum _{n=0}^{\infty } a_n t^n\)) work only for barely divergent series like \(a_n = (1)^n\). The series that arise in eigenvalue perturbation theory are usually badly divergent but, fortunately, there are some methods that work even in that case. Two that have been shown to work for suitable eigenvalue problems are Padé and Borel summability.
Theorem 3.10
If \(\{a_n\}_{n=0}^\infty \) is a series of Stieltjes, then for each \(j \in {\mathbb {Z}}\), the diagonal Padé approximates, \(f^{[N,N+j]}(z)\), converge as \(N \rightarrow \infty \) for all \(z \in {\mathbb {C}}{\setminus } [0,\infty )\) to a function \(f_j(z)\) given by (3.42) with \(\mu \) replaced by \(\mu _j\) which obeys (3.41) (with \(\mu =\mu _j\)). The \(f_j\) are either all equal or all different depending on whether (3.41) has a unique solution, \(\mu \), or not.
It follows from results of Loeffel et al. [434, 435] that if \(E_m(\beta )\) is an eigenvalue of \(p^2+x^2+\beta x^4\) for \(\beta \in [0,\infty )\), then \(E_m(\beta )\) has an analytic continuation to \({\mathbb {C}}{\setminus } [0,\infty )\) with a positive imaginary part in the upper half plane. Results of Simon [568] imply that \(E_m(\beta ) \le C(1+\beta )^{1/3}\). A Cauchy integral formula then implies that \((E_m(0)E_m(\beta ))/\beta \) has a representation of the form (3.42). Thus, by [435], the diagonal Padé approximates converge. Moreover, it is a fact (related to the above mentioned theorem of Carleman) that if \(\{a_n\}_{n=0}^\infty \) is the set of moments of a measure on \([0,\infty )\) with \(a_n \le CD^n(kn)!\) with \(k \le 2\), then the solution to the moment problem is unique [612, Problem 5.6.2]. This implies that for the \(x^4\) anharmonic oscillator, the diagonal Padé approximates converge to the eigenvalues. The same is true for the \(x^6\) oscillator but for the \(x^8\) oscillator, it is known (Graffi–Grecchi [200]) that, while the diagonal Padé approximates converge, they have different limits and none is the actual eigenvalue!
The key convergence result for Borel sums is a theorem of Watson [673]; see Hardy [219] for a proof:
Theorem 3.11
Graffi–Grecchi–Simon [204] proved that this theorem is applicable to the \(x^4\) anharmonic oscillator. They did numeric calculations making an unjustified use of Padé approximation to analytically continue g to all of \([0,\infty )\) and found more rapid convergence than Padé on the original series. By conformally mapping a subset of the union of \({\mathbb {D}}\) and \(\Lambda \) containing \([0,\infty )\) onto the disk, one can do the analytic continuation by summing a mapped power series and so do numerics without an unjustified Padé; see Hirsbrunner and Loeffel [249].
Avron–Herbst–Simon [25, 26, 27, 28, Part III] proved that for the Zeeman effect in arbitrary atoms, the perturbation series of the discrete eigenvalues is Borel summable. The Schwinger functions of various quantum field theories have been proven to have Borel summable Feynman perturbation series: \(P(\phi )_2\) [133], \(\phi ^4_3\) [438], \(Y_2\) [513, 514], \(Y_3\) [439].
In general, Padé summability is hard to prove because it requires global information, so it has been proven to work only in very limited situations (for example a higher dimensional quartic anharmonic oscillator is known to be Borel summable but nothing about Padé is known). Clearly, when it can be proven, Borel summability is an important improvement over the mere asymptotic series that concerned Kato.
4 Eigenvalue perturbation theory, III: spectral concentration
Titchmarsh proved that the Green’s kernel for h, originally defined for energies in \({\mathbb {C}}_+\), had a continuation onto the lower half plane with a pole near the discrete eigenvalues of h(g, z, f) and he identified the real part of the pole with perturbation theory up to second order. He conjectured that the imaginary part of the pole was exponentially small in 1 / f. He then showed in a certain sense that the spectrum of \(h(g,z,f \ne 0)\) as \(f \downarrow 0\) concentrated near the real parts of his poles [647, 648, 649, 650, 651, Part V].
Kato discussed things in terms of what he called pseudoeigenvalues and pseudoeigenvectors. He later realized that these notions imply a concentration of spectrum like that used by Titchmarsh. In his book [345], he emphasized what he formally defined as spectral concentration and linked the two approaches. In this section, I’ll begin by defining spectral concentration and then prove, following Kato, that it is implied by the existence of pseudoeigenvectors. Finally, I’ll discuss the complex scaling theory of resonances and how it extends and illuminates the theory of spectral concentration.
Consider first the case where \(A(\beta )\) converges to \(A_0\) as \(\beta \downarrow 0\) in srs and \(E_0\) is a discrete simple eigenvalue of \(A_0\). Let T be a closed interval with \(\sigma (A_0) \cap T = \{E_0\}\). By Theorem 3.7, for any \(\epsilon > 0\), we have that \(P_{T{\setminus } (E_0\epsilon ,E_0+\epsilon )} (A(\beta )) \overset{s}{\rightarrow } 0\). Thus, in a sense, the spectrum of \(A(\beta )\) in T is concentrated near \(E_0\). In the above, if we could replace \((E_0\epsilon ,E_0+\epsilon )\) by \((E_0+a_1\beta \beta ^{3/2},E_0+a_1\beta + \beta ^{3/2})\), we’d be able to claim that the spectrum was concentrated near \(E_0+a_1\beta \) in a way that would determine \(a_1\).
Taking into account that we may want to also have T shrink in cases like Example 3.2, we make the following definition. Let \(T(\beta ), S(\beta )\) be Borel sets in \({\mathbb {R}}\) given for \(0< \beta < B\) so that if \(0< \beta ' < \beta \), then \(T(\beta ') \subset T(\beta ), S(\beta ') \subset S(\beta )\) and so that for all \(\beta \), \(S(\beta ) \subset T(\beta )\). We say that the spectrum of \(A(\beta )\) in \(T(\beta )\) is asymptotically concentrated in \(S(\beta )\) if and only if \(P_{T(\beta ){\setminus } S(\beta )} \overset{s}{\rightarrow } 0\).
If \(E_0\) is a simple eigenvalue of \(A_0\) and \(\{a_j\}_{j=1}^N\) are real numbers, we say the spectrum near \(E_0\) is asymptotically concentrated near \(E_0+\sum _{j=0}^{N} a_j \beta ^j\) if there exist positive functions f and g obeying \(f(\beta ) \rightarrow 0,\, f(\beta )/\beta \rightarrow \infty ,\, g(\beta )/\beta ^N \rightarrow 0\) as \(\beta \downarrow 0\) so that the spectrum of \(A(\beta )\) in \((E_0f(\beta ),E_0+f(\beta ))\) is asymptotically concentrated in \((E_0+\sum _{j=0}^{N} a_j \beta ^j g(\beta ), E_0+\sum _{j=0}^{N} a_j \beta ^j + g(\beta ))\). It is easy to see if that happens, it determines the \(a_j,\, j=1,\ldots ,n\).
Theorem 4.1
Remarks
 1.
Riddell also has a converse.
 2.
Both papers consider the situation where \(E_0\) has multiplicity \(k < \infty \) and there are k orthonormal pairs obeying (4.3) and they prove spectral concentration on a union of k intervals of size \(\text {o}(\beta ^N)\) about the \(\lambda _j\).
 3.The proof isn’t hard. One picks \(g(\beta ) = \text {o}(\beta ^N)\) so that \((A(\beta )\lambda (\beta ))\varphi (\beta )/g(\beta ) \rightarrow 0\). This implies that if \(Q(\beta ) = P_{(\lambda (\beta )g(\beta ),\lambda (\beta )+g(\beta ))}(A(\beta ))\), then \((1Q(\beta ))\varphi (\beta ) \rightarrow 0\). By (4.4), this implies thatIf \(d < {\mathrm{dist}}(E_0,\sigma (A){\setminus }\{E_0\})\), Theorem 3.7 implies that \(P_{(E_0d,E_0+d)}(A(\beta ))\psi \rightarrow P_{E_0}(A_0)\psi \) for any \(\psi \). Thus by (4.5), \(\left[ P_{(E_0d,E_0+d)}(A(\beta ))Q(\beta )\right] \psi \rightarrow 0\) which is the required spectral concentration$$\begin{aligned} Q(\beta )P_{E_0}(A_0) \rightarrow 0 \end{aligned}$$(4.5)
These ideas were used by Friedrichs and Rejto [175] to prove spectral concentration in Example 3.5 (i.e. \(A_0\) of rank 1 and B multiplication by x). They assumed the function \(\psi (x)\) of (3.18) is strictly positive on \({\mathbb {R}}\) and Hölder continuous and prove that \(A(\beta )\) has no point eigenvalues and has a weak spectral concentration (of order \(\beta ^p\) for some \(0<p<1\)). Riddell [515] proved spectral concentration to all orders for the Stark effect for Hydrogen using pseudoeigenvectors and Rejto [501, 502] proved the analog for Helium (see below for more on spectral concentration for the Stark effect).
As noted Titchmarsh related spectral concentration to second sheet poles of Green’s functions for certain differential operators. This theme was developed by James Howland, a student of Kato, in 5 papers [255, 256, 257, 258, 259]. Howland discussed two situations. One was where \(A_0\) was finite rank and whose nonzero eigenvalues are washed away much like Example 3.5. The other was where \(A_0\) has eigenvalues embedded in continuous spectrum and B is finite rank, so related to the Friedrichs model mentioned at the end of Example 3.2.
In both cases, there is a finite dimensional space, \({\mathcal {V}}\), where the finite rank operator lives and Howland considered \({\{\langle \varphi ,(A(\beta )z)^{1}\psi \rangle ,\, \varphi ,\psi \in {\mathcal {V}}\}}\) and proved (under suitable conditions) that these functions initially defined on \({\mathbb {C}}_+\) have meromorphic continuations through \({\mathbb {R}}\) into a neighborhood of \(E_0\), a finite multiplicity eigenvalue of \(A_0\). These continuations had second sheet poles at \(E_j(\beta )\) converging as \(\beta \downarrow 0\) to \(E_0\). The number of poles is typically the multiplicity of \(E_0\) as an eigenvalue of \(A_0\).
In the case where \(A_0\) has a discrete eigenvalue, Howland showed that \(\text {Im}\,E(\beta ) = \text {O}(\beta ^\ell )\) for all \(\ell \) and was able to use this to prove spectral concentration to all orders. But in cases where \(A_0\) had an embedded eigenvalue, it was typically true that \(\text {Im}\, E(\beta ) = a_k \beta ^k + \text {o}(\beta ^k)\) for some k and some \(a_k < 0\); indeed Howland often proved a Fermi golden rule with \(a_2 \ne 0\). In that case, he showed there was spectral concentration of order \(k1\) but not k so spectral concentration couldn’t specify a perturbation series to all orders.
Howland also discovered that even when \(A_0\) and B were selfadjoint, an eigenvalue could turn into a second order pole whose perturbation series could have nontrivial fractional power series in the asymptotic expression, i.e. Rellich’s theorem fails for resonance energies.
Howland also introduced what I’ve called Howland’s razor (see the discussion of Example 3.3) and he gave one possible answer: it often happened that the embedded eigenvalue turned into a resonance, i.e. second sheet pole, for real values of \(\beta \) but for suitable complex \(\beta \), it was a pole in \({\mathbb {C}}_+\) and so a normal discrete eigenvalue of \(A(\beta )\). Thus the resonance energy could be interpreted as the analytic continuation of a perturbed eigenvalue.
Perhaps the most successful approach to the study of resonances, one that handles problems in atomic physics like Examples 3.2 and 3.3, is the method of complex scaling, initially called dilation or dilatation analyticity (the name change to complex scaling was by quantum chemists when they took up the method for numerical calculation of molecular resonances). The idea appeared initially in a technical appendix of a never published note by J. M. Combes who realized the potential of this idea and then published papers with coauthors: Aguilar–Combes [6] on the two body problem and Balslev–Combes [42] on Nbody problems (Eric Balslev was Kato’s first Berkeley student); see Simon [573] for extensions and simplifications and [497, Sections XIII.10 and XII.6] for a textbook presentation. Combes and collaborators knew that the formalism, which they used to prove the absence of singular continuous spectrum, provided a possible definition of a resonance. It was Simon [574] who realized that the formalism was ideal for studying eigenvalues embedded in the continuous spectrum like autoionizing states. We will not discuss an extension needed for molecules in the limit of infinite nuclear masses where one uses exterior complex scaling or a close variant, see Simon [594], Hunziker [263] and Gérard [187].
Discrete eigenvalues are given by analytic functions, \(E_j(\theta )\). Since changing \(\text {Re}\,\theta \) provides unitarily equivalent H’s, \(E_j(\theta )\) is constant under changes of \(\text {Re}\,\theta \), so constant by analyticity. We conclude that so long as discrete eigenvalues avoid \(S_\theta \), they remain discrete eigenvalues of \(H(\theta )\). In particular, negative eigenvalues of H are eigenvalues of \(H(\theta )\) if \(\text {Im}\,\theta  < \tfrac{\pi }{2}\). An additional argument shows that embedded positive eigenvalues become discrete eigenvalues of \(H(\theta )\) for \(\text {Im}\,\theta \in (0,\tfrac{\pi }{2})\).
By this persistence, \(H(\theta )\) for \(\theta \) with \(\text {Im}\,\theta \in (0,\tfrac{\pi }{2})\), there can’t be any eigenvalues in \(\{z\,\, \arg z \in (0,2\pi 2\text {Im}\,\theta ){\setminus }\{\pi \}\}\) (for taking \(\theta \) back to zero would result in nonreal eigenvalues of H) but there isn’t any reason there can’t be for z with \(\arg z \in (2\text {Im}\,z,0)\). That is, moving \(\text {Im}\,\theta \) can uncover eigenvalues in \({\mathbb {C}}_\) which we interpret as resonances (but see the discussion below).
