Abstract
A solution of the problems raised in Introduction requires appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces, especially of unbounded symmetric and self-adjoint operators. We remind the reader of basic notions and facts from the theory of Hilbert spaces and of linear operators in such spaces which are relevant to the subject of the present book.
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- 1.
Finite-dimensional Hilbert spaces (or Euclidean spaces) are also encountered in QM as spaces of states, e.g., in QM of two-level systems, finite spin systems, and so on. Finite-dimensional spaces are free from the problems that are examined in the present book.
- 2.
We use “iff” in its standard usage for “if and only if.” For brevity, the arrow ⇒ stands for “implies.”
- 3.
A Hilbert space is a particular case of a normed and metric space in which a norm and a metric (distance) satisfying standard requirements are generated by a scalar product; see [9].
- 4.
In short, a Hilbert space is complete with respect to a metric generated by a scalar product.
- 5.
We hope that there will be no confusion with the similar symbols for complex conjugation and closure; they refer to different notions, the first involving complex numbers, and the second, sets.
- 6.
We are forced to use the symbol (ξ1 /ξ2) for a two-component column (“spinor”) instead of the conventional symbol \(\left (\begin{array}{c} {\xi }_{1} \\ {\xi }_{2} \end{array} \right )\) for reasons of space.
- 7.
When speaking about some function belonging to L 2(a, b) and possessing some additional specific properties like absolute continuity, we actually mean the representative of the corresponding equivalence class.
- 8.
By local integrability on an interval (a, b), we mean the (absolute) integrability on any finite interval [9] belonging to (a, b), a ≤ α < β ≤ b, where the equality signs are meaningful for finite endpoints; by local integrability in an interval (a, b), we mean the integrability on any finite interval [9] within (a, b), a < α < β < b.
- 9.
In the Russian mathematical literature, a smooth compactly supported function is known as a “finitnaya” function.
- 10.
A subtlety is that the set of powers of x, {x k}0 ∞, is a complete sequence in L 2 − 1, 1, but it does not form a basis [9].
- 11.
Encountered unbounded linear functionals cannot be defined in the whole Hilbert space: an unbounded linear functional defined everywhere is equal to zero almost everywhere. The requirement of boundedness is often included in the definition of a linear functional.
- 12.
In what follows, we often omit the symbol for the argument of a function if it is clear from context.
- 13.
An unbounded operator cannot in general be defined in all of \(\mathfrak{H}\).
- 14.
It may be that ξ∉D f if \(\hat{f}\) is not closed; a continuous \(\hat{f}\) can be nonclosed, but is always closable (see below).
- 15.
In particular, criteria for closability and methods for constructing the closure are formulated in terms of them.
- 16.
The latter term is used to avoid confusion with the kernel of an integral operator.
- 17.
We separate Lemmas 2.32 and 2.29 for our later convenience.
- 18.
In fact, the spectrum of any operator is closed [125].
- 19.
We can also ask whether the subspace \(\mathcal{E}{\mathbb{G}}_{f}\) is a graph. In fact, we already know the answer: it is easy to see that \(\mathcal{E}{\mathbb{G}}_{f}\) is a graph iff \(\hat{f}\) is invertible, and if so, \(\mathcal{E}{\mathbb{G}}_{f}\) determines the operator \(-\hat{{f}}^{-1}\), \(\mathcal{E}{\mathbb{G}}_{f} = {\mathbb{G}}_{-{f}^{-1}}.\)
- 20.
Some of the assertions are already known; we collect them for future reference.
- 21.
Without loss of generality, we can assume \(\hat{f}\) to be closed; then the domain of \(\hat{\mathcal{R}}\left (z\right )\) is closed.
- 22.
If \(\hat{\mathcal{R}}\left (z\right )\) is densely defined, the point z is a regular point, and \(\hat{\mathcal{R}}\left (z\right )\) is the resolvent of \(\hat{f}\) at the point z.
- 23.
If two operators \(\hat{g}\) and \(\hat{h}\) are related by \(\hat{g} =\hat{ h}\hat{s}\), where \(\hat{s}\) is defined everywhere and invertible and \(\hat{s}\) − 1 is also defined everywhere, the kernels of these operators are related by \(\ker \hat{g} =\hat{ {s}}^{-1}\ker \hat{h}\) and \(\ker \hat{h} =\hat{ s}\ker \hat{g}\), which implies that the kernels are of the same dimension.
- 24.
According to Theorem 2.44; see also Lemma 2.67.
- 25.
We recall that unitary operators are bounded and defined everywhere, and the notion of commutativity for such operators is unambiguous; see Sect. 2.3.3.
- 26.
It may be that an s.a. operator has no eigenvectors, in which case its spectrum is continuous.
- 27.
In contrast to the general closed operator, whose spectrum can be empty [128]. We also note that the spectrum of any bounded operator is not empty.
- 28.
This classification is sufficient for our purposes. A more advanced classification can be found in [128].
- 29.
It may be an exotic situation whereby the point spectrum is dense in the continuous spectrum and the eigenvectors corresponding to the point spectrum form a complete orthonormalized set.
- 30.
For the identity operator \(\hat{I}\), the point and continuous spectra coincide, reducing to the single eigenvalue λ = 1 of infinite multiplicity.
- 31.
The convenience of the minus sign in front of the vector ζ becomes clear below.
References
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Gitman, D.M., Tyutin, I.V., Voronov, B.L. (2012). Linear Operators in Hilbert Spaces. In: Self-adjoint Extensions in Quantum Mechanics. Progress in Mathematical Physics, vol 62. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4662-2_2
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