Solvable models in quantum mechanics. With an appendix by Pavel Exner. 2nd revised ed. (English) Zbl 1078.81003

Albeverio, Sergio; Gesztesy, Friedrich; Høegh-Krohn, Raphael; Holden, Helge

Providence, RI: AMS Chelsea Publishing (ISBN 0-8218-3624-2/hbk). xiv, 488 p. (2005).

The self-adjoint operator \(H\) in \(L^2(\mathbb{R}^d)\) describes a point interaction at the point \(a\) if \(Hf= -\Delta f\) for every smooth function \(f\) with compact support away from \(a\). The authors obtain fairly general results on the approximation in the norm resolvent sense of certain point-interaction-Hamiltonians (so-called \(\delta\)-interaction-Hamiltonians) by regular Schrödinger operators. Moreover, they derive explicit formulas for the resolvent, the eigenvalues, the eigenfunctions, the resonances, the scattering lengths, and the essential, the absolutely continuous and the singular continuous spectra of \(\delta\)-interaction-Hamiltonians and so-called \(\delta'\)-interaction-Hamiltonians. These results are then extended to point interactions at finite sets and at lattices; random point-interaction-Hamiltonians are discussed in detail, too. The authors also describe applications of the mathematical theory in a wide variety of models in quantum mechanics.
The second edition contains an appendix where Pavel Exner sketches various related results achieved after the publication of the first edition [(Texts and Monographs in Physics, Springer, New York) (1988; Zbl 0679.46057)].

Reviewer: Johannes F. Brasche (Clausthal)

MSC:

81-02	Research exposition (monographs, survey articles) pertaining to quantum theory
47N50	Applications of operator theory in the physical sciences
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81U05	\(2\)-body potential quantum scattering theory
81U15	Exactly and quasi-solvable systems arising in quantum theory

Keywords:

point interactions; zero-range potentials; eigenvalues; eigenfunctions; resonances; regularization; spectra; scattering length

Citations:

Zbl 0679.46057