Example 3.2 revisited. (following [574]) The thresholds are \(\left\{ \tfrac{1}{4n^2}\right\} _{n=1}^\infty \) so the eigenvalue at \(E_{2,2} = \tfrac{1}{8}\) is not a threshold. Thus it is an isolated eigenvalue of \(A(1/Z,0,\theta )\) if \(i\theta \in (0,\tfrac{\pi }{2})\). It follows that the Kato–Rellich theory applies so, for 1 / Z small, there is an eigenvalue, \(E_{2,2}(1/Z,\theta )\) independent of \(\theta \) (although it is only an eigenvalue if \(\arg (E_{2,2}(1/Z)+\tfrac{1}{4}) < \text {Im}\,\theta \). This first implies there is a convergent perturbation series (i.e. time–dependent perturbation theory, suitably defined, converges). One can compute the perturbation coefficients which are \(\theta \) independent for \(i\theta \in (0,\tfrac{\pi }{2})\) and then take \(i\theta \) to 0. One gets a suitable limit of \((V\varphi ,SV\varphi )\) where S is a reduced resolvent. Using the fact that the distribution limit of \(1/(x+i\epsilon )\) is \({\mathcal {P}}\left( \tfrac{1}{x}\right) i\pi \delta (x)\), Simon [574] computed \(\text {Im}\, a_2\) as given by the Fermi golden rule.
In spite of this accepted wisdom, a quantum chemist, Bill Reinhardt, did calculations for the Stark problem using complex scaling [498] and got sensible results. Motivated by this, Herbst [241] was able to define complex scaling for a class of two body Hamiltonians including the Hydrogen Stark problem. He discovered that for \(F \ne 0\), and \(0< \arg \theta < \pi /3\), \(H_0(\theta ,F)\) has empty spectrum (!), i.e. \((H_0(\theta ,F)  z)\) is invertible for all z. It is a theorem that elements in Banach algebras and, in particular, bounded operators on any Banach space, have nonempty spectrum but that is only for bounded operators. In some sense, \(H_0(\theta ,F)\) has only \(\infty \) in its spectrum—specifically \(\sigma [(H_0(\theta ,F)z)^{1}] = \{0\}\) for all z.
Herbst–Simon [245] studied the analytic properties of \(E(F,Z,\theta )\) and proved analyticity for \(F^2 \in \{z \,\, z < R\} \cap ({\mathbb {C}}{\setminus } (\infty ,0])\) and used this to prove Borel summability that recovers \(E(F,Z,\theta )\) directly for \(\text {Re}\,(F^2)>0\) (which doesn’t include any real F). The physical value is then determined by analytic continuation. Graffi–Grecchi [198] had proven Borel summability slightly earlier using very different methods. Graffi–Grecchi [202] and Herbst–Simon [245] also proved Borel summability for discrete eigenvalues of general atoms.
For Hydrogen, Herbst–Simon conjectured (3.14) noting that it was implied by their analyticity results and the then unproven Oppenheimer formula. Shortly thereafter, Harrell–Simon [222] proved the Oppenheimer formula for the complex scaled defined Stark resonance and so also (3.14). They used similar arguments to prove the Bender–Wu formula for the anharmonic oscillator. Later Helffer–Sjöstrand [234] proved Bender–Wu formulae for higher dimensional oscillators.
We have not discussed in detail various subtleties that are dealt with in the quoted papers: among them, Herbst [241] showed that \(A(F,Z,\theta )\) is of type(A) with domain \(D(\Delta )\cap D(z)\) on \(\{(F,Z,\theta ) \,\, F>0, \text {Im}\,\theta \in (0,\pi /3)\}\) by proving a quadratic estimate. The proof of stability of the eigenvalues of \(A(F=0,Z,\theta )\) for \(\text {Im}\,\theta \in (0,\pi /3)\) uses ideas from [25, 26, 27, 28, Part I]. While the free Stark problem has scaled Hamiltonians with empty spectrum when there is one positive charge and N particles of equal mass and equal negative charge, there are charges and masses, where the spectrum is not empty.
Sigal [559, 560, 561, 562] and Herbst–Møller–Skibsted [242] have further studied Stark resonances in multielectron atoms proving that the widths are strictly positive and exponentially small in 1 / F.
We end this discussion by noting that I have reason to believe that, at least at one time, Kato had severe doubts about the physical relevance of the complex scaling approach to resonances. [222] was rejected by the first journal it was submitted to. The editor told me that the world’s recognized greatest expert on perturbation theory had recommended rejection so he had no choice. I had some of the report quoted to me. The referee said that the complex scaling definition of resonance was arbitrary and physically unmotivated with limited significance.
There is at least one missing point in a reply to this criticism: however it is defined, a resonance must correspond to a pole of the scattering amplitude. While this is surely true for resonances defined via complex scaling, as of this day, it has not been proven for the models of greatest interest. So far, resonance poles of scattering amplitudes in quantum systems have only been proven for two and three cluster scattering with potentials decaying faster (often much faster) than Coulomb and not for Stark scattering; see Babbitt–Balslev [33], Balslev [39, 40, 41], Hagedorn [214], Jensen [285] and Sigal [555, 558]. This is a technically difficult problem which hasn’t drawn much attention. That said, following [222] and others, we note the following in support of the notion that eigenvalues of \(H(\theta )\) that lie in \({\mathbb {C}}_\) are resonances:
(1) Going back to Titchmarsh [647, 648, 649, 650, 651, 654], poles of the diagonal (i.e. \(x=y\)) Green’s function (integral kernel, G(x, y; z) of \((Hz)^{1})\) are viewed as resonances for one dimensional problems. In dimension \(\nu \ge 2\), G(x, y; z) diverges as \(x \rightarrow y\) so it is natural to consider poles of \(\langle \varphi ,(Hz)^{1}\varphi \rangle \). Howland’s razor implies that you can’t look at all \(\varphi \in L^2({\mathbb {R}}^\nu , d^\nu x)\) but a special class of functions which are smooth in x and p space would be a reasonable replacement for \(x=y\). One can show (see [497, Section XIII.10]) that if \(\varphi \) is a polynomial times a Gaussian, then \(\langle \varphi ,(Hz)^{1}\varphi \rangle \) has a meromorphic continuation across \({\mathbb {R}}\) between thresholds with poles exactly at the eigenvalues of \(H(\theta )\).
(2) In the autoionizing case, E is an analytic function of 1 / Z and in the Stark case, analytic for \(F^2\) in a cut disk about 0. For the physically relevant values, 1 / Z real or F real, E has \(\text {Im}\,E < 0\) and these resonances are on the second sheet and disappear at \(\theta =0\). But for 1 / Z or F pure imaginary, the corresponding E is in \({\mathbb {C}}_+\) and so persists when \(\text {Im}\,\theta \downarrow 0\), i.e. E for these unphysical values of the parameters is an eigenvalue of these corresponding H. Thus resonances can be viewed as analytic continuations of actual eigenvalues from unphysical to physical values of the parameters.
(3) It is connected to the sum or Borel sum of a suitable perturbation series, see [78, 79].
(4) It yields information on asymptotic series and spectral concentration in a particularly clean way and, in particular, a proof of a Bender–Wu type formula for the asymptotics of the perturbation coefficients in the Stark problem.
While we’ve focused on the complex scaling approach to resonances, there are other methods. One called distortion analyticity works sometimes for potentials which are the sum of a dilation analytic potential and a potential with exponential decay (but not necessarily any xspace analyticity). The basic papers include Jensen [285], Sigal [557], Cycon [100], and Nakamura [456, 457]. Some approaches for nonanalytic potentials include Cattaneo–Graf–Hunziker [85], Cancelier–Martinez–Ramond [80] and Martinez–Ramond–Sjöstrand [443]. There is an enormous literature on the theory of resonances from many points of view. It would be difficult to attempt a comprehensive discussion of this literature and given that the subject is not central to Kato’s work, I won’t even try. But I should mention a beautiful set of ideas about counting asymptotics of resonances starting with Zworski [714]; see Sjöstrand [619] for unpublished lectures that include lots of references, a recent review of Zworski [715] and forthcoming book of Dyatlov–Zworski [127]. The form of the Fermi Golden Rule at Thresholds is discussed in Jensen–Nenciu [290] (see Sect. 16). A review of the occurrence of resonances in NR Quantum Electrodynamics and of the smooth Feshbach–Schur map is Sigal [563] and a book on techniques relevant to some approaches to resonances is Martinez [442].
5 Eigenvalue perturbation theory, IV: pairs of projections
Recall [616, Section 2.1] that a (bounded) projection on a Banach space, X, is a bounded operator with \(P^2=P\). If \(Y = {\mathrm{ran}}(P)=\ker (1P)\) and \(Z = {\mathrm{ran}}(1P)=\ker (P)\), then Y and Z are disjoint closed subspaces and \(Y+Z=X\) and that \((y,z) \mapsto y+z\) is a Banach space linear homeomorphism of \(Y \oplus Z\) and X. There is a oneone correspondence between such direct sum decompositions and bounded projections. We saw in Sect. 2 that the following is important in eigenvalue perturbation theory:
Theorem 5.1
 (a)
For P fixed, U(P, Q) is analytic in Q in that it is a norm limit, uniformly in each ball \(\{Q\,\,PQ < 1\epsilon \}\), of polynomials in Q.
 (b)
If X is a Hilbert space and P, Q are selfadjoint projections, then U is unitary.
Remarks
 1.
We don’t require \(U(P,P) = {\varvec{1}}\) which might seem natural because, below, when P and Q are selfadjoint, we’ll find a U for which (5.19) holds and it can be shown that is inconsistent with \(U(P,P) = {\varvec{1}}\). Of course, given any \(U_0(P,Q)\) obeying (5.1), \(U(P,Q)=U_0(P,Q)U_0(P,P)^{1}\) also obeys (5.1) and has \(U(P,P)={\varvec{1}}\) so it is no great loss. Both the U’s we construct below also obey \(U(Q,P) = U(P,Q)^{1}\).
 2.
U is actually jointly analytic in P, Q and the proof easily implies if P is fixed and \(\beta \mapsto Q(\beta )\) is analytic (resp. continuous, \(C^k\), \(C^\infty \)) in \(\beta \), then so is U.
First Proof of Theorem 5.1
Since \((1A^2)^{1/2}\) is a norm limit of polynomials in P and Q, so is U proving (a). If X is a Hilbert space and \(P^*=P, Q^*=Q\), then \(\widetilde{U} = U^*\), so by (5.15) U is unitary, proving (b). \(\square \)
Theorem 5.1 for the selfadjoint Hilbert space case goes back to SzNagy [454] who was interested in the result because of its application to the convergent perturbation theory of eigenvalues. His formula for U looks more involved than (5.2)/(5.14). Wolf [689] then extended the result to general Banach spaces but needed \(P^2PQ < 1\) and \(1P^2PQ < 1\) which is a strictly stronger hypothesis.
In [320], Kato proved that if \(\beta \mapsto P(\beta )\) is a real analytic family of projections on a Banach space for \(\beta \in [0,B]\), then there exists a real analytic family of invertible maps, \(U(\beta )\) so that \(U(\beta )P(\beta )U(\beta )^{1}=P(0)\). He did this using the same formalism he had developed for his treatment of the adiabatic theorem (Kato [313] and Sect. 17 in Part 2). In 1955, in an unpublished report [324], Kato presented all of the algebra above (except for \(AB+BA=0\)) and used it to prove Theorem 5.1 exactly as we do above.
After Avron et al [31] found and exploited \(AB+BA=0\) (see below), Kato told me that he had found this relation about 1972 but didn’t have an application. Because [324] isn’t widely available, the standard reference for his approach to pairs of projections is his book [345]. In [324], Kato noted that his expression was equal to the object found by SzNagy [454] but in the Banach space case, one could get better estimates from his formula for the object. In that note, he also remarked that when \(PQ < 1\), one can find a smooth, one parameter family of projections, \(P(t),\, 0 \le t \le 1\) with \(P(0) = P\) and \(P(1) = Q\) so that the U obtained via his earlier method of solving a differential equation was identical to the U of (5.2)/(5.14).
While this concludes Kato’s contribution to the subject of pairs of projections, I would be remiss if I didn’t say more about the rich structure of this simple setting, especially when \(PQ \ge 1\) (in the selfadjoint Hilbert space setting one has that \(PQ \le 1\) but for nonselfadjoint projections and the general case of Banach spaces, one often has \(PQ > 1\)). There are two approaches. The one we’ll discuss first is due to Avron–Seiler–Simon [31] and uses algebraic relations, especially (5.6). Since \(AB+BA=0\) is the signature of supersymmetry, we’ll call this the supersymmetric approach. Here is a typical use of this method:
Theorem 5.2
 (a)If \(\lambda \ne \pm 1\), then(5.16)
 (b)For such \(\lambda \), we have that$$\begin{aligned} \dim {\mathcal {H}}_{\lambda } = \dim {\mathcal {H}}_{\lambda } \end{aligned}$$(5.17)
 (c)If \(PQ\) is trace class, then$$\begin{aligned} {\mathrm{Tr}}(PQ) \in {\mathbb {Z}}\end{aligned}$$(5.18)
 (d)If \(PQ < 1\), then \(U \equiv \mathrm {sgn}(B)\) is a unitary operator obeying (5.1). Indeed,$$\begin{aligned} UPU^{1} = Q, \qquad UQU^{1} = P \end{aligned}$$(5.19)
Remarks
 1.By \(\mathrm {sgn}(B)\), we mean f(B) defined by the functional calculus [616, Section 5.1] whereThis is unitary because \(A < 1\) and \(B^2 = 1A^2\) implies that \(\ker B = \{0\}\). One can also write$$\begin{aligned} f(x) = \left\{ \begin{array}{ll} \,\,\,\, 1, &{} x > 0\\ 1, &{} x<0 \\ \,\,\,\, 0, &{} x=0 \end{array} \right. \end{aligned}$$$$\begin{aligned} U = B(1A^2)^{1/2} \end{aligned}$$(5.20)
 2.If we use (5.20) to define U in the general Banach space case when \(PQ < 1\), the same proof shows that we have (5.19). Indeed, since \([A^2,B]=0\), we have that \(U^2={\varvec{1}}\) so (5.1) implies \(UQU^{1}=P\). So we get another proof of Theorem 5.1 in the general Banach space case. However if \(P=Q\), then \(B=12P\) and \(A=0\) so by (5.20)Thus, \(U(P,P) \ne {\varvec{1}}\) but see the remarks after Theorem 5.1.$$\begin{aligned} U = {\varvec{1}} 2P \end{aligned}$$(5.21)
 3.
