Graduate Studies in Mathematics, Volume 157, Amer. Math. Soc., Providence, 2014.

Mathematical Methods in Quantum Mechanics

With Applications to Schrödinger Operators

Gerald Teschl

This manuscript provides a brief introduction to mathematical methods in quantum mechanics (spectral theory) with applications to Schrödinger operators. The first part covers mathematical foundations of quantum mechanics from self-adjointness, the spectral theorem, quantum dynamics (including Stone's and the RAGE theorem) to perturbation theory for self-adjoint operators.

The second part starts with a detailed study of the free Schrödinger operator respectively position, momentum and angular momentum operators. Then we develop Weyl-Titchmarsh theory for Sturm-Liouville operators and apply it to spherically symmetric problems, in particular to the hydrogen atom. Next we investigate self-adjointness of atomic Schrödinger operators and their essential spectrum, in particular the HVZ theorem. Finally we have a look at scattering theory and prove asymptotic completeness in the short range case.

MSC2000: 81-01, 81Qxx, 46-01, 34Bxx, 47B25
Keywords: Schrödinger operators, quantum mechanics, unbounded operators, spectral theory.

Ordering Details
Cover Publisher:American Mathematical Society
Series: Graduate Studies in Mathematics, ISSN: 1065-7338
Volume: 157
Publication Year: 2014
ISBN: 978-1-4704-1704-8
Paging: 356 pp; hardcover
List Price: $67
Member Price: $53.60
Itemcode: GSM/157
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Table of contents

    Part 0: Preliminaries

    A first look at Banach and Hilbert spaces
    1. Warm up: Metric and topological spaces
    2. The Banach space of continuous functions
    3. The geometry of Hilbert spaces
    4. Completeness
    5. Bounded operators
    6. Lebesgue Lpspaces
    7. Appendix: The uniform boundedness principle

    Part 1: Mathematical Foundations of Quantum Mechanics

  1. Hilbert spaces
    1. Hilbert spaces
    2. Orthonormal base
    3. The projection theorem and the Riesz lemma
    4. Orthogonal sums and tensor products
    5. The C* algebra of bounded linear operators
    6. Weak and strong convergence
    7. Appendix: The Stone-Weierstraß theorem
  2. Self-adjointness and spectrum
    1. Some quantum mechanics
    2. Self-adjoint operators
    3. Quadratic forms and the Friedrichs extension
    4. Resolvents and spectra
    5. Orthogonal sums of operators
    6. Self-adjoint extensions
    7. Appendix: Absolutely continuous functions
  3. The spectral theorem
    1. The spectral theorem
    2. More on Borel measures
    3. Spectral types
    4. Appendix: Herglotz-Nevanlinna functions
  4. Applications of the spectral theorem
    1. Integral formulas
    2. Commuting operators
    3. Polar decomposition
    4. The min-max theorem
    5. Estimating eigenspaces
    6. Tensor products of operators
  5. Quantum dynamics
    1. The time evolution and Stone's theorem
    2. The RAGE theorem
    3. The Trotter product formula
  6. Perturbation theory for self-adjoint operators
    1. Relatively bounded operators and the Kato-Rellich theorem
    2. More on compact operators
    3. Hilbert-Schmidt and trace class operators
    4. Relatively compact operators and Weyl's theorem
    5. Relatively form bounded operators and the KLMN theorem
    6. Strong and norm resolvent convergence
  7. Part 2: Schrödinger Operators

  8. The free Schrödinger operator
    1. The Fourier transform
    2. Sobolev spaces
    3. The free Schrödinger operator
    4. The time evolution in the free case
    5. The resolvent and Green's function
  9. Algebraic methods
    1. Position and momentum
    2. Angular momentum
    3. The harmonic oscillator
    4. Abstract commutation
  10. One-dimensional Schrödinger operators
    1. Sturm-Liouville operators
    2. Weyl's limit circle, limit point alternative
    3. Spectral transformations I
    4. Inverse spectral theory
    5. Absolutely continuous spectrum
    6. Spectral transformations II
    7. The spectra of one-dimensional Schrödinger operators
  11. One-particle Schrödinger operators
    1. Self-adjointness and spectrum
    2. The hydrogen atom
    3. Angular momentum
    4. The eigenvalues of the hydrogen atom
    5. Nondegeneracy of the ground state
  12. Atomic Schrödinger operators
    1. Self-adjointness
    2. The HVZ theorem
  13. Scattering theory
    1. Abstract theory
    2. Incoming and outgoing states
    3. Schrödinger operators with short range potentials
  14. Part 3: Appendix

  15. Almost everything about Lebesgue integration
    1. Borel measures in a nut shell
    2. Extending a premeasure to a measure
    3. Measurable functions
    4. How wild are measurable objects
    5. Integration - Sum me up, Henri
    6. Product measures
    7. Transformation of measures and integrals
    8. Vague convergence of measures
    9. Decomposition of measures
    10. Derivatives of measures