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 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 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.

Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics 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.

- Preface
- Warm up: Metric and topological spaces
- The Banach space of continuous functions
- The geometry of Hilbert spaces
- Completeness
- Bounded operators
- Lebesgue
*L*spaces^{p} - Appendix: The uniform boundedness principle
- Hilbert spaces
- Hilbert spaces
- Orthonormal base
- The projection theorem and the Riesz lemma
- Orthogonal sums and tensor products
- The
*C*algebra of bounded linear operators^{*} - Weak and strong convergence
- Appendix: The Stone-Weierstraß theorem

- Self-adjointness and spectrum
- Some quantum mechanics
- Self-adjoint operators
- Quadratic forms and the Friedrichs extension
- Resolvents and spectra
- Orthogonal sums of operators
- Self-adjoint extensions
- Appendix: Absolutely continuous functions

- The spectral theorem
- The spectral theorem
- More on Borel measures
- Spectral types
- Appendix: Herglotz-Nevanlinna functions

- Applications of the spectral theorem
- Integral formulas
- Commuting operators
- Polar decomposition
- The min-max theorem
- Estimating eigenspaces
- Tensor products of operators

- Quantum dynamics
- The time evolution and Stone’s theorem
- The RAGE theorem
- The Trotter product formula

- Perturbation theory for self-adjoint operators
- Relatively bounded operators and the Kato-Rellich theorem
- More on compact operators
- Hilbert-Schmidt and trace class operators
- Relatively compact operators and Weyl’s theorem
- Relatively form bounded operators and the KLMN theorem
- Strong and norm resolvent convergence

- The free Schrödinger operator
- The Fourier transform
- Sobolev spaces
- The free Schrödinger operator
- The time evolution in the free case
- The resolvent and Green’s function

- Algebraic methods
- Position and momentum
- Angular momentum
- The harmonic oscillator
- Abstract commutation

- One-dimensional Schrödinger operators
- Sturm-Liouville operators
- Weyl’s limit circle, limit point alternative
- Spectral transformations I
- Inverse spectral theory
- Absolutely continuous spectrum
- Spectral transformations II
- The spectra of one-dimensional Schrödinger operators

- One-particle Schrödinger operators
- Self-adjointness and spectrum
- The hydrogen atom
- Angular momentum
- The eigenvalues of the hydrogen atom
- Nondegeneracy of the ground state

- Atomic Schrödinger operators
- Self-adjointness
- The HVZ theorem

- Scattering theory
- Abstract theory
- Incoming and outgoing states
- Schrödinger operators with short range potentials

- Almost everything about Lebesgue integration
- Borel measures in a nut shell
- Extending a premeasure to a measure
- Measurable functions
- How wild are measurable objects
- Integration – Sum me up, Henri
- Product measures
- Transformation of measures and integrals
- Vague convergence of measures
- Decomposition of measures
- Derivatives of measures

#### Part 0: Preliminaries

A first look at Banach and Hilbert spaces

#### Part 1: Mathematical Foundations of Quantum Mechanics

#### Part 2: Schrödinger Operators

#### Part 3: Appendix

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