That \({\mathrm{Tr}}(PQ) \in {\mathbb {Z}}\) was first proven by Effros [134] and can also be proven using the Krein spectral shift [616, Problem 5.9.1]. It is also true if P, Q are not necessarily selfadjoint projections in a Hilbert space and for suitable Banach space cases; see below.
Proof
If \(\psi \in {\mathcal {H}}_{\lambda }\), then, by the above, \(\varphi \equiv (1\lambda ^2)^{1}B\psi \in {\mathcal {H}}_\lambda \) and \(B\varphi = \psi \) so Open image in new window is all of \({\mathcal {H}}_{\lambda }\) and thus V is unitary.
(b) is immediate from (a)
Theorem 5.3
 (a)
\(\lambda \in \sigma (A){\setminus }\{1,1\} \Rightarrow \lambda \in \sigma (A)\)
 (b)For such \(\lambda \), we have that$$\begin{aligned} \dim {\mathcal {H}}_{\lambda } = \dim {\mathcal {H}}_{\lambda } \end{aligned}$$(5.31)
 (c)
If \(\pm 1 \notin \sigma (A)\), then there exists an invertible map U so that (5.19) holds.
 (d)
If A obeys Lidskii’s theorem, then \({\mathrm{Tr}}(PQ) \in {\mathbb {Z}}\).
Remark
(d) was proven by Kalton [304] using different methods. The results (a)–(c) and the proof we give of (d) is new in the present paper.
Proof
Suppose \(\lambda \ne \pm 1\). Since A leaves \({\mathcal {H}}_\lambda \) invariant and Open image in new window , we have that \((1A^2)=(1A)(1+A)\) restricted to \({\mathcal {H}}_\lambda \) has an inverse R. Thus RB is a left inverse to B as a map of \({\mathcal {H}}_\lambda \rightarrow {\mathcal {H}}_{\lambda }\) so B as a map between those spaces is 1–1. This implies that \(\dim {\mathcal {H}}_\lambda \le \dim {\mathcal {H}}_{\lambda }\). By interchanging \(\lambda \) and \(\lambda \), we see that (5.31) holds which implies (a) and (b).
(d) From Lidskii’s theorem and (5.31), we see that (5.24) holds.
Theorem 5.4
Remarks
 1.
In (5.36), both sides may be infinite.
 2.
If \(\pm 1\) are isolated points of the spectrum of A and are discrete eigenvalues, then \(K:{\mathrm{ran}}\, P \rightarrow {\mathrm{ran}}Q\) by Open image in new window is a Fredholm operator [616, Section 3.15], both sides of (5.36) are finite and their difference is the index of K. So, in this case, the theorem says that U obeying (5.19) exists if and only if index\((K)=0\). This special case is in [31].
 3.
The general case of this theorem is due to Wang, Du and Dou [694] whose proof used the Halmos representation discussed below. Our proof here is from Simon [617]. Two recent papers [70, 124] classify all solutions of (5.19)
 4.
Proof
If U exists, it is easy to see that U must be a unitary map of \({\mathcal {K}}_{P,Q}\) to \({\mathcal {K}}_{1P,1Q}\), so (5.36) must hold.
So it suffices to prove the result for \({\mathcal {H}}_2\), i.e. in the special case that \({\mathcal {K}}_{P,Q} = {\mathcal {K}}_{1P,1Q} = \{0\}\). If that holds, we have that \(\ker (1A^2) = \{0\}\), so \(\ker (B) = \{0\}\) and \(U_2 \equiv \mathrm {sgn}(B)\) is unitary. Since \(U_2A_2=A_2U_2,\, U_2B_2=B_2U_2\), we get that \(U_2P_2U_2^{1} = Q_2,\, U_2Q_2U_2^{1} = P_2\) by (5.27). Clearly, also \(U_2^2={\varvec{1}},\, U_2^*=U_2\).
Our final big topic in this section concerns the Halmos representation. As a first step, we note that
Proposition 5.5
 (a)
\({\mathcal {K}}_{P,Q} = \{\varphi \,\,A\varphi =\varphi \}, \qquad {\mathcal {K}}_{1P,1Q} = \{\varphi \,\,A\varphi =\varphi \}\)
 (b)
\({\mathcal {K}}_{P,1Q} = \{\varphi \,\,B\varphi =\varphi \},\qquad {\mathcal {K}}_{1Q,P} = \{\varphi \,\,B\varphi =\varphi \}\)
 (c)
\({\mathcal {K}}_{P,1Q}\oplus {\mathcal {K}}_{1Q,P}= \{\varphi \,\,A\varphi =0\}\) \({\mathcal {K}}_{P,Q}\oplus {\mathcal {K}}_{1P,1Q}= \{\varphi \,\,B\varphi =0\}\)
 (d)
These four spaces are mutually orthogonal.
 (e)
All four spaces are \(\{0\}\) if and only if \(\ker A=\ker B = \{0\}\).
Proof
 (a)
\(P \le {\varvec{1}},\,Q \ge 0\) so \(A\varphi =\varphi \Rightarrow \varphi ^2 \ge \langle \varphi ,P\varphi \rangle = \varphi ^2+\langle \varphi ,Q\varphi \rangle \Rightarrow \langle \varphi ,Q\varphi \rangle = 0 \Rightarrow \langle Q\varphi ,Q\varphi \rangle = 0 \Rightarrow Q\varphi = 0 \Rightarrow \) (since \((PQ)\varphi =\varphi \)) \(P\varphi =\varphi \Rightarrow \varphi \in {\mathcal {K}}_{P,Q}\). Conversely, \( \varphi \in {\mathcal {K}}_{P,Q} \Rightarrow P\varphi =\varphi \& Q\varphi =0 \Rightarrow A\varphi =\varphi \). The proof of the second statement is similar.
 (b)
Similar to (a) using \(B=(1P)Q\).
 (c)
The two spaces in the first statement are orthonormal by (b) and the mutual orthogonality of eigenspaces. Since \(A^2\varphi =(1B^2)\varphi \), that direct sum is \(\ker A^2=\ker A\). Conversely, if \(A\varphi = 0\), then \((1B^2)\varphi =A^2\varphi =0\). If \(\varphi _\pm = \tfrac{1}{2}(1 {\mp } B)\varphi \), then \(\varphi _\pm \in \ker (1\pm B)\) and \(\varphi =\varphi _+ + \varphi _\), so by (b), \(\varphi \in {\mathcal {K}}_{P,1Q}\oplus {\mathcal {K}}_{1Q,P}\). The second relation has a similar proof.
 (d)
Immediate from the orthogonality of different eigenspaces of a selfadjoint operator.
 (e)
Immediate from (c). \(\square \)
We say that two orthogonal projections are in generic position if \(\ker A=\ker B = \{0\}\), equivalently if \({\mathcal {K}}_{P,Q}, {\mathcal {K}}_{1P,1Q}, {\mathcal {K}}_{P,1Q}, {\mathcal {K}}_{1Q,P}\) are all \(\{0\}\). The Halmos two projection theorem says
Theorem 5.6
Remarks
2. C and S stand, of course, for \(\text {cosine}\) and \(\text {sine}\). One often defines an operator, \(\Theta \) with spectrum in \([0,\pi /2]\) so that \(C=\cos (\Theta ),\,S=\sin (\Theta )\). While 0 and/or 1 may lie in the spectrum of \(\Theta \), they cannot be eigenvalues.
3. This result is due to Halmos [216]. There were earlier related results by Krein et. al. [391], Dixmier [120] and Davis [106]. The proof we give here is due to Amrein–Sinha [13].
Proof
Böttcher–Spitkovsky [69] is a review article on lots of applications of the Halmos representation. We mention also Lenard [421] who computes the joint numerical range (i.e. \(\{(\langle \varphi ,P\varphi \rangle ,\langle \varphi ,Q\varphi \rangle )\,\,\varphi =1\}\)) for pairs of projections in terms of the operator \(\Theta \) of remark 2 to Theorem 5.6. This range is a union of certain ellipses.
Finally, we mention one result that Kato proved in 1960 [332] that turns out to be connected to pairs of selfadjoint projections, although Kato didn’t himself mention or exploit this connection.
Theorem 5.7
Kato has this as a Lemma in a technical appendix to [332], but it is now regarded as a significant enough result that Szyld [632] wrote an article to advertise it and explain myriad proofs ([69] also discusses proofs). Del Pasqua [111] and Ljance [433] found proofs slightly before Kato but the methods are different and independent; indeed, for many years, no user of the result seemed to know of more than one of these three papers.
Del Pasqua [111] noted that (5.47) might fail in general Banach spaces—indeed, it is now known [211] that if (5.47) holds for all projections in a Banach space, X, then its norm comes from an inner product.
6 Eigenvalue perturbation theory, V: Temple–Kato inequalities
While strictly speaking the central material in this section is not so much about perturbation theory as variational methods, the subjects are related as Kato mentioned in several places, so we put it here. In fact, following Kato, we’ll see the inequalities proven here can be used to prove certain irregular perturbations yield asymptotic perturbation series. Kato also had several other papers about variational methods for scattering phase shifts [311, 317, 318] and for an aspect of Thomas–Fermi theory [267] (not the energy variational principle central to TF theory but one concerning a technical issue connected to the density at the nucleus). But none of these other papers had the impact of the work we discuss in this review, so we will not discuss them further.
In 1949, Kato [307] (with an announcement in Physical Review [312]) in one of his little gems found a simple proof of Temple’s inequality and also extended the result to any eigenvalue. Here is his theorem:
Theorem 6.1
Remarks
 1.As we’ll see, a version of (6.8) holds even if we don’t suppose there is only one point in \(\sigma (A) \cap (\alpha ,\zeta )\), namely ifthen \(\sigma (A) \cap (\alpha ,\kappa _0] \ne \emptyset \) and \(\sigma (A) \cap [\gamma _0,\zeta ) \ne \emptyset \)$$\begin{aligned} \gamma _0 = \eta _\varphi  \frac{\epsilon _\varphi ^2}{\zeta \eta _\varphi }; \qquad \kappa _0 = \eta _\varphi + \frac{\epsilon _\varphi ^2}{\eta _\varphi \alpha } \end{aligned}$$(6.10)
 2.
If we take \(\alpha \rightarrow \infty \) and \(\zeta = \eta _\varphi +1\), the upper bound in (6.8) is just the Rayleigh bound (6.1) and if we take \(\zeta =\mu \), then the lower bound in (6.8) is just Temple’s inequality (6.4).
 3.If \(0<\alpha < 1\), thenso (6.9) implies that$$\begin{aligned} 22(1\alpha ^2)^{1/2}&= \left[ \frac{44(1\alpha ^2)}{2+2(1\alpha ^2)^{1/2}}\right] \\&\le \frac{4\alpha ^2}{4(1\alpha ^2)^{1/2}} = \left[ \frac{\alpha }{(1\alpha ^2)^{1/4}}\right] ^2 \end{aligned}$$which is how Kato writes it in Kato [321] (see Knyazev [382] for refined versions of these types of estimates).$$\begin{aligned} \varphi \psi  \le \frac{\epsilon }{\delta }\left( 1\frac{\epsilon ^2}{\delta ^2}\right) ^{1/4} \end{aligned}$$(6.11)
The proof we’ll give follows Kato’s approach (see also Harrell [221]). The key to this proof is what Temple [640] calls Kato’s Lemma:
Lemma 6.2
Proof
The spectral theorem (see [616, Chapter V and Section 7.2]) says that A is a direct sum of multiplications by x on \(L^2({\mathbb {R}}{\setminus } (\alpha ,\zeta ), d\mu (x))\). Since \((x\alpha )(x\zeta ) \ge 0\) for \(x \in {\mathbb {R}}{\setminus } (\alpha ,\zeta )\), we see that \((A\alpha )(A\zeta ) \ge 0\).
Remark
While we use the Spectral Theorem (as Kato did), all we need is a spectral mapping theorem, i.e. if \(f(x) = (x\alpha )(x\zeta )\), then \(\sigma (f(A))=f[\sigma (A)]\) and the fact that an operator with spectrum in \([0,\infty )\) is positive. The spectral mapping theorem for polynomials holds for elements of any Banach algebra and the proof in [616, Theorem 2.2.6] extends to unbounded operators. That this lemma follows from considerations of resolvents only was noted by Temple [640].
Taking contrapositives in (6.12), we get the following Corollary (if Lemmas are allowed to have Corollaries):
Corollary 6.3
The final preliminary of the proof is
Lemma 6.4
Proof
Proof of Theorem 6.1
 (a)We have thatby (6.7). By Corollary 6.3, we see that \(\sigma (A) \cap (\alpha ,\zeta ) \ne \emptyset \).$$\begin{aligned} \langle \varphi ,(A\alpha )(A\zeta )\varphi \rangle&= \langle \varphi ,(A\eta _\varphi )^2\varphi \rangle +\langle \varphi ,\left[ \eta _\varphi ^2+\alpha \zeta (\alpha +\zeta )A\right] \varphi \rangle \nonumber \\&= \epsilon _\varphi ^2(\eta _\varphi \alpha )(\zeta \eta _\varphi ) < 0 \end{aligned}$$(6.19)
 (b)As in the proof of (6.19), for any \(\gamma , \kappa \), we have thatFix \(\kappa =\zeta \). Then, using \(\zeta > \eta _\varphi \):$$\begin{aligned} \langle \varphi ,(A\gamma )(A\kappa )\varphi \rangle = \epsilon _\varphi ^2(\eta _\varphi \gamma )(\kappa \eta _\varphi ) \end{aligned}$$(6.20)(with \(\gamma _0\) given by (6.10)) so by Corollary 6.3,$$\begin{aligned} \text {RHS of } (6.20)< 0 \iff \gamma < \gamma _0 \end{aligned}$$(6.21)Since \(\sigma (A)\) is closed, this implies that$$\begin{aligned} \gamma < \gamma _0 \Rightarrow \sigma (A) \cap (\gamma ,\zeta ) \ne \emptyset \end{aligned}$$(6.22)Similarly,$$\begin{aligned} \sigma (A) \cap [\gamma _0,\zeta ) \ne \emptyset \end{aligned}$$(6.23)In particular, if there is a single point, \(\lambda \), in \((\alpha ,\zeta )\), we must have that \(\lambda \in (\alpha ,\kappa _0] \cap [\gamma _0,\zeta ) = [\gamma _0,\kappa _0]\) which is (6.8).$$\begin{aligned} \sigma (A) \cap (\alpha ,\kappa _0] \ne \emptyset \end{aligned}$$(6.24)
 (c)
This is Lemma 6.4.
