Self-adjoint Extensions in Quantum Mechanics

After reviewing quantization problems emphasizing non-triviality of consistent operator construction, the book shows how problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems.

This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces. It begins by considering quantization problems in general, emphasizing the non-triviality of consistent operator construction by presenting paradoxes to the naive treatment. It then builds the necessary mathematical background following it by the theory of self-adjoint extensions. By considering several problems such as the one-dimensional Calogero problem, the Aharonov-Bohm problem, the problem of delta-like potentials and relativistic Coulomb problemIt then shows how quantization problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems. In the end, related problems in quantum field theory are briefly introduced. This well organized text is most suitable for students and post graduates interested in deepening their understanding of mathematical problems in quantum mechanics. However, scientists in mathematical and theoretical physics and mathematicians will also find it useful.

Provides a consistent treatment of certain quantization problems in quantum mechanics with several examples Covers necessary mathematical background Clear organization Ends with a interesting discussion related to similar quantum field theory problems

Contenu

Introduction.- Linear Operators in Hilbert Spaces.- Basics of Theory of s.a. Extensions of Symmetric Operators.- Differential Operators.- Spectral Analysis of s.a. Operators.- Free One-Dimensional Particle on an Interval.- One-Dimensional Particle in Potential Fields.- Schrödinger Operators with Exactly Solvable Potentials.- Dirac Operator with Coulomb Field.- Schrödinger and Dirac Operators with Aharonov-Bohm and Magnetic-Solenoid Fields.