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Solvable Models in Quantum Mechanics: Second Edition
S. Albeverio, University of Bonn, Germany, F. Gesztesy, University of Missouri, Columbia, MO, R. Høegh-Krohn, H. Holden, Norwegian University of Science & Technology, Trondheim, Norway, and an appendix by P. Exner, Czech Academy of Sciences, Prague, Czech Republic
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AMS Chelsea Publishing
1988; 488 pp; hardcover
Volume: 350
ISBN-10: 0-8218-3624-2
ISBN-13: 978-0-8218-3624-8
List Price: US$76 Member Price: US$68.40
Order Code: CHEL/350.H

This monograph presents a detailed study of a class of solvable models in quantum mechanics that describe the motion of a particle in a potential having support at the positions of a discrete (finite or infinite) set of point sources. Both situations-where the strengths of the sources and their locations are precisely known and where these are only known with a given probability distribution-are covered.

The authors present a systematic mathematical approach to these models and illustrate its connections with previous heuristic derivations and computations. Results obtained by different methods in disparate contexts are thus unified and a systematic control over approximations to the models, in which the point interactions are replaced by more regular ones, is provided.

The first edition of this book generated considerable interest for those learning advanced mathematical topics in quantum mechanics, especially those connected to the Schrödinger equations. This second edition includes a new appendix by Pavel Exner, who has prepared a summary of the progress made in the field since 1988. His summary, centering around two-body point interaction problems, is followed by a bibliography focusing on essential developments made since 1988. appendix by Pavel Exner, who has prepared a summary of the progress made in the field since 1988. His summary, centering around two-body point interaction problems, is followed by a bibliography focusing on essential developments made since 1988.

The material is suitable for graduate students and researchers interested in quantum mechanics and Schrödinger operators.

Graduate students and research mathematicians interested in quantum mechanics and Schrödinger operators.

Reviews

"There is a wealth of very pretty examples of Schrödinger operators here which could be presented ... in an elementary quantum mechanics course."

-- MathSciNet

• Introduction
The one-center point interaction
• The one-center point interaction in three dimensions
• Coulomb plus one-center point interaction in three dimensions
• The one-center $$\delta$$-interaction in one dimension
• The one-center $$\delta$$'-interaction in one dimension
• The one-center point interaction in two dimensions
Point interactions with a finite number of centers
• Finitely many point interactions in three dimensions
• Finitely many $$\delta$$-interactions in one dimension
• Finitely many $$\delta$$'-interactions in one dimension
• Finitely many point interactions in two dimensions
Point interactions with infinitely many centers
• Infinitely many point interactions in three dimensions
• Infinitely many $$\delta$$-interactions in one dimension
• Infinitely many $$\delta$$'-interactions in one dimension
• Infinitely many point interactions in two dimensions
• Random Hamiltonians with point interactions
Appendices
• Self-adjoint extensions of symmetric operators
• Spectral properties of Hamiltonians defined as quadratic forms
• Schrödinger operators with interactions concentrated around infinitely many centers
• Boundary conditions for Schrödinger operators on $$(0,\infty)$$
• Time-dependent scattering theory for point interactions
• Dirichlet forms for point interactions
• Point interactions and scales of Hilbert spaces
• Nonstandard analysis and point interactions
• Elements of probability theory
• Relativistic point interactions in one dimension
• References
• Author Index
• Subject Index
• Seize ans après
• Bibliography