##
**Solvable models in quantum mechanics.**
*(English)*
Zbl 0679.46057

Texts and Monographs in Physics. New York etc.: Springer-Verlag. xiv, 452 p. DM 158.00 (1988).

This monograph presents a detailed and systematic study of models of a particle with \(\delta\)-function interaction described by the Schrödinger operator with potential supported on a discrete (finite or infinite) set of points. The dimensions \(d\) of the space are \(d=1,2,3.\)

These models have been discussed extensively, particularly in solid state physics (e.g. the Kronig-Penney model of a crystal), atomic and nuclear physics (describing short-range nuclear forces or low-energy phenomena) and electromagnetism (propagation in dielectric media).

The main purpose of this monograph is to present the mathematical approaches developed in recent years and to systematize results obtained earlier by different and often heuristic methods in disparate contexts.

Exposition is divided into 3 parts, namely point interactions with one center (Part I), finitely many centers (Part II) and infinitely many centers (Part III). Chapter III.5 of Part III, stochastic potentials are discussed.

The spectrum, the eigenfunctions, resonances, and scattering quantities are explicitly discussed. Convergences of their approximations are also treated.

These models have been discussed extensively, particularly in solid state physics (e.g. the Kronig-Penney model of a crystal), atomic and nuclear physics (describing short-range nuclear forces or low-energy phenomena) and electromagnetism (propagation in dielectric media).

The main purpose of this monograph is to present the mathematical approaches developed in recent years and to systematize results obtained earlier by different and often heuristic methods in disparate contexts.

Exposition is divided into 3 parts, namely point interactions with one center (Part I), finitely many centers (Part II) and infinitely many centers (Part III). Chapter III.5 of Part III, stochastic potentials are discussed.

The spectrum, the eigenfunctions, resonances, and scattering quantities are explicitly discussed. Convergences of their approximations are also treated.

Reviewer: H.Araki

### MSC:

46N50 | Applications of functional analysis in quantum physics |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |

35P25 | Scattering theory for PDEs |

81U05 | \(2\)-body potential quantum scattering theory |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

81V70 | Many-body theory; quantum Hall effect |

03H10 | Other applications of nonstandard models (economics, physics, etc.) |