Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians (Springer Monographs in Mathematics)
Language: English
Published by Springer, 2024
- Softcover
- New
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- Title
- Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians (Springer Monographs in Mathematics)
- Author
- Gallone, Matteo; Michelangeli, Alessandro
- Publisher
- Springer
- Publication year
- 2024
- Condition
- New
- Binding
- Soft cover
- Language
- English
- ISBN 10
- 3031108876
- ISBN 13
- 9783031108877
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.
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About the Author
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.
"About the title" may belong to another edition of this title.
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