Spectral Theory and Quantum Mechanics

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 Schrödinger’s 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.