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 selfcontained, 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 selfcontained 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 onesemester 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^{p}spaces
 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 StoneWeierstraß theorem
 Selfadjointness and spectrum
 Some quantum mechanics
 Selfadjoint operators
 Quadratic forms and the Friedrichs extension
 Resolvents and spectra
 Orthogonal sums of operators
 Selfadjoint extensions
 Appendix: Absolutely continuous functions
 The spectral theorem
 The spectral theorem
 More on Borel measures
 Spectral types
 Appendix: HerglotzNevanlinna functions
 Applications of the spectral theorem
 Integral formulas
 Commuting operators
 Polar decomposition
 The minmax 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 selfadjoint operators
 Relatively bounded operators and the KatoRellich theorem
 More on compact operators
 HilbertSchmidt 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
 Onedimensional Schrödinger operators
 SturmLiouville operators
 Weyl’s limit circle, limit point alternative
 Spectral transformations I
 Inverse spectral theory
 Absolutely continuous spectrum
 Spectral transformations II
 The spectra of onedimensional Schrödinger operators
 Oneparticle Schrödinger operators
 Selfadjointness and spectrum
 The hydrogen atom
 Angular momentum
 The eigenvalues of the hydrogen atom
 Nondegeneracy of the ground state
 Atomic Schrödinger operators
 Selfadjointness
 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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