Kato exploited what are now called the Temple–Kato inequalities in his thesis to prove results on asymptotic perturbation theory. Below are two typical results whose proofs are very much in the spirit of this work of Kato—see Sect. 3 for what it means for an eigenvalue to be stable.
Theorem 6.5
Proof
Theorem 6.6
As Kato noted in his thesis, this idea shows if all the terms for the nth order formal series for the eigenvector lie in \({\mathcal {H}}\), then one gets asymptotic series for the energy with errors of order O\((\beta ^{2n})\), i.e. the 2n coefficients \(E_0,\ldots ,E_{2n1}\) but the method doesn’t handle odd powers. Indeed in [316], he said: “However, there has been a serious gap in the series of these conditions; for all of them had in common the property that they give the expansion of the eigenvalues up to even orders of approximation, and there was no corresponding theorem giving an expansion up to an odd order.” Personally, I think “serious” is a bit strong given that he handles the case of infinite order (for me the most important) and first order results but it shows he was frustrated by a problem he tried to solve without initial success. But in [309], he put in a Note Added in Proof announcing he had solved the problem! The solution appeared in [322]. For example, if \(A_0 \ge 0, B \ge 0\), he proved that if \(\varphi \in Q(B)\), then \(E(\beta ) = E_0+E_1\beta +\text {o}(\beta )\) and if \(B^{1/2}\varphi \in Q(B^{1/2}A_0^{1}B^{1/2})\), he proved that \(E(\beta ) = E_0+E_1\beta +E_2\beta ^2+\text {o}(\beta ^2)\). Not surprisingly, in addition to estimates of Temple–Kato type, the proofs use a variant of quadratic form methods. I note that Kato did not put any of these results in his book where his discussion of asymptotic series applies to general Banach space settings and not just positive operators and the ideas are closer to what we put in Sect. 3.
Besides the original short paper on Temple–Kato inequalities, Kato returned to the subject several times. In two papers [321, 357], he considered the fact that in some applications of interest, the natural trial vector has \(\varphi \in Q(A)\), not D(A). Trial functions only in Q(A) are fine for the Rayleigh upper bound but if \(\varphi \notin D(A)\), then \(\epsilon _\varphi ^A = \infty \), so \(\varphi \) cannot be used for Temple’s inequality or the Temple–Kato inequality. Of course, one could look at the Temple–Kato inequality for \(\sqrt{A}\) if \(A \ge 0\) but calculation of \(\langle \varphi ,\sqrt{A}\varphi \rangle \) may not be easy for, say, a second order differential operator where \(\sqrt{A}\) is a pseudodifferential operator. But such operators can often be written \(A=T^*T\) where T is a first order differential operator. Variants of the Temple–Kato inequality for operators of this form are the subject of two papers of Kato [321, 357]. Kato et al. [359] studies an application of these ideas.
Interesting enough, while Kato’s work was 20 years after Temple, Temple was young when he did that work and was still active in 1949 and he reacted to Kato’s paper with two of his own [639, 640]. George Frederick James Temple (1901–1992) was a mathematician with a keen interest in physics—he wrote two early books on quantum mechanics in 1931 and 1934. He spent much of his career at King’s College, London although for the last fifteen years of it, he held the prestigious Sedleian Chair of Natural Philosophy at Oxford, the chair going back to 1621. He was best known in British circles for a way of discussing distributions as equivalence classes of approximating smooth functions, an idea that was popular because the old guard didn’t want to think about the theory of topological vector spaces central to Schwartz’ earlier approach. His other honors include a knighthood (CBE, for War work), a fellowship in and the Sylvester Medal of the Royal Society. At age 82, he became a benedictine monk and spent the last years of his life in a monastery on the Isle of Wright. The long biographical note of his life written for the Royal Society [370] doesn’t even mention Temple’s inequality!
Davis [105] extended what he calls “the ingenious method of Kato” by replacing the single interval \((\alpha ,\zeta )\) by a finite union of intervals. Thirring [643] has discussed Temple’s inequality as a consequence of the Feshbach [149, 150] projection method (which mathematicians call the method of Schur [545] complements). Turner [658] and Harrell [221] have extensions to the case where A is normal rather than selfadjoint and Kuroda [400] to n commuting selfadjoint operators (and so including the normal case). Cape et al. [81] apply Temple–Kato inequalities to graph Laplacians. Golub–van der Vost [195] have a long review on eigenvalue values bounds mentioning that by the time of their review in 2000, Temple–Kato inequalities had become a standard part of linear algebra.
7 Selfadjointness, I: Kato’s theorem
This is the first of four sections on selfadjointness issues. We assume the reader knows the basic notions, including what an operator closure and an operator core are and the meaning of essential selfadjointness. A reference for these things is [616, Section 7.1].
This section concerns the Kato–Rellich theorem and its application to prove the essential selfadjointness of atomic and molecular Hamiltonians. The quantum mechanical Hamiltonians typically treated by this method are bounded from below. Section 8 discusses cases where \(V(x) \ge cx^2d\) like Stark Hamiltonians. Section 9 discusses Kato’s contribution to the realization that the positive part of V can be more singular than the negative part without destroying essential selfadjointness and Sect. 10 turns to Kato’s contribution to the theory of quadratic forms. To save ink, in this article, I’ll use “esa” as an abbreviation for “essentially selfadjoint” or “essential selfadjointness” and “esa\(\nu \)” for “essentially selfjoint on \(C_0^\infty ({\mathbb {R}}^\nu )\).”.
Kato’s big 1951 result was
Theorem 7.1
(Kato’s Theorem [314], First Form) Let \(\nu =3\). Let each \(V_{ij}\) in (7.1) lie in \(L^2({\mathbb {R}}^3)+L^\infty ({\mathbb {R}}^3)\). Then the Hamiltonian of (7.1) is selfadjoint on \(D(H) = D(\Delta )\) and esa(3N).
Remarks
 1.
The same results holds with the terms in (7.2) added so long as each \(V_j\) lies in \(L^2({\mathbb {R}}^3)+L^\infty ({\mathbb {R}}^3)\).
 2.Kato also notes the exact description of \(D(\Delta )\) on \(L^2({\mathbb {R}}^\nu )\) in terms of the Fourier transform (see [612, Chapter 6]) \(\hat{\varphi }(k) = (2\pi )^{\nu /2} \int e^{ik\cdot x} \varphi (x) d^\nu x\):$$\begin{aligned} D(\Delta ) = \{ \varphi \in L^2({\mathbb {R}}^\nu ) \,\, \int (1+k^2)^2 \hat{\varphi }(k)^2 d^\nu k < \infty \} \end{aligned}$$(7.4)
 3.
The proof shows that the graph norms of H and \(\Delta \) on \(D(\Delta )\) are equivalent, so any operator core for \(\Delta \) is a core for H. Since it is easy to see that \(C_0^\infty ({\mathbb {R}}^{3N})\) is a core for \(\Delta \), the esa result follows from the selfadjointness claim, so we concentrate on the latter.
 4.
Kato didn’t assume that \(V \in L^2({\mathbb {R}}^3)+L^\infty ({\mathbb {R}}^3)\) but rather the stronger hypothesis that for some \(R < \infty \), one has that \(\int _{x< R} V(x)^2 d^3x < \infty \) and \(\sup _{x \ge R} V(x) < \infty \), but his proof extends to \(L^2({\mathbb {R}}^3)+L^\infty ({\mathbb {R}}^3)\).
 5.
Kato didn’t state that \(C_0^\infty ({\mathbb {R}}^{3N})\) is a core but rather that \(\psi \)’s of the form \(P(x) e^{\tfrac{1}{2}x^2}\) with P a polynomial in the coordinates of x is a core (He included the \(\tfrac{1}{2}\) so the set was invariant under Fourier transform.) His result is now usually stated in terms of \(C_0^\infty \).
If \(v(x) = 1/x\) on \({\mathbb {R}}^3\), then \(v \in L^2({\mathbb {R}}^3)+L^\infty ({\mathbb {R}}^3)\), so Theorem 7.1 has the important Corollary, which includes the Hamiltonians of atoms and molecules:
Theorem 7.2
Remark
This result assures that the time dependent Schrödinger equation \(\dot{\psi }_t = iH\psi _t\) has solutions (since selfadjointness means that \(e^{itH}\) exists as a unitary operator). The analogous problem for Coulomb Newton’s equation (i.e. solvability for a.e. initial condition) is open for \(N \ge 5\)!
 (1)
The Kato–Rellich theorem which reduces the proof to showing that each \(V_{ij}\) is relatively bounded for Laplacian on \({\mathbb {R}}^3\) with relative bound 0.
 (2)
A proof that any function in \(L^2({\mathbb {R}}^3)+L^\infty ({\mathbb {R}}^3)\), as an operator on \(L^2({\mathbb {R}}^3)\), is \(\Delta \)bounded with relative bound 0. This relies on a simple Sobolev estimate.
 (3)
A piece of simple kinematics that says that the two body estimate in step 2 extends to one for \(v_{ij}(x_ix_j)\) as an operator on \(L^2({\mathbb {R}}^{3N})\).
Theorem 7.3
Remarks
 1.
This result is due to Rellich [504, 505, 506, 507, 508, Part III]. Kato found it in 1944, when he was unaware of Rellich’s work, so it is independently his.
 2.
The proof uses von Neumann’s criteria: a closed symmetric operator, C, on D(C) is selfadjoint if and only if for some \(\kappa \in (0,\infty )\), one has that \({\mathrm{ran}}(C\pm i\kappa )={\mathcal {H}}\). For C closed implies that \({\mathrm{ran}}(C\pm i\kappa )\) are closed subspaces with \({\mathrm{ran}}(C\pm i\kappa )^\perp = \ker (C^*{\mp } i\kappa )\). Thus, if C is selfadjoint, then \(\ker (C^*{\mp } i\kappa ) = \{0\}\) proving one direction. For the other direction, suppose that \({\mathrm{ran}}(C\pm i\kappa ) = {\mathcal {H}}\). Given \(\psi \in D(C^*)\), find \(\varphi \in D(C)\) with \((C+i\kappa )\varphi = (C^*+i\kappa )\psi \) (since \({\mathrm{ran}}(C+i\kappa )={\mathcal {H}}\)). Thus \((C^*+i\kappa )(\varphi \psi ) = 0\). Since \({\mathrm{ran}}(Ci\kappa ) = {\mathcal {H}}= \ker (C^*+i\kappa )^\perp \), we have that \(\varphi \psi = 0\). Thus \(D(C^*) = D(C)\) and C is selfadjoint.
 3.For the rest of the proof, use \((C+i\kappa )\varphi ^2 = C\varphi ^2+\kappa ^2\varphi ^2\) to see thatIt follows from this (with \(C=A\)) that when (7.7) holds, one has that$$\begin{aligned} C(C\pm i\kappa )^{1} \le 1, \qquad (C\pm i\kappa )^{1} \le \kappa ^{1} \end{aligned}$$(7.8)Since \(a < 1\), we can be sure that if \(\kappa \) is very large, then \({B(A\pm i\kappa )^{1} < 1}\) so using a geometric series, we have that \({1 + B(A\pm i\kappa )^{1}}\) is invertible which implies that it maps \({\mathcal {H}}\) onto \({\mathcal {H}}\). Since \((A\pm i\kappa )\) maps D(A) onto \({\mathcal {H}}\), we see that$$\begin{aligned} B(A\pm i\kappa )^{1} \le a + b\kappa ^{1} \end{aligned}$$(7.9)maps D(A) onto \({\mathcal {H}}\). Thus by von Neumann’s criterion, \(A+B\) is selfadjoint on D(A). By a simple argument, \(A\cdot +\cdot \) is an equivalent norm to \((A+B)\cdot +\cdot \) which proves the esa result.$$\begin{aligned} (A+B\pm i\kappa ) =(1+B(A\pm i\kappa )^{1})(A\pm i\kappa ) \end{aligned}$$(7.10)
 4.The case \(B=A\) shows that one can’t conclude selfadjointness of \(A+B\) on D(A) if (7.7) holds with \(a=1\) but Kato [345] proved that \(A+B\) is esa on D(A) in that case and Wüst [690] proved the stronger result of esa on D(A) if one has for all \(\varphi \in D(A)\)$$\begin{aligned} B\varphi ^2 \le A\varphi ^2 + b\varphi ^2 \end{aligned}$$(7.11)
 5.
In some of my early papers, I called B Kato small if B was \(Abounded\) with relative bound less than 1 and Kato tiny if the relative bound was 0. I am pleased to say that while many of my names (hypercontractive, almost Mathieu, Berry’s phase, Kato class,...) have stuck, this one has not!
Kato states in the paper that he had found the results by 1944. Kato originally submitted the paper to Physical Review. Physical Review transferred the manuscript to the Transactions of the AMS where it eventually appeared. They had trouble finding a referee and in the process the manuscript was lost (a serious problem in preXerox days!). Eventually, von Neumann got involved and helped get the paper accepted. I’ve always thought that given how important he knew the paper was, von Neumann should have suggested Annals of Mathematics and used his influence to get it published there. The receipt date of October 15, 1948 on the version published in the Transactions shows a long lag compared to the other papers in the same issue of the Transactions which have receipt dates of Dec., 1949 through June, 1950. Recently after Kato’s widow died and left his papers to some mathematicians (see the end of Sect. 1) and some fascinating correspondence of Kato with Kemble and von Neumann came to light. There are plans to publish an edited version [181].
