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Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians
eBook - Springer Monographs in Mathematics
ISBN/EAN: 9783031108853
Umbreit-Nr.: 9081317
Sprache:
Englisch
Umfang: 0 S., 9.40 MB
Format in cm:
Einband:
Keine Angabe
Erschienen am 04.04.2023
Auflage: 1/2023
E-Book
Format: PDF
DRM: Digitales Wasserzeichen
- Zusatztext
- This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and KreinVishikBirman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the readers convenience).<p>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.</p><p>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, DiracCoulomb 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.</p><p>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.</p>
- Autorenportrait
- <b>M</b><b>atteo Gallone</b> 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.<p><b>Alessandro Michelangeli </b>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.</p>