Advertisement

Perturbations of Self-adjoint Operators

  • Konrad Schmüdgen
Part of the Graduate Texts in Mathematics book series (GTM, volume 265)

Abstract

Chapter 8 begins with differential operators with constant coefficients on ℝ d . The Fourier transform allows us to give an elegant approach to these operators and their spectral properties. Then the self-adjointness of the sum A+B of self-adjoint operators under relatively bounded perturbations is studied, and the theorems of Kato–Rellich and Wüst are derived. The essential spectrum of a self-adjoint operator is investigated, and versions of Weyl’s theorem on the invariance of the essential spectrum under relatively compact perturbations are proved. The main motivation for these investigations stems from quantum mechanics, where A+B=−Δ+V is a Schrödinger operator. The operator-theoretic results of this chapter are applied for studying the self-adjointness and the essential spectrum of Schrödinger operators.

Keywords

Compact Operator Essential Spectrum Borel Function Jacobi Operator Compact Perturbation 
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.

References

Books

  1. [AG]
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Ungar, New York (1961) Google Scholar
  2. [BS]
    Birman, M.S., Solomyak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Kluwer, Dordrecht (1987) Google Scholar
  3. [EE]
    Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Clarendon Press, Oxford (1987) Google Scholar
  4. [K2]
    Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966) Google Scholar
  5. [RS1]
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, New York (1972) Google Scholar
  6. [RS4]
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York (1978) Google Scholar
  7. [We]
    Weidmann, J.: Linear Operators in Hilbert Spaces. Springer-Verlag, Berlin (1987) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Konrad Schmüdgen
    • 1
  1. 1.Dept. of MathematicsUniversity of LeipzigLeipzigGermany

Personalised recommendations