Mathematical Methods in Quantum Mechanics – Gerald Teschl – 1st Edition

Quantum mechanics and the of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This is a brief, but self-contained, to the of quantum mechanics, with a view towards applications to Schrodinger operators.

Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution.

Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.

This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced and a one-semester introduction to are required. In particular, no functional and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.

Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.


    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* 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
Title: Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators
Author: Gerald Teschl
Edition: 1st Edition
ISBN: 1470417049 | 978-1470417048
Type: eBook
Language: English

No Comments

  • Can you please leave feedback and comments here

    Your opinions and comments would be greatly appreciated. If you have comments or questions we've added this section so that we might have a dialogue with you.

Complete all fields

one × five =