Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians
Springer International Publishing (Verlag)
978-3-031-10884-6 (ISBN)
Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics.Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling.
The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include 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.
Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.
Matteo Gallone is a researcher in mathematical physics at the Mathematics department of the University of Milan. His scientific interests lie between theoretical physics, mathematical physics and functional analysis. He has studied energy localisation in classical mechanical systems with many degrees of freedom, operator-theoretic problems stemming from quantum mechanical models, and his interests have recently also extended to quasi-periodic systems in statistical mechanics using the techniques of the constructive renormalisation group.Alessandro Michelangeli is an Alexander von Humboldt Senior Researcher at the Institute for Applied Mathematics of the University of Bonn and at the Hausdorff Center for Mathematics, Bonn, and a member of the Institute of Theoretical Quantum Technologies (TQT), Trieste, having also held positions at the LMU Munich and the SISSA Trieste. His research is at the interface of analysis, mathematical physics, and theoretical physics, with expertise in functional analysis, operator theory, spectral theory, non-linear partial differential equations, and quantum mechanics. His more than 60 publications include two authored books and three edited monographs.
- Part I Theory. - 1. Generalities on Symmetric and Self-Adjoint Operators on Hilbert Space. - 2. Classical Self-Adjoint Extension Schemes. - Part II Applications. - 3. Hydrogenoid Spectra with Central Perturbations. - 4. Dirac-Coulomb Hamiltonians for Heavy Nuclei. - 5. Quantum Particle on Grushin Structures. - 6. Models of Zero-Range Interaction for the Bosonic Trimer at Unitarity. - Appendix A: Physical Requirements Prescribing Self-Adjointness of Quantum Observables. - Appendix B: References to Pedagogical Examples.
"In this book, the extension theory of symmetric operators is discussed. The presentation is modern and clear, and the book can be seen, in particular, as a standard reference for the Krein-Vishik-Birman extension theory. While in the first part of the book, the abstract framework from the operator and extension theory of symmetric operators is presented, in the second part, that takes the largest part of the book, various applications are presented in detail." (Markus Holzmann, zbMATH 1533.47001, 2024)
| Erscheinungsdatum | 07.04.2023 |
|---|---|
| Reihe/Serie | Springer Monographs in Mathematics |
| Vorwort | Sergio Albeverio |
| Zusatzinfo | XXV, 538 p. 13 illus. in color. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 966 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik | |
| Schlagworte | Dirac-Coulomb Hamiltonians • Hydrogenoid Hamiltonians with Central Perturbations • Krein Extension Theory • perturbation theory • Quantum Confinement on Grushin Manifolds • Self-adjointness in Quantum Mechanics • self-adjoint operators • spectral theory • unbounded operators on Hilbert space • Unitary Gases • Vishik-Birman Extension Theory • von Neumann Extension Theory • Zero-range Interaction Quantum Systems |
| ISBN-10 | 3-031-10884-1 / 3031108841 |
| ISBN-13 | 978-3-031-10884-6 / 9783031108846 |
| Zustand | Neuware |
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