It is a puzzle why it took so long for this theorem to be found. One factor may have been von Neumann’s attitude. Bargmann told me of a conversation several young mathematicians had with von Neumann around 1948 in which von Neumann told them that selfadjointness for atomic Hamiltonians was an impossibly hard problem and that even for the Hydrogen atom, the problem was difficult and open. This is a little strange since, using spherical symmetry, Hydrogen can be reduced to a direct sum of one dimensional problems. For such ODEs, there is a powerful limit point–limit circle method named after Weyl and Titchmarsh (although it was Stone, in his 1932 book, who first made it explicit). Using this, it is easy to see (there is one subtlety for \(\ell =0\) since the operator is limit circle at 0) that the Hydrogen Hamiltonian is selfadjoint and this appears at least as early as Rellich [509]. Of course, this method doesn’t work for multielectron atoms. In any event, it is possible that von Neumann’s attitude may have discouraged some from working on the problem.
Still it is surprising that neither Friedrichs nor Rellich found this result. In exploring this, it is worth noting that there is an alternate to step 2:
Rellich used Hardy’s inequality in his perturbation theory papers [504, 505, 506, 507, 508] in a closely related context. Namely he used (7.19) and (7.20) for \(C=r^{1}\) to show that \(r^{1} \le 4\epsilon (\Delta ) + \tfrac{1}{4} \epsilon ^{1}\) to note the semiboundedness of the Hydrogen Hamiltonian. Since Rellich certainly knew the Kato–Rellich theorem, it appears that he knew steps 1 and 2\('\).
In a sense, it is pointless to speculate why Rellich didn’t find Theorem 7.2, but it is difficult to resist. It is possible that he never considered the problem of esa of atomic Hamiltonians, settling for a presumption that using the Friedrichs extension suffices (as Kato suggests in [1]) but I think that unlikely. It is possible that he thought about the problem but dismissed it as too difficult and never thought hard about it. Perhaps the most likely explanation involves Step 3: once you understand it, it is trivial, but until you conceive that it might be true, it might elude you.
Theorem 7.4
In exploring extensions of Theorem 7.1, it is very useful to have simple selfadjointness criteria for \(\tfrac{d^2}{dx^2}+q(x)\) on \(L^2(0,\infty )\) which then translate to criteria for \(\Delta +V(x)\) if \(V(x) = q(x)\) is a spherically symmetric potential. If \(q \in L^2_{loc}(0,\infty )\), for each \(z \in {\mathbb {C}}\), the set of solutions of \(u''+qu=zu\) (in the sense that u is \(C^1\), \(u'\) is absolutely continuous and \(u''\) is its \(L^1_{loc}\) derivative) is two dimensional. If all solutions are \(L^2\) at \(\infty \) (resp. 0), we say that \(\tfrac{d^2}{dx^2}+q(x)\) is limit circle at \(\infty \) (resp. 0). If it is not limit circle, we say it is limit point. It is a theorem that whether one is limit point or limit circle is independent of z. However, in the limit point case, whether the set of \(L^2\) solutions near infinity is 0 or 1 dimensional can be z dependent. One has the basic
Theorem 7.5
(Weyl limit point–limit circle theorem) Let \(q \in L^2_{loc}(0,\infty )\). Then \(\tfrac{d^2}{dx^2}+q(x)\) is esa on \(C_0^\infty (0,\infty )\) if and only if \(\tfrac{d^2}{dx^2}+q(x)\) is limit point at both 0 and \(\infty \).
Remarks
 1.
This result holds for any interval \((a,b) \subset {\mathbb {R}}\) where a can be \(\infty \) and/or b can be \(\infty \).
 2.
If it is limit point at only one of 0 and \(\infty \) and limit circle at the other point, the deficiency indices (see [616, Section 7.1] for definitions) are (1, 1) and if it is limit circle at both 0 and \(\infty \), they are (2, 2). In particular if it is limit point at \(\infty \) and \(\int _{0}^{1} V(x) dx < \infty \), then the deficiency indices are (1,1) and the extensions are described by boundary conditions \(\cos \theta \, u'(0)+\sin \theta \, u(0) = 0\).
 3.
The ideas behind much of the theorem go back to Weyl [677, 678, 680] in 1910 and predate the notion of selfadjointness. It was Stone [630] who first realized the implications for selfadjointness and proved Theorem 7.5. [616, Thm 7.4.12] has a succinct proof. Titchmarsh [653] reworked the theory so much that it is sometimes called Weyl–Titchmarsh theory. For additional literature, see [94, 131, 423].
Example 7.6
(\(x^{2}\) on \((0,\infty )\)) Let \(q(x) = \beta x^{2}\). Trying \(x^\alpha \) in \(u''+\beta x^{2} u = 0\), one finds that \(\alpha (\alpha 1)=\beta \) is solved by \(\alpha _\pm = \tfrac{1}{2}(1 \pm \sqrt{1+4\beta })\). For \(\beta \ne \tfrac{1}{4}\), this yields two linearly independent solutions, so a basis. The larger solution (and sometimes both) is not \(L^2\) at infinity, so it is always limit point there.
Proposition 7.7
 (1)Bounded from below$$\begin{aligned} H_\beta ^{(\nu )} \ge 0 \iff \beta \ge \frac{(\nu 2)^2}{4} \end{aligned}$$(7.34)
 (2)\(H_\beta ^{(\nu )}\) is esa on \(C_{00}^\infty ({\mathbb {R}}^\nu )\) if and only if$$\begin{aligned} \beta \ge \frac{\nu (\nu 4)}{4} \end{aligned}$$(7.35)
Remarks
 1.
This uses \(\tfrac{(\nu 1)(\nu 3)}{4}\tfrac{1}{4} = \tfrac{(\nu 2)^2}{4}\) and \(\tfrac{(\nu 1)(\nu 3)}{4}+\tfrac{3}{4} = \tfrac{\nu (\nu 4)}{4}\).
 2.
 3.
By (7.28), if \(\nu \ge 4\) and V is spherically symmetric and obeys \(V(x) \ge \tfrac{\nu (\nu 4)}{4x^2}\), then \(\Delta +V\) is esa\(\nu \) (discussed further in Sect. 9).
 4.
In particular, \(C_{00}^\infty ({\mathbb {R}}^\nu )\) is an operator core for \(\Delta \) if and only if \(\nu \ge 4\) and a form core for \(\Delta \) if and only if \(\nu \ge 2\).
 5.
By (7.28), if \(\gamma > 2\), then \(\Delta +\lambda x^{\gamma }\) (\(\lambda >0\)) is esa on \(C_{00}^\infty ({\mathbb {R}}^\nu )\). If \(\nu \ge 5\) and \(2< \gamma < \nu /2\), we have that \(x^{\gamma } \in L^2({\mathbb {R}}^\nu )+L^\infty ({\mathbb {R}}^\nu )\), so one can define \(T \equiv \Delta +\lambda x^{\gamma }\) on \(C_{0}^\infty ({\mathbb {R}}^\nu )\) and it is easy to see that T is symmetric. It follows by general principles [616, Section 7.1] that T is esa\(\nu \).
 6.
There is an intuition to explain why one loses selfadjointness of \(\Delta x^{\gamma }\) when \(\gamma > 2\). If \(\gamma <2\), in classical mechanics there is an \(\tfrac{\ell ^2}{x^2}\) barrier which dominates the \(x^{\gamma }\), so for almost every initial condition, the classical particle avoids the singularity at the origin. But when \(\gamma > 2\), every negative energy initial condition will fall into the origin in finite time so in classical mechanics, one needs to supplement with a rule about what happens when the particle is captured by the singularity. The quantum analog is the loss of esa. There is of course a difference at \(\gamma = 2\) where classically there is a problem no matter the coupling but not in quantum mechanics. This is associated with the uncertainty principle. In the next section, we’ll see that this intuition is also useful to understand what happens with V’s going to \(\infty \) at spatial infinity.
We summarize in
Example 7.8
\(\lambda B\) is Abounded with relative bound \(<1\) if and only if \(0 \le \lambda < \lambda _1\).
\(A+\lambda B\) is esa on D(A) if \(0 \le \lambda < \lambda _2\) and not if \(\lambda > \lambda _2\).
\(A+\lambda B\) is bounded from below if \(0 \le \lambda < \lambda _3\) and not if \(\lambda > \lambda _3\)
We call p, \(\nu \)canonical if \(p=2\) for \(\nu \le 3\), \(p > 2\) if \(\nu =4\) and \(p=\nu /2\) if \(p \ge 5\). The optimal \(L^p\) extension of Theorem 7.1 is
Theorem 7.9
Let p be \(\nu \)canonical. Then \(V \in L^p({\mathbb {R}}^\nu )+L^\infty ({\mathbb {R}}^\nu )\) is \(\Delta \)bounded with relative bound zero. If \(\nu \ge 5\), \(V \in L^p_w({\mathbb {R}}^\nu )+L^\infty ({\mathbb {R}}^\nu )\) is \(\Delta \)bounded on \(L^2({\mathbb {R}}^\nu )\).
Remarks
 1.
In the \(L^p_w\) case, the relative bound may not be zero; for example \(V(x) = x^{2}\) as discussed above. Since any \(L^p\) function can be written as the sum of a bounded function and an \(L^p_w\) function of arbitrarily small \(\cdot _{p, w}^*\), the second sentence implies the first.
 2.
One proof of the \(\nu \ge 5\) result uses a theorem of Stein–Weiss [624] (see [615, Section 6.2]) that if \(f \in L^{\nu /2}_w({\mathbb {R}}^\nu )\) and \(g \in L^{\nu /(\nu 2)}_w({\mathbb {R}}^\nu )\), then \(h \mapsto g*(fh)\) maps \(L^2\) to \(L^2\). Another proof uses Rellich’s inequality and Brascamp–Lieb–Luttinger inequalities (see [71, 520, 521, 522] or [611]).
 3.
That one can use \(p = \nu /2\) rather than \(p > \nu /2\) when \(\nu \ge 5\) was noted first by Faris [146].
 (a)
Since \(\int _{w \le 1} w^{\beta +\nu } dw_{\kappa +1}\ldots dw_\nu \sim (w_1,\ldots .w_\kappa )^{\beta +\kappa }\) where the tilde means comparable in terms of upper and lower bounds, extra variables go through directly and there is no need for step 3 in Kato’s proof.
 (b)
As we’ve seen, it is uniformly local, i.e. to be in a Stummel class rather than \(L^p\), one only needs \(L^p_{unif}\).
 (c)
By Young’s inequality [612, Theorem 6.6.3], the Brownell \(L^p\) condition implies Stummel’s condition, so Stummel’s result is stronger.
Theorem 7.10
One key to the proof is a simple necessary and sufficient condition
Theorem 7.11
([101]; Section 1.2) A multiplication operator, V, is in \(S_\nu \) if and only if \(\lim _{E \rightarrow \infty } (\Delta +E)^{2}V^2_{\infty ,\infty } = 0\) where \(\cdot _{p,p}\) is the operator norm from \(L^p({\mathbb {R}}^\nu )\) to itself.
Theorem 7.12
This result is essentially due to Rellich [509] in 1943. He proved it using spherical symmetry and applying the Weyl limit–limit circle theory (Theorem 7.5). We say “essentially” because at the time he did this, the Weyl theory had not been proven for systems and (7.47) is a system. This theory for systems was established by Kodaira [385] in 1951 (see also Weidmann [675]) so Theorem 7.12 should be regarded as due to Rellich–Kodaira. Interestingly enough, Kato seems to have been unaware of this result when he wrote his book (second edition was 1976).
Theorem 7.13
Let V obey (7.49) where \(\mu <\tfrac{1}{2}\sqrt{3}\). Then \(T_0+V\) is esa on \(C^\infty _0({\mathbb {R}}^3;{\mathbb {C}}^4)\).
We’ll return to Dirac operators in Sect. 8 and at the end of Sect. 10. Having mentioned a result of Schmincke, I should mention that in the 1970s and early 1980s there was a lively school founded by Günter Hellwig that produced a cornucopia of results on esa questions for Schrödinger and Dirac operators. Among the group were H. Cycon, H. Kalf, U.W. Schmincke, R. Wüst and J. Walter.
This said, there is a sense in which Kato’s critical value \(\mu =\tfrac{1}{2}\) is connected to loss of esa. Arai [15, 16] has shown that for any \(\mu > \tfrac{1}{2}\) there is a symmetric matrix valued potential Q(x) with \(Q(x)=\mu x^{1}\) for all x so that \(T_0+Q\) is not esa on \(C^\infty _0({\mathbb {R}}^3;{\mathbb {C}}^4)\), so Theorems 7.12 and 7.13 depend on scalar potentials.
8 Selfadjointness, II: the Kato–Ikebe paper
Kato was clearly aware that his great 1951 paper didn’t include the Stark Hamiltonian where H isn’t bounded from below, and in fact \(\eta (x) \equiv \int _{xy \le 1} \min (V(y),0)^2 dy \rightarrow \infty \) if one takes \(x \rightarrow \infty \) in a suitable direction. For esa, one needs restrictions on the growth of \(\eta \) at infinity (whereas, we’ll see in Sect. 9, if \(\min (V(y),0)\) is replaced by \(\max (V(y),0)\), no restriction is needed). To understand this, it is useful to first consider one dimension. Suppose that \(V(x) \rightarrow \infty \) as \(x \rightarrow \infty \). In classical mechanics, if a particle of mass m starts at \(x=c\) with zero speed, \(V(c) = 0\) and \(V'(x) < 0\) on \((c,\infty )\), the particle will move to the right. By conservation of energy, the speed when the particle is at point \(x > c\) will be \(v(x) = \sqrt{V(x)}\) if \(\tfrac{1}{2}m = 1\). The time to get from c to \(x_0 > c\) is thus \(\int _{c}^{x_0} \tfrac{dx}{\sqrt{V(x)}}\). Thus the key issue is whether \(\int _{c}^{\infty } \tfrac{dx}{\sqrt{V(x)}}\) is finite or not. If it is finite, the particle gets to infinity in finite time and the motion is incomplete. One expects that the quantum mechanical equivalent is that \(\tfrac{d^2}{dx^2}+V(x)\) is esa if and only if \(\int _{c}^{\infty } \tfrac{dx}{\sqrt{V(x)}} =\infty \). In particular, if \(V(x) = \lambda x^\alpha \), this suggests esa if and only if \(\alpha \le 2\).
Theorem 8.1
(Nilsson–Wienholtz) If V(x) is a continuous function of \({\mathbb {R}}^\nu \) obeying (8.1), then \(\Delta +V\) is esa\(\nu \).
