Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators (Second Edition)
Gerald Teschl
180 x 240 mm
Year of Publishing
Territorial Rights
Universities Press

Quantum mechanics and the theory 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 book is a brief, but self-contained, introduction to the mathematical methods 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 Schrödinger 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 calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. This new edition has additions and improvements throughout the book to make the presentation more student friendly. The book is written in a very clear and compact style. It is well suited for self-study and includes numerous exercises (many with hints).

Gerald Teschl is University Professor at the Institute for Mathematics, University of Vienna, Vienna, Austria.

Part 0. Preliminaries 
 Chapter 0. A first look at Banach and Hilbert spaces 
Appendix: The uniform boundedness principle 
Part 1. Mathematical Foundations of Quantum Mechanics 
Chapter 1. Hilbert spaces 
Appendix: The Stone–Weierstraß theorem  
Chapter 2. Self-adjointness and spectrum 
Appendix: Absolutely continuous functions 
Chapter 3. The spectral theorem 
Appendix: Herglotz–Nevanlinna functions 
Chapter 4. Applications of the spectral theorem 
Chapter 5. Quantum dynamics 
Chapter 6. Perturbation theory for self-adjoint operators 
Part 2. Schrodinger Operators 
Chapter 7. The free Schrodinger operator 
Chapter 8. Algebraic methods 
Chapter 9. One-dimensional Schrodinger operators 
Chapter 10. One-particle Schrodinger operators 
Chapter 11. Atomic Schrodinger operators 
Chapter 12. Scattering theory 
Part 3. Appendix 
Appendix A Almost everything about Lebesgue integration 
Bibliographical notes 
Glossary of notation 

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