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＜E-Book＞
Self-adjoint Extensions in Quantum Mechanics : General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials / by D.M. Gitman, I.V. Tyutin, B.L. Voronov
(Progress in Mathematical Physics. ISSN:21971846 ; 62)

Edition	1st ed. 2012.
Publisher	(Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser)
Year	2012
Language	English
Size	XIII, 511 p. 3 illus : online resource
Authors	*Gitman, D.M author Tyutin, I.V author Voronov, B.L author SpringerLink (Online service)
Subjects	LCSH:Mathematical physics LCSH:Operator theory LCSH:Quantum physics LCSH:Mathematics FREE:Mathematical Physics FREE:Mathematical Methods in Physics FREE:Operator Theory FREE:Quantum Physics FREE:Applications of Mathematics
Notes	Introduction -- Linear Operators in Hilbert Spaces -- Basics of Theory of s.a. Extensions of Symmetric Operators -- Differential Operators -- Spectral Analysis of s.a. Operators -- Free One-Dimensional Particle on an Interval -- One-Dimensional Particle in Potential Fields -- Schrödinger Operators with Exactly Solvable Potentials -- Dirac Operator with Coulomb Field -- Schrödinger and Dirac Operators with Aharonov-Bohm and Magnetic-Solenoid Fields Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and their spectral analysis.  Though a “naïve”  treatment exists for dealing with such problems, it is based on finite-dimensional algebra or even infinite-dimensional algebra with bounded operators, resulting in paradoxes and inaccuracies.   A proper treatment of these problems requires invoking certain nontrivial notions and theorems from functional analysis concerning the theory of unbounded self-adjoint operators and the theory of self-adjoint extensions of symmetric operators. Self-adjoint Extensions in Quantum Mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naïve treatment.  The necessary mathematical background is then built by developing the theory of self-adjoint extensions.  Through examination of  various quantum-mechanical systems, the authors show how quantization problems associated with the correct definition of observables and their spectral analysis can be treated consistently for comparatively simple quantum-mechanical systems.  Systems that are examined include free particles on an interval, particles in a number of potential fields including delta-like potentials, the one-dimensional Calogero problem, the Aharonov–Bohm problem, and the relativistic Coulomb problem. This well-organized text is most suitable for graduate students and postgraduates interested in deepening their understanding of mathematical problems in quantum mechanics beyond the scope of those treated in standard textbooks.  The book may also serve as a useful resource for mathematicians and researchers in mathematical andtheoretical physics HTTP:URL=https://doi.org/10.1007/978-0-8176-4662-2

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