Wienholtz had also considered first order terms but didn’t write it in the form (8.2) which is the right form for quantum physics; \(a_j(x)\) is the vector potential, i.e. B=da is the magnetic field. Ikebe–Kato had the important realization that one needs no global hypothesis on a, i.e. any growth at \(\infty \) of a is allowed. While they had too strong a local hypothesis on local behavior of a (see Sect. 9), their discovery on behavior at \(\infty \) was important.
Rather than discuss their techniques, I want to sketch two approaches to Wienholtz’s result which allow local singularities and are of especial elegance. For one of them, Kato made an important contribution. The first approach is due to Chernoff [90, 92] as modified by Kato [341] and the second approach is due to Faris–Lavine [148]. Interesting enough, each utilizes a selfadjointness criterion of Ed Nelson but two different criteria that he developed in different contexts. Here is the criteria for Chernoff’s method (which Nelson developed in his study of the relation between unitary group representations and their infinitesimal generators).
Theorem 8.2
(Chernoff–Nelson Theorem) Let A be a selfadjoint operator and \(U_t=e^{itA},\,t \in {\mathbb {R}}\), the induced unitary group. Suppose that \({\mathcal {D}}\) is a dense subspace of \({\mathcal {H}}\) with \({\mathcal {D}}\subset D(A^\ell )\) for some \(\ell = 1,2,\ldots \) and suppose that for all t, we have that \(U_t[{\mathcal {D}}] \subset {\mathcal {D}}\). Then \({\mathcal {D}}\) is a core for \(A,A^2,\ldots ,A^\ell \).
Remarks
 1.
Recall that Stone’s theorem [616, Theorem 7.3.1] says there is a oneone correspondence between oneparameter unitary groups and selfadjoint operators, via \(U_t=e^{itA},\,t \in {\mathbb {R}}\).
 2.
Chernoff considers the case \({\mathcal {D}}\subset D^\infty (A) \equiv \cap _\ell D^\ell (A)\) in which case \({\mathcal {D}}\) is a core for \(A^\ell \) for all \(\ell \).
 3.
Nelson [458] did the case \(\ell =1\) and Chernoff [90] noted his argument can be used for general \(\ell \).
 4.The argument is simple. Let Open image in new window for some \(k=1,\ldots ,\ell \). Suppose that \(B^*\psi = i\psi \). Let \(\varphi \in {\mathcal {D}}\) and let \(f(t) = \langle \psi ,U_t\varphi \rangle \). Then since \(U_t\varphi \in D(A^k)\), we have that f is a \(C^k\) function andIf \(g(t) = e^{i\alpha t}\), then g solves (8.3) if and only if \((i\alpha )^k = i^{k+1}\), i.e. \(\alpha ^k = i\). No solution of this is real, so g is a linear combination of exponentials which grow at different rates at either \(+\infty \) or \(\infty \), so the only bounded solution is 0. Since \(f(t) \le \psi \varphi \), we conclude that \(f(0)=0\) so \(\psi \perp {\mathcal {D}}\). Since \({\mathcal {D}}\) is dense, \(\psi = 0\), i.e. \(\ker (B^*i)=\{0\}\). Similarly, \(\ker (B^*+i)=\{0\}\), so B is esa.$$\begin{aligned} f^{(k)}(t)&= \langle \psi ,(iA)^kU_t\varphi \rangle = i^k \langle \psi ,BU_t\varphi \rangle \nonumber \\&= i^k \langle B^*\psi ,U_t\varphi \rangle = i^{k+1} f(t) \end{aligned}$$(8.3)
Kato proved his famous selfadjointness result to be able to solve the time dependent Schrödinger equation, \(\dot{\psi }_t = iH\psi _t\). Chernoff turned this argument around! If one can solve the equation \(\dot{\psi }_t = iA\psi _t\) for a dense set \({\mathcal {D}}\) in \(D^\infty (A)\) and prove that \(\psi _{t=0} \in {\mathcal {D}}\Rightarrow \psi _t \in {\mathcal {D}}\), then by Theorem 8.2, all powers of A are esa on \({\mathcal {D}}\). He combined this with existence and smoothness results of Friedrichs [173] and Lax [418] for hyperbolic equations plus finite propagation speed to show that if A is a hyperbolic equation, then the solution map takes \(C_0^\infty \) to itself.
In particular, since the Dirac equation is hyperbolic, Chernoff proved
Theorem 8.3
(Chernoff [90]) If \(T_0\) is the free Dirac operator, (7.45), and V is a \(C^\infty ({\mathbb {R}}^3)\) function, then \(T = T_0+V\) and all its powers are esa on \(C_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\).
Notice that there are no restrictions on the growth of V at \(\infty \). This is an expression of the fact that for the Dirac equation, no boundary condition is needed at infinity—intuitively, this is because the particle cannot get to infinity in finite time because speeds are bounded by the speed of light! Several years after his initial paper, Chernoff [92] used results on solutions of singular hyperbolic equations and proved the following version of the fact that Dirac equations have no boundary condition at infinity:
Theorem 8.4
Let \(T_0\) be the free Dirac equation and \(V \in L^2_{loc}({\mathbb {R}}^3)\) (so \(T_0+V\) is defined on \(C_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\)). Suppose for each \(x_0 \in {\mathbb {R}}^3\), there is a \(V^{(x_0)}\) equal to V in a neighborhood of \(x_0\) and so that \(T_0+V^{(x_0)}\) is esa on \(C_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\). Then \(T_0+V\) is esa on \(C_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\).
Combining this with Schmincke’s result (Theorem 7.13) one gets
Corollary 8.5
 (a)There are constants \(\mu _j < \sqrt{3}/2\) and \(C_j\) so that for x near \(x_j\), say x obeys \(xx_j \le \tfrac{1}{2} \min _{k\ne j} x_jx_k\), one has that$$\begin{aligned} V(x) \le \mu _j xx_j^{1} + C_j \end{aligned}$$(8.4)
 (b)
V is locally bounded near any \(x \notin \{x_j\}_{j=1}^N\).
Then \(T_0+V\) is esa on \(C_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\).
Other results on esa for Dirac operators which are finite sums of Coulomb potentials include [293, 305, 374, 408, 409, 410, 463].
Theorem 8.6
Remarks
 1.
This proof of the result appeared in Chernoff [90], but the result itself appeared earlier in Povzner [485] and Wienholtz [683].
 2.
In his second paper, Chernoff [92] handled singular V’s and also used the idea of Kato we’ll describe shortly and also Kato’s inequality ideas (see Sect. 9). He proved that \(\Delta +V\) is esa\(\nu \) if \(V=UW\) with \(U, W \ge 0\), \(U \in L^2_{loc}({\mathbb {R}}^\nu ), W \in L^p_{loc}({\mathbb {R}}^\nu )\) (with p \(\nu \)canonical) and \(\Delta +V+cx^2\) bounded from below for some \(c>0\).
In [341], Kato showed how to modify Chernoff’s argument to extend Theorem 8.6 to replace the condition that \(\Delta +V\) is bounded from below by the condition that for some \(c>0\), one has that \(\Delta +V+cx^2\) is bounded from below (and thereby gets a Wienholtz–Ikebe–Kato type of result). Kato’s idea (when \(c=1\)) was to solve \(\tfrac{\partial ^2 u}{\partial t^2} = (\Delta V)u4t^2u\). He was able to prove that \(u(t)_2\) (which is bounded in the case \(\Delta +V\) is bounded below) doesn’t grow worse than \(t^3\) and then push through a variant of the Chernoff–Nelson argument (since a \(t^3\) bound can eliminate exponential growth).
This completes our discussion of the Chernoff approach. The underlying selfadjointness criterion of Nelson needed for the Faris–Lavine approach is
Theorem 8.7
Remarks
 1.
The name comes from the fact that \(\langle A\psi ,N\varphi \rangle  \langle N\psi ,A\varphi \rangle = \langle \psi , [N,A] \varphi \rangle \) if \(N\varphi \in D(A)\) and \(A\varphi \in D(N)\).
 2.
Nelson [462] was motivated by Glimm–Jaffe [193] which also required bounds on [N, [N, A]] which would not apply to the Faris–Lavine choices without extra conditions on V.
 3.
To illustrate the use of this theorem, here is a special case of the Faris–Lavine theorem (see Faris–Lavine [148] or Reed–Simon [495, Theorem X.38] for the full theorem) that gives a \(V(x) \ge x^2\) type of result:
Theorem 8.8
Proof
By a simple argument, we can assume \(c=1, d=0\). Let \(N=\Delta +V+2x^2\) by which we mean the closure of that sum on \(C_0^\infty ({\mathbb {R}}^\nu )\). Let A be the operator closure of Open image in new window . By Theorem 9.1 below, N is selfadjoint. \(NA = 2x^2 \ge 0\) while \(N+A = \Delta + (2V(x)+2x^2) \ge 0\) so \(\pm A \le N\) which is (8.10).
The same method that proved (3.33) implies an estimate \(x^2\varphi  \le aN\varphi \) on \(C_0^\infty ({\mathbb {R}}^\nu )\) so \(\varphi \in D(N) \Rightarrow x^2\varphi \in L^2 \Rightarrow \varphi \in D(A)\). Thus \(D(N) \subset D(A)\).
9 Selfadjointness, III: Kato’s inequality
This section will discuss a selfadjointness method that appeared in Kato [340] based on a remarkable distributional inequality. Its consequences is a subject to which Kato returned often with at least seven additional papers [73, 343, 348, 349, 351, 355, 356]. It is also his work that most intersected my own—I motivated his initial paper and it, in turn, motivated several of my later papers. Throughout this section, we’ll use quadratic form ideas that we’ll only formally discuss in Sect. 10 (see [616, Section 7.5]).
To explain the background, recall that in Sect. 7, we defined p to be \(\nu \)canonical (\(\nu \) is dimension) if \(p=2\) for \(\nu \le 3\), \(p > 2\) for \(\nu = 4\) and \(p = \nu /2\) for \(\nu \ge 5\). For now, we focus on \(\nu \ge 5\) so that \(p=\nu /2\). As we saw, if \(V \in L^p({\mathbb {R}}^\nu )+L^\infty ({\mathbb {R}}^\nu )\), then \(\Delta +V\) is esa\(\nu \). The example \(V(x) =  \lambda x^{2}\) for \(\lambda \) sufficiently large shows that \(p=\nu /2\) is sharp. That is, for any \(2 \le q \le \nu /2\), there is a \(V \in L^q({\mathbb {R}}^\nu )+L^\infty ({\mathbb {R}}^\nu )\), so that \(\Delta +V\) is defined on but not esa on \(C_0^\infty ({\mathbb {R}}^\nu )\).
Theorem 9.1
(Kato [340]) If \(V \ge 0\) and \(V \in L^2_{loc}({\mathbb {R}}^\nu )\), then \(\Delta +V\) is esa\(\nu \).
Remark
As we’ll see later, this extends, for example, to \(V_+\in L^2_{loc}, V_\in L^p_{unif}\) with p \(\nu \)canonical
Kato’s result was actually a conjecture that I made on the basis of a slightly weaker result that I had proven:
Theorem 9.2
(Simon [575]) If \(V \ge 0\) and \(V \in L^2({\mathbb {R}}^\nu ,e^{cx^2}\,d^\nu x)\) for some \(c > 0\), then \(\Delta +V\) is esa\(\nu \).
Of course this covers pretty wild growth at infinity but Theorem 9.1 is the definitive result since one needs that \(V \in L^2_{loc}({\mathbb {R}}^\nu )\) for \(\Delta +V\) to be defined on all functions in \(C_0^\infty ({\mathbb {R}}^\nu )\).
Segal [548, 549] then proved that these same hypotheses imply that \(A_0+V\) is esa on \(D(A_0)\cap D(V)\) (for the field theory case Glimm–Jaffe [192] and Rosen [527] using Nelson’s estimates but additional properties had earlier proven esa for this specific situation).
Within a few weeks of my sending out a preprint with Theorem 9.2 and the conjecture of Theorem 9.1, I received a letter from Kato proving the conjecture by what appeared to be a totally different method. Over the next few years, I spent some effort understanding the connection between Kato’s work and semigroups. I will begin the discussion here by sketching a semigroup proof of Theorem 9.1, then give Kato’s proof of this theorem, then discuss semigroup aspects of Kato’s inequality and finally discuss some other aspects of Kato’s paper [340].
After the smoke cleared, it was apparent that my failure to get the full Theorem 9.1 in 1972 was due to my focusing on \(L^p\) properties of semigroups on probability measure spaces rather than on \(L^p({\mathbb {R}}^\nu ,d^\nu x)\). As a warmup to the semigroup proof of Theorem 9.1, we prove (we use quadratic form ideas only discussed in Sect. 10)
Theorem 9.3
(Simon [595]) Let \(V \ge 0\) be in \(L^1_{loc}({\mathbb {R}}^\nu ,d^\nu x)\) and let \(a \in L^2_{loc}({\mathbb {R}}^\nu ,d^\nu x)\) be an \({\mathbb {R}}^\nu \) valued function. Let \(Q(D_j^2) = \{\varphi \in L^2({\mathbb {R}}^\nu ,d^\nu x)\,\, (\nabla _jia_j)\varphi \in L^2({\mathbb {R}}^\nu ,d^\nu x)\}\) with quadratic form \(\langle \varphi ,D_j^2\varphi \rangle = (\nabla _jia_j)\varphi ^2\). Let h be the closed form sum \(\sum _{j=1}^{\nu } D_j^2+V\). Then \(C_0^\infty ({\mathbb {R}}^\nu )\) is a form core for h.
Remarks
 1.
For \(a=0\), this result was first proven by Kato [343], although [616] mistakenly attributes it to Simon.
 2.
Kato [348] proved this result if \(a \in L^2_{loc}\) is replaced by \(a \in L^\nu _{loc}\) and he conjectured this theorem.
 3.
Since \(a_j \in L^2_{loc}\), we have that \(a_j\varphi \in L^1_{loc}\) so \((\nabla _jia_j)\varphi \) is a well defined distribution and it makes sense to say that it is in \(L^2\).
 4.
