Spectral and scattering theory for the Aharonov-Bohm operators. (English) Zbl 1230.81054

Pankrashkin, Konstantin; Richard, Serge

Rev. Math. Phys. 23, No. 1, 53-81 (2011).

The Aharonov-Bohm (A-B) model has been introduced by Y. Aharonov and D. Bohm [Phys. Rev., II. Ser. 115, 485–491 (1959; Zbl 0099.43102)] to describe the motion of a charged particle in a magnetic field concentrated at a single point. This model is considered as one of the systems in mathematical physics for which the spectral and the scattering properties can be completely computed.
The theme of this paper is to provide the spectral and the scattering analysis of A-B operators on \(\mathbb R^2\) for all possible boundary conditions.
This work is motivated by the recent result that the A-B wave operators can be expressed in terms of explicit functions of the generator of dilations and of the Laplacian. However, the analysis in this paper is carried in the modern theory of self-adjoint extensions and rigorous expressions for the wave operators and the scattering operator are derived using the stationary approach of scattering theory in [D. R. Yafaev, Mathematical scattering theory. Analytic theory. Mathematical Surveys and Monographs 158. Providence, RI: American Mathematical Society (AMS). (2010; Zbl 1197.35006)].

Reviewer: Miyeon Kwon (Platteville)

MSC:

81U20	\(S\)-matrix theory, etc. in quantum theory
47A