This textbook presents the spectral theory of self-adjoint operators on Hilbert space and its applications in quantum mechanics. Based on a course taught by the author in Paris, the book not only covers the mathematical theory but also provides its physical interpretation, offering an accessible introduction to quantum mechanics for students with a background in mathematics. The presentation incorporates numerous physical examples to illustrate the abstract theory. The final two chapters present recent findings on Schrödingers equation for systems of particles.
While primarily designed for graduate courses, the book can also serve as a valuable introduction to the subject for more advanced readers. It requires no prior knowledge of physics, assuming only a graduate-level understanding of mathematical analysis from the reader.
1 Introduction to quantum mechanics: the hydrogen atom.- 2
Self-adjointness.- 3 Self-adjointness criteria: Rellich, Kato & Friedrichs.-
4 Spectral theorem and functional calculus.- 5 Spectrum of self-adjoint
operators.- 6 N-particle systems, atoms, molecules.- 7 Periodic Schrödinger
operators, electronic properties of materials.- Appendix A: Sobolev spaces.-
Appendix B: Problems.
Mathieu Lewin is CNRS research director (Directeur de Recherche) at Paris-Dauphine University. He specialises in spectral and variational methods in quantum systems.