Just as \(V \in L^2_{\mathrm{loc}}\) is necessary for \(H\varphi \) to lie in \(L^2\) for all \(\varphi \in C_0^\infty ({\mathbb {R}}^\nu )\), \(V \in L^1_{loc}\) and \(a \in L^2_{loc}\) are necessary for \(C_0^\infty \subset V_h\).
 5.
There is an analog of Theorem 9.1 with magnetic field. If \(V \ge 0\), one needs to have \(V \in L^2_{loc}, \, a \in L^4_{loc}\) and \(\nabla \cdot \overrightarrow{a} \in L^2_{loc}\) for H to be defined as an operator on \(C_0^\infty \). It is a theorem of Leinfelder–Simader [420] that this is also sufficient for esa\(\nu \) (see [101, Section 1.4] for a proof along the lines discussed below for the current theorem).
 6.Kato [343] has a lovely way of interpreting that \(C_0^\infty \) is a form core. A natural maximal operator domain for the operator associated with h is \(H_{max}\) defined on (here \(V_h=Q(V)\cap \bigcap _{j=1}^\nu Q(D_j^\nu )\))Since \(\varphi \in V_h\), we have that \(D_j\varphi \in L^2\) which implies that \(a_jD_j\varphi \in L^1_{loc}\) and \(\nabla _j D_j\varphi \) makes sense as a distribution. Also \(\varphi \in V_h \Rightarrow V^{1/2}\varphi \in L^2 \Rightarrow V\varphi = V^{1/2}(V^{1/2}\varphi ) \in L^1_{loc}\) so \(D_j^2\varphi +V\varphi \) is a well defined distribution. What Kato shows is that if H is the operator associated to the closed form, h, then \(H_{max}\) symmetric \(\iff H_{max} = H \iff C_0^\infty \) is a form core for h.$$\begin{aligned} D(H_{max}) = V_h \cap \{\varphi \,\, \sum _{j=1}^{\nu } D_j^2\varphi +V\varphi \in L^2({\mathbb {R}}^\nu )\} \end{aligned}$$(9.6)
Here is a sketch of a proof of Theorem 9.3 following [595]
Step 3. Let \(g \in C_0^\infty ({\mathbb {R}}^\nu )\). Then \(\varphi \mapsto g\varphi \) maps Q(H) to itself. Moreover, if \(g(x) = 1\) for \(x \le 1\) and \(g_n(x)=g(x/n)\), then for any \(\varphi \in Q(H)\) we have that \(g_n\varphi \rightarrow \varphi \) in the form norm of H. Since \(V^{1/2}\varphi \in L^2 \Rightarrow gV^{1/2}\varphi \in L^2\) and \((g_n1)V^{1/2}\varphi _2 \rightarrow 0\), we see that the V pieces behave as claimed. Moreover, \(D_j(g\varphi ) = gD_j\varphi +(\partial _jg)\varphi \) as distributions, so \(D_j\varphi ,\varphi \in L^2 \Rightarrow D_j(g\varphi ), g\varphi \in L^2\) and since \(\partial _j g_n_\infty \le Cn^{1}\), we get the required convergence.
Step 4. Since \(e^{t\Delta }\) maps \(L^2\) to \(L^\infty \), by (9.7), we have that \(e^{H}[L^2]\), which is a form core for H, lies in \(L^\infty \). We conclude by step 3 that \(\{\varphi \in Q(H)\,\, \varphi \in L^\infty \) and \(\varphi \) has compact support\(\}\) is a core for H.
Step 5. We haven’t yet used \(V \in L^1_{loc}\) in that the above arguments work, for example, if \(V(x) = x^{\beta }\) for any \(\beta > 0\). We now want to look at \(k*\varphi \) for \(k \in C_0^\infty ({\mathbb {R}}^\nu )\) and for \(\beta > \nu \) it is easy to see that \(\varphi \mapsto k*\varphi \) does not leave \(Q(x^{\beta })\) invariant (since such functions must vanish at \(x=0\)).
If \(\varphi \) is bounded with compact support and \(V \in L^1_{loc}\) it is easy to see that for \(k \in C_0^\infty ({\mathbb {R}}^\nu )\), we have that \(V^{1/2}(k*\varphi ) \in L^2\) and if \(k_n\) is an approximate identity, that \(V^{1/2}(k_n*\varphi )V^{1/2}\varphi \rightarrow 0\). Similarly, if \((\partial _jia_j)\varphi \in L^2\) and \(\varphi \) bounded with compact support, then \(\partial _j\varphi \in L^2\) so \(D_j(k*\varphi ) \in L^2\) and if \(k_n\) is an approximate identity, then \({D_j(k_n*\varphi )D_j\varphi  \rightarrow 0}\). It follows that \(C_0^\infty ({\mathbb {R}}^\nu )\) is a form core concluding this sketch of the proof of Theorem 9.3.
We next turn to Kato’s original approach to proving his theorem, Theorem 9.1. He proved
Theorem 9.4
Remarks
 1.
What we call \(\mathrm {sgn}(u)\), Kato calls \(\mathrm {sgn}(\bar{u})\).
 2.
We should pause to emphasize what a surprise this was. Kato was a long established master of operator theory. He was 55 years old. Seemingly from left field, he pulled a distributional inequality out of his hat. It is true, like other analysts, that he’d been introduced to distributional ideas in the study of PDEs, but no one had ever used them in this way. Truly a remarkable discovery.
At first sight, Kato’s proof seems to have nothing to do with the semigroup ideas used in the proof of Theorem 9.2 and our first proof of Theorem 9.1. But in trying to understand Kato’s work, I found the following abstract result:
Theorem 9.5
 (a)(\(e^{tA}\) is positivity preserving)$$\begin{aligned} \forall u \in L^2,\, u\ge 0, t \ge 0 \Rightarrow e^{tA}u \ge 0 \end{aligned}$$
 (b)(Beurling–Deny criterion) \(u \in Q(A) \Rightarrow u \in Q(A)\) and$$\begin{aligned} q_A(u) \le q_A(u) \end{aligned}$$(9.21)
 (c)(Abstract Kato Inequality) \(u \in D(A) \Rightarrow u \in Q(A)\) and for all \(\varphi \in Q(A)\) with \(\varphi \ge 0\), one has that$$\begin{aligned} \langle A^{1/2}\varphi ,A^{1/2}u \rangle \ge {{\mathrm{Re}}}\langle \varphi ,\mathrm {sgn}(u) Au \rangle \end{aligned}$$(9.22)
The equivalence of (a) and (b) for M a finite set (so A is a matrix) is due to Beurling–Deny [54]. For a proof of the full theorem (which is not hard), see Simon [582] or [616, Theorem 7.6.4].
The issue with Nelson’s proof is that at the time, the Feynman–Kac–Ito was only known for smooth a’s. One can obtain the Feynman–Kac–Ito for more general a’s by independently proving a suitable core result. Simon [582] and then Kato [348] obtained results for more and more singular a’s until Simon [595] proved
Theorem 9.6
(Simon [595]) (9.30) holds for \(V \ge 0\), \(V \in L^1_{loc}({\mathbb {R}}^\nu )\) and \(\overrightarrow{a} \in L^2_{loc}\).
Indeed, our proof of (9.7) above implies this if we don’t use (9.8) but keep \(e^{tV}\) (equivalently, if we just use (9.9)).
As with Theorem 9.5, there is an abstract two operator Kato inequality result (originally conjectured in Simon [582]):
Theorem 9.7
 (a)For all \(\varphi \in L^2\) and all \(t \ge 0\), we have that$$\begin{aligned} e^{tB}\varphi  \le e^{tA}\varphi  \end{aligned}$$
 (b)\(\psi \in D(B) \Rightarrow \psi  \in Q(A)\) and for all \(\varphi \in Q(A)\) with \(\varphi \ge 0\) and all \(\psi \in D(B)\) we have that$$\begin{aligned} \langle A^{1/2}\varphi ,A^{1/2}\psi  \rangle \le {{\mathrm{Re}}}\langle \varphi ,\mathrm {sgn(\psi ) B\psi } \rangle \end{aligned}$$(9.31)
For a proof, see the original papers or [616, Theorem 7.6.7].
As one might expect, the ideas in Kato [340] have generated an enormous literature. Going back to the original paper are two kinds of extensions: replace \(\Delta \) by \(\sum _{i,j=1}^{\nu } \partial _i a_{ij}(x) \partial _j\) and allowing \(q(x) \rightarrow \infty \) as \(x \rightarrow \infty \) with lower bounds of the Wienholtz–Ikebe–Kato type as discussed in Sect. 8. Some papers on these ideas include Devinatz [117], Eastham et al. [130], Evans [132], Frehse [166], Güneysu–Post [210], Kalf [299], Knowles [379, 380, 381], Milatovic [445] and Shubin [553]. There is a review of Kato [351]. For applications to higher order elliptic operators, see Davies–Hinz [104], Deng et al. [114] and Zheng–Yao [711]. There are papers on V’s obeying \(V(x) \ge \nu (\nu 4)x^{2}; \, \nu \ge 5\), some using Kato’s inequality by Kalf–Walter [302], Schmincke [539], Kalf [297], Simon [576], Kalf–Walter [303] and Kalf et. al. [300].
Kato himself applied these ideas to complex valued potentials in three papers [73, 349, 355]. In particular, Brézis–Kato [355] has been used extensively in the nonlinear equation literature as part of a proof of \(L^p\) regularity of eigenfunctions.
Kato used \(K_\nu \) to discuss local (and global) singularities of the negative part of V. Ironically, \(K_\nu \) is not maximal for such considerations. If \(\nu \ge 3\) and \(V(x) = x^{2} \log (x^{1})^{\delta }\) (for \(x < \tfrac{1}{2})\), then \(V \in K_\nu \iff \delta > 1\) but V is form bounded if and only if \(\delta > 0\). However, Aizenman–Simon [7] have proven the following showing the naturalness of Kato’s class for semigroup considerations:
Theorem 9.8
10 Selfadjointness, IV: quadratic forms
Hilbert, around 1905, originally discussed operators on inner product spaces in terms of (bounded) quadratic forms, not surprising given Hilbert’s background in number theory. F. Riesz emphasized the operator theory point of view starting in 1913 and von Neumann’s approach to unbounded operators in 1929 also emphasized the operator point of view which has dominated most of the discussion since. In the 1930s and 1940s, there was work in which the quadratic form point of view was implicit but it was only in the 1950s that forms became explicitly discussed objects and Kato was a major player in this development. In this section, we’ll first describe the basic theory and give a Kato–centric history and then discuss two special aspects in which Kato had seminal contributions: first, the theory of monotone convergence for forms and secondly, the theory of pseudoFriedrichs extensions and its application to the Dirac Coulomb problem, as well as some other work of Kato on the Dirac Coulomb problem.
In his delightful reminisces of Kato, Cordes [97] quotes Kato as saying “there is no decent Banach space, except Hilbert space.” While this ironic given Kato’s development of eigenvalue perturbation theory and semigroup theory in general Banach spaces, it is likely he had in mind the spectral theorem and the subject of this section.
An elementary fact is:
Theorem 10.1
 (a)If (V, Q) is a sesquilinear form, define a quadratic form, q, by(so \(V_q=V\)).$$\begin{aligned} q(\varphi ) = \left\{ \begin{array}{ll} Q(\varphi ,\varphi ) &{} \hbox { if } \varphi \in V \\ \infty , &{} \hbox { if } \varphi \notin V \end{array} \right. \end{aligned}$$(10.6)
 (b)If q is a quadratic form, take \(V = V_q\) and define a map, Q on \(V \times V\) by$$\begin{aligned} Q(\varphi ,\psi ) = \tfrac{1}{4}[q(\varphi +\psi )q(\varphi \psi )+i q(\varphi i\psi ) i q(\varphi +i\psi )] \end{aligned}$$(10.7)
If \(q:{\mathcal {H}}\rightarrow (\infty ,\infty ]\) so that there is an \(\alpha \) so that \(\widetilde{q}(\varphi )=q(\varphi )+\alpha \varphi ^2\) is a (positive) quadratic form, we say that q is a semibounded quadratic form. Theorem 10.1 extends and we speak of semibounded sesquilinear forms (where \(Q(\varphi ,\varphi ) \ge 0\) is replaced by \(Q(\varphi ,\varphi ) \ge \alpha \varphi ^2\)). For any semibounded sesquilinear form, we define \(\beta =\inf _{\varphi \in V,\varphi \ne 0} Q(\varphi ,\varphi )/\varphi ^2\) to be the lower bound of Q.
We say that a quadratic form, q, is closable if and only if q has a closed extension. One can show that there is then a smallest closed extension, \(\bar{q}\) (in that if t is another closed extension of q, it is also an extension of \(\bar{q}\)).
Example 10.2
Example 10.3
Example 10.4
Given A as in the last example and \(g:\sigma (A) \rightarrow [0,\infty )\) which is continuous and bounded and obeys \(\lim _{t \rightarrow \infty } g(t)=0\), we define g(A) on \({\mathcal {H}}\) by setting it to the spectral theorem g(A) on \({\mathcal {K}}\) and to 0 on \({\mathcal {K}}^\perp \). If \(A=0\) on \({\mathcal {K}}\) (and in some sense \(\infty \) on \({\mathcal {K}}^\perp \)), then for any \(t>0\), we have that \(e^{tA}\) is the orthogonal projection onto \({\mathcal {K}}\).
What makes quadratic forms so powerful is that, in a sense, Example 10.3 has a converse. Here are two versions of this result:
Theorem 10.5
Let q be a closed quadratic form. Let \({\mathcal {K}}=\overline{V_q}\). Then there is a unique positive selfadjoint operator, A, on \({\mathcal {K}}\) so that \(q=q_A\).
Remark
The closure in \(\overline{V_q}\) means closure in the Hilbert space topology (which in many cases is the entire Hilbert space).
Theorem 10.6
 (a)
\(D(A) \subset V_q\)
 (b)If \(\varphi \in D(A), \psi \in V_q\), thenMoreover, D(A) is a form core for A.$$\begin{aligned} Q_q(\psi ,\varphi ) = \langle \psi ,A\varphi \rangle \end{aligned}$$(10.13)
Remarks
 1.
