**About the Book** 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).

**Table of Contents**Preface

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

Bibliography

Glossary of notation

Index