Trace Class Perturbations of Spectra of Self-adjoint Operators

Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 265)

Abstract

Chapter 9 deals with trace class perturbations of parts of the spectrum of self-adjoint operators. In the first section, the decomposition of a self-adjoint operator and its spectrum into a pure point part, a singularly continuous part, and an absolutely continuous part is obtained. Let A be a self-adjoint operator, and D a self-adjoint trace class operator acting on the same Hilbert space, and set B=A+D. Krein’s spectral shift associated with such a pair {B,A} is a function on the real line which involves an integral representation formula for the trace of the difference of functions of A and B. The spectral shift and its properties and the proof of Krein’s trace formula are central themes of this chapter. We follow M.G. Krein’s original approach to the spectral shift which was based on perturbation determinants. A short introduction into perturbation determinants is given in a separate section. Another main topic of Chap. 9 is the spectral theory of rank one perturbations B=A+α〈⋅,uu of the operator A, where α is a real number, and u is a unit vector. We prove the Aronszajn–Donoghue theorem. It characterizes the various parts of the spectrum of the self-adjoint operator B in terms of boundary values on ℝ  of the holomorphic function 〈R z (A)u,u〉.

Keywords

Spectral Shift Trace Formula Continuous Part Trace Class Trace Class Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Classical Articles

1. [K1]
Kato, T.: Fundamental properties of Hamiltonians of Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951)
2. [Re]
Rellich, F.: Störungstheorie der Spektralzerlegung II. Math. Ann. 116, 555–570 (1939)
3. [W1]
Weyl, H.: Über beschränkte quadratische Formen, deren Differenz vollstetig ist. Rend. Circ. Mat. Palermo 27, 373–392 (1909)

Books

1. [BG]
Berenstein, C.A., Gay, R.: Complex Variables. Springer-Verlag, New York (1991)
2. [Fe]
Federer, H.: Geometric Measure Theory. Springer-Verlag, New York (1969)
3. [GGK]
Gokhberg, I.C., Goldberg, S., Krupnik, N.: Traces and Determinants of Linear Operators. Birkhäuser-Verlag, Basel (2000)
4. [GK]
Gokhberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Am. Math. Soc., Providence (1969)
5. [Kr6]
Krein, M.G.: Topics in Differential and Integral Equations and Operator Theory, pp. 107–172. Birkhäuser-Verlag, Basel (1983)
6. [RS3]
Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. Scattering Theory. Academic Press, New York (1979)
7. [RS4]
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York (1978)
8. [Sm1]
Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979)
9. [Wr]
Wiener, N.: The Fourier Integral and Certain of Its Applications. Cambridge Univ. Press, New York (1933) Google Scholar
10. [Yf]
Yafaev, D.R.: Mathematical Scattering Theory: General Theory. Am. Math. Soc., Providence, RI (1992)

Articles

1. [Ar]
Aronszajn, N.: On a problem of Weyl in the theory of Sturm–Liouville equations. Am. J. Math. 79, 597–610 (1957)
Aronzajn, N., Donoghue, W.F.: On exponential representations of analytic functions in the upper half-plane with positive imaginary part. J. Anal. Math. 5, 321–388 (1957)
3. [BK]
Birman, M.Sh., Krein, M.G.: On the theory of wave operators and scattering operators. Dokl. Akad. Nauk SSSR 144, 475–478 (1962)
4. [BP]
Birman, M.Sh., Pushnitski, A.B.: Spectral shift function, amazing and multifaceted. Integral Equ. Oper. Theory 30, 191–199 (1998)
5. [BY]
Birman, M.Sh., Yafaev, D.R.: The spectral shift function. The work of M.G. Krein and its further development. St. Petersburg Math. J. 4, 833–870 (1993)
6. [Dn2]
Donoghue, W.F.: On the perturbation of spectra. Commun. Pure Appl. Math. 18, 559–579 (1965)
7. [GMN]
Gesztesy, F., Makarov, K.A., Naboko, S.N.: The spectral shift operator. Oper. Theory Adv. Appl. 108, 59–99 (1999)
8. [Kp]
Koplienko, L.S.: On the trace formula for perturbations of non-trace class type. Sib. Mat. Zh. 25, 72–77 (1984)
9. [Kr4]
Krein, M.G.: On the trace formula in perturbation theory. Mat. Sb. 33, 597–626 (1953)
10. [Kr5]
Krein, M.G.: On perturbation determinants and the trace formula for unitary and self-adjoint operators. Dokl. Akad. Nauk SSSR 144, 268–273 (1962)
11. [Lf]
Lifshits, I.M.: On a problem in perturbation theory. Usp. Mat. Nauk 7, 171–180 (1952)
12. [Sm2]
Simon, B.: Notes on infinite determinants. Adv. Math. 24, 244–273 (1977)
13. [Sm3]
Simon, B.: Spectral analysis of rank one perturbations and applications. CRM Lecture Notes, vol. 8, pp. 109–149. Am. Math. Soc., Providence (1995). Google Scholar
14. [SW]
Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math. 39, 75–90 (1986)
15. [Str]
Strichartz, R.: Multipliers on fractional Sobolev spaces. J. Math. Mech. 16, 1031–1060 (1967)
16. [Wt]
Wüst, R.: Generalizations of Rellich’s theorem on perturbations of (essentially) self-adjoint operators. Math. Z. 119, 276–280 (1971)