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Linear Operators in Hilbert Spaces

  • Alessandro Teta
Chapter
Part of the UNITEXT for Physics book series (UNITEXTPH)

Abstract

We recall some fundamental notions of the theory of linear operators in Hilbert spaces which are required for a rigorous formulation of the rules of Quantum Mechanics in the one-body case. In particular, we introduce and discuss the main properties of bounded and unbounded operators, adjoint operators, symmetric and self-adjoint operators, self-adjointness criterion and stability of self-adjointness under small perturbations, spectrum, isometric and unitary operators, spectral theorem, unitary group, decomposition of the spectrum, and Weyl’s theorem on the essential spectrum. The above topics are treated with the help of examples and exercises and avoiding complete generality. In some cases, e.g., for the spectral theorem, the results are formulated without proofs.

References

  1. 1.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I: Functional Analysis. Academic Press, New York (1980)Google Scholar
  2. 2.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)Google Scholar
  3. 3.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV: Analysis of Operators. Academic Press, New York (1978)Google Scholar
  4. 4.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover Publ, New York (1993)Google Scholar
  5. 5.
    Amrein, W.O.: Hilbert Space Methods in Quantum Mechanics. EPFL Press, Lausanne (2009)Google Scholar
  6. 6.
    Blanchard, P., Brüning, E.: Mathematical Methods in Physics. Birkhäuser, Boston (2015)Google Scholar
  7. 7.
    Birman, M.S., Solomjak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. D. Reidel Publ. Co., Dordrecht (1987)Google Scholar
  8. 8.
    Dell’Antonio G.: Lectures on the Mathematics of Quantum Mechanics. Atlantis Press (2015)Google Scholar
  9. 9.
    Gustafson, S.J., Sigal, I.M.: Mathematical Concepts of Quantum Mechanics. Springer, Berlin (2011)Google Scholar
  10. 10.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)Google Scholar
  11. 11.
    Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Dover Publ, New York (1999)Google Scholar
  12. 12.
    de Oliveira, C.R.: Intermediate Spectral Theory and Quantum Dynamics. Birkhäuser, Basel Boston Berlin (2009)Google Scholar
  13. 13.
    Prugovecki, E.: Quantum Mechanics in Hilbert Space. Academic Press, New York (1981)Google Scholar
  14. 14.
    Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Springer, Dordrecht (2012)Google Scholar
  15. 15.
    Teschl, G.: Mathematical Methods in Quantum Mechanics. American Mathematical Society, Providence (2009)Google Scholar
  16. 16.
    Thirring, W.: Quantum Mechanics of Atoms and Molecules. Springer, New York (1981)Google Scholar
  17. 17.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2011)Google Scholar
  18. 18.
    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, Rhode Island (1998)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica Guido CastelnuovoUniversità degli Studi di Roma “La Sapienza”RomeItaly

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