In his book [345], Kato calls Theorem 10.6 the first representation theorem and Theorem 10.5 the second representation theorem. He puts Theorem 10.6 first because it is the version going back to the 1930s (see below). I put Theorem 10.5 first because I think that it is the fundamental result—indeed, it is the only variant in Reed–Simon [494] and Simon [616].
 2.
Example 10.7
If B is not bounded, one can show that \(\widetilde{q_B}\) is never closed but one can prove [616, Theorem 7.5.19] that it is always closable. If \(q^\#\) is its closure, there is a selfadjoint A with \(q^\#=q_A\). One can show (it is immediate from Theorem 10.6) that A is an operator extension of B so B has a natural selfadjoint extension. It is called the Friedrichs extension, \(B_F\). Unless B is esa, there are lots of other selfadjoint extensions as we’ll see. It can happen (but usually doesn’t) that B is not esa but has a unique positive selfadjoint extension.
There is a form analog of the Kato–Rellich theorem:
Theorem 10.8
Remarks
 1.
 2.
If formally \(q(\varphi )=\langle \varphi ,A\varphi \rangle , R(\psi ,\varphi )=\langle \psi ,C\varphi \rangle \), then since s is closed, we have that \(s=q_D\). Then \(Db{\varvec{1}}\) gives a selfadjoint meaning to the formal sum \(A+C\). It is called the form sum.
 3.
The proof is really simple. If \(\cdot  _{+1,q}\) and \(\cdot  _{+1,s}\) are the \(\cdot  _{+1}\) for q and s, then (10.15) implies that the two norms are equivalent so one is complete if and only if the other one is.
Example 10.9
The following is elementary to prove but useful
Theorem 10.10
The sum of two closed quadratic forms is closed
Remarks
 1.
This allows a definition of a selfadjoint sum of any two positive selfadjoint operators.
 2.
It is obvious that \(V_{q_1+q_2} = V_{q_1} \cap V_{q_2}\).
 3.
There is a similar result for n arbitrary closed forms.
 4.
The simplest proof is to use the Davies–Kato characterization (below) that closedness is equivalent to lower semicontinuity.
 \(\textcircled {1}\).

There are closed symmetric operators which are not selfadjoint but every closed quadratic form is the form of a selfadjoint operator.
 \(\textcircled {2}\).

Every symmetric operator has a smallest closed extension but there exist quadratic forms with no closed extensions.
 \(\textcircled {3}\).
 If A and B are selfadjoint operators and B is an extension of A (i.e. \(D(A) \subset D(B)\) and Open image in new window ), then \(A=B\). But there exist closed quadratic forms \(q_1\) and \(q_2\) where \(q_2\) is an extension of \(q_1\) but \(q_1 \ne q_2\). For example, let \({\mathcal {H}}= L^2([0,1],dx)\) and \(q_0\) given byHere \(C^\infty ([0,1])\) means the functions infinitely differentiable on [0, 1] with one sided derivatives at the end points. Let \(q_1\) be the closure of the restriction of \(q_0\) to \(C_0^\infty (0,1)\) and \(q_2\) the closure of \(q_0\). Then \(q_1\) is the quadratic form of \(\tfrac{d^2}{dx^2}\) with Dirichlet boundary conditions and \(q_2\) the quadratic form of \(\tfrac{d^2}{dx^2}\) with Neumann boundary conditions (see [616, Examples 7.5.25 and 7.5.26]) and \(q_2\) is an extension of \(q_1\).$$\begin{aligned} q_0(\varphi ) = \left\{ \begin{array}{ll} \int _{0}^{1} \varphi '(x)^2\, dx, &{} \hbox { if } \varphi \in C^\infty ([0,1]) \\ \infty , &{} \hbox { otherwise} \end{array} \right. \end{aligned}$$
Friedrichs [168, 169] (long before Krein) provided the first proof of von Neumann’s conjecture (Stone [630] had a proof at about the same time) by a construction related to the method behind Theorem 10.6. A followup paper of Freudenthal [167] did Friedrichs extension in something close to form language. In the 1950s, work on parabolic PDEs and NRQM by Kato [323], Lax–Milgram [419], Lions [431] and Nelson [459] led to a systematic general theory. In particular, Kato’s lecture notes [323] had considerable impact.
We will need the following result of Simon [586] (see also [616, Theorem 7.5.15])
Theorem 10.11
Any quadratic form q has an associated closed quadratic form, \(q_r\), which is the largest closed form less than q, i.e. \(q_r \le q\) and if t is closed with \(t \le q\), then \(t \le q_r\).
Remarks
 1.
One defines \(q_s=qq_r\). More precisely, \(V_{q_s} = V_q\) and for \(\varphi \in V_q\) we have that \(q_s(\varphi )=q(\varphi )q_r(\varphi )\). “r” is for regular and “s” for singular.
 2.Let \(\mu \) and \(\nu \) be two probability measures on a compact space, X, and \(d\nu = fd\mu +d\nu _s\) with \(d\nu _s\) singular wrt \(d\mu \) the Lebesgue decomposition (see [612, Theorem 4.7.3]). If \({\mathcal {H}}=L^2(X,d\mu )\) and if \(q_\nu \) is defined with \(V_{q_\nu }=C(X)\) and for \(\varphi \in C(X)\)then [616, Problem 7.5.7] \((q_\nu )_r\) is the closure of the form (on C(X))$$\begin{aligned} q_\nu (\varphi ) = \int \varphi (x)^2 d\nu (x) \end{aligned}$$(10.21)whose associated operator is multiplication by f(x) (on the operator domain of those \(\varphi \) with \(\int f(x)^2\varphi (x)^2 d\mu < \infty \)). \(V_{q_s} = C(X)\). For \(\varphi \in C(X)\), \(q_s\) is given by (10.21) with \(d\nu \) replaced by \(d\nu _s\). In particular, if q is the form of (10.11), then \(q_r=0\).$$\begin{aligned} \varphi \mapsto \int f(x)\varphi (x)^2 d\mu \end{aligned}$$(10.22)
The two monotone convergence theorems for (positive) quadratic forms are
Theorem 10.12
Theorem 10.13
Remarks
 1.
For proofs, see [616, Theorem 7.5.18].
 2.
Let \(q_n\) be the form of \(\tfrac{1}{n}\tfrac{d^2}{dx^2}+\delta (x)\) as defined in Example 10.9. Then \(q_n\) is decreasing and \(q_\infty \) is the form \(\delta (x)\) so that \((q_\infty )_r=0\). This shows that in the decreasing case, the limit need not be closed or even closable.
Theorems of this genre appeared first in Kato’s book [345] (already in the first edition). He only considered cases where all \(V_{q_n}\) are dense. In the increasing case, he assumed there was a \(\tilde{q}\) with \(V_{\tilde{q}}\) dense so that for all n, one has that \(q_n \le \tilde{q}\). In both cases, he proved there was a selfadjoint operator, \(A_\infty \), with \(A_n\) converging to \(A_\infty \) in srs. He considered the form \(q_\infty (\varphi ) = \lim _{n} q_n(\varphi )\). In the decreasing case, he proved that if \(q_\infty \) is closable, its closure is the form of \(A_\infty \). In the increasing case, he said it was an open question whether \(q_\infty \) was the form of \(A_\infty \). This material from the 1966 first edition was unchanged from the 1976 second edition.
In 1971, Robinson [518] proved Theorem 10.12. He noted that \(q_\infty \) was closed by writing \(q_n = \sum _{j=1}^{n}s_j\) where \(s_1=q_1, s_j=q_jq_{j1}\) if \(j \ge 2\). Then \(q_\infty =\sum _{j=1}^{\infty } s_j\) and he says that the proof that \(q_\infty \) is closed is the same as the proof that an infinite direct sum of Hilbert spaces is complete; see Bratteli–Robinson [72, Lemma 5.2.13] for a detailed exposition of the proof. In 1975, Davies [102] also proved this theorem. His proof relied on lower semicontinuity being equivalent to q being closed (see below). Robinson seems to have been aware of the results in Kato’s book. While Davies quotes Kato’s book for background on quadratic forms, he may have been unaware of the monotone convergence results which are in a later chapter (Chapter VIII) than the basic material on forms (Chapter VI). When Kato published his second edition, he was clearly unaware of their work.
The lower semicontinuity fits in nicely with even then well known work on variational problems that used the weak lower semicontinuity of Banach space norms so it was not surprising. Indeed Davies mentions it in passing in his paper without proof. To add to the historical confusion, in his 1980 book [103], when Davies quoted this result, he seems to have forgotten that it appeared first explicitly in his paper and attributes it to the 1966 first edition of Kato [345] where it doesn’t appear!
Shortly after this second edition, I wrote and published [586] which had the notion of \((q)_r\) and the full versions of Theorems 10.12 and 10.13. I noted that these extended and complemented what was in Kato’s book. At the time I wrote the preprint, I was unaware of the relevant work of Davies and Robinson although I knew each of them personally. In response to my preprint, Kato wrote to me that he had an alternate proof that in the increasing case, \(q_\infty \) was always closed. He stated a lovely result.
Theorem 10.14
A quadratic form is closed if and only if it is lower semicontinuous as a function from \({\mathcal {H}}\) to \([0,\infty ]\).
Remarks
 1.
For a proof, see [616, Theorem 7.5.2]
 2.
This theorem provides a quick proof that \(\delta (x)\) is not closable. It is easy to find a \(C_0^\infty ({\mathbb {R}})\) function \(\varphi \) with \(\varphi (0)=1\) and a sequence \(\varphi _n \in C_0^\infty \) with \(\varphi _n(0)=0,\,\varphi _n \le \varphi \) and \(\varphi _n \rightarrow \varphi \) in \(L^2\). Given this convergent sequence with \(\lim \delta (\varphi _n) =0 < \delta (\varphi ) = 1\), there cannot be a lower semicontinuous function that agrees with \(\delta \) on \(C_0^\infty \).
Given the theorem, it is immediate that \(q_\infty \) is closed in the increasing case, since an increasing limit of lower semicontinuous functions is lower semicontinuous. I note that in precisely this context, Theorem 10.14 was also found by Davies [102]. Kato told me that he had no plans to publish his remark and approved my writing [587] that explores consequences of Theorem 10.14. However, in 1980, Springer published an “enlarged and corrected” printing of the second edition of Kato’s book and one of the few changes was a completely reworked discussion of monotone convergence theorems! In particular, he had the full Theorem 10.12 using Theorem 10.14. In the Supplemental Notes, he quotes [586] and [587] but neither of the papers of Davies and Robinson, despite the fact that in response to their writing to me after the preprint, I added a Note Added in Proof to [586] referencing their work.
Theorem 10.15
Kato called C the pseudoFriedrichs extension. Kato remarked that this had little to do with quadratic forms (which for him were positive) but the constructions shared elements of Friedrichs’ construction of his extension. Faris [147] has a presentation that uses sesquilinear forms and makes this closer to the KLMN theorem.
So while the book is given as the source for the inequality, the standard place given for the proof is a lovely paper of Herbst [239] who computes the norm of \(x^{\alpha }p^{\alpha }\) as an operator on \(L^p({\mathbb {R}}^\nu )\) when \(1< p < \nu \alpha ^{1}\) (that the operator is bounded on \(L^p\) is a theorem of Stein–Weiss [624]). This has as special cases the optimal constants for Kato’s, Hardy’s and Rellich’s inequalities. Herbst notes that this operator commutes with scaling, so after applying the Mellin transform, it commutes with translations and so, it is a convolution operator in Mellin transform space. The function it is convolution with is positive function so the norm is related to the computable integral of this explicit function. Five later publications on the optimal constant are Beckner [45], Yafaev [702], Frank–Lieb–Seiringer [162], Frank–Seiringer [164] and Balinsky–Evans [38, pgs 48–50].
That said, Kato’s ideas stimulated later work which picked out a natural extension for all \(\mu \) with \(\mu  < 1\). Among the papers on the subject are Schmincke [541], Wüst [691, 692, 693], Nenciu [463], Kalf et. al. [300], Estaban–Loss [144] and Estaban–Lewin–Séré [143]. Domain conditions motivated by Kato’s pseudoFriedrichs extension are common. Typical is the following result of Nenciu [463] (which is a variant of Schmincke [541]):
Theorem 10.16
(10.32) uses the fact that, by the above mentioned inequality of Kato, if \(\psi \in D(T_0^{1/2})\), then \(\psi \in D(r^{1/2})\).
In 1983, Kato wrote a further paper on the Dirac Coulomb problem [353] (see also [354]) which seems to be little known (I only learned of it while preparing this article). To understand Kato’s idea, return to \(\Delta \beta r^{2}\) on \(L^2({\mathbb {R}}^\nu ), \nu \ge 5\) as discussed in Proposition 7.7 above. If \(0 < \beta \le \tfrac{\nu (\nu 4)}{4}\), then \(H(\beta )\) can be defined as the operator closure of the operator on \(C_0^\infty ({\mathbb {R}}^\nu )\). It is selfadjoint and except at the upper end, we know the domain is that of \(\Delta \). For \(\tfrac{\nu (\nu 4)}{4} < \beta \le \tfrac{(\nu 2)^2}{4}\), there is a Friedrichs extension since \(\Delta \beta r^{2} \ge 0\) on \(C_0^\infty ({\mathbb {R}}^\nu )\). Kato notes that the Friedrichs extension is natural from the following point of view: \(H(\beta )\) is an analytic family of operators for \(0< \beta < \tfrac{(\nu 2)^2}{4}\) and is the unique analytic family from the esa region—it is type (A) if \(\beta \in (0,\tfrac{\nu (\nu 4)}{4})\) and type (B) if \(\beta \in (0,\tfrac{(\nu 2)^2}{4})\). (In fact, it can proven that as a holomorphic family, there is a square root singularity at \(\beta = \tfrac{(\nu 2)^2}{4}\) and in the variable \(m = \sqrt{\beta \tfrac{(\nu 2)^2}{4}}\), one has a holomorphic family in \(\text{ Re }(m) > 1\); see Bruneau–Dereziński–Georgescu [76]).
In the same way, Kato showed that the distinguished selfadjoint extension of the Dirac operator in (10.26) found by others for \(\mu  < 1\) is an analytic family for \(\mu \in (1,1)\) and is the unique analytic continuation from the Kato–Rellich region \(\mu \in (\tfrac{1}{2},\tfrac{1}{2})\).
Notes
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