Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians

This book discusses the self-adjoint extension problem for symmetric operators on Hilbert space, presenting classical von Neumann and Krein-Vishik-Birman extension schemes and analyzing applications beyond standard pedagogical examples. It emphasizes the importance of self-adjointness for quantum-mechanical modeling, particularly in declaring additional physical states and describing different physics. The analysis covers hydrogenoid Hamiltonians, Dirac-Coulomb Hamiltonians, geometric quantum confinement, and few-body quantum particles with zero-range interaction.

Format: Hardback
Length: 538 pages
Publication date: 05 April 2023
Publisher: Springer International Publishing AG

This comprehensive book delves into the realm of the self-adjoint extension problem for symmetric operators on Hilbert space, presenting both the classical von Neumann and Krein-Vishik-Birman extension schemes in their modern forms and from a historical perspective. It offers a detailed analysis of a wide range of applications beyond the standard pedagogical examples, which are conveniently indexed in a final appendix for the reader's convenience. The self-adjointness of operators on Hilbert space, particularly those representing quantum observables like quantum Hamiltonians, is essential for ensuring real-valued energy levels, unitary evolution, and the development of a self-consistent theory. Physical heuristics often lead to candidate Hamiltonians that are symmetric, necessitating their extension to larger domains of self-adjointness to encompass additional physical states and maintain consistency in physics. Distinct self-adjoint extensions arise, describing different aspects of the underlying phenomena. Realizing observables self-adjointly is considered the foundational problem in quantum-mechanical modelling.

The applications discussed in this book are of significant interest in modern mathematical physics, particularly in the context of classifying self-adjoint realizations of certain Hamiltonians and studying their spectral and scattering properties. These applications encompass a diverse range of topics, including hydrogenoid Hamiltonians, Dirac-Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction. The analysis presented in the book addresses intermediate technical questions such as characterizing the corresponding operator closures and adjoints, contributing to the advancement of the field.

This book is particularly valuable for graduate students and researchers seeking to deepen their understanding of the self-adjoint extension problem and its applications in mathematical physics. It provides a comprehensive and up-to-date treatment of the subject, making it an essential resource for scholars and practitioners in the field.

Weight: 1004g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031108846
Edition number: 1st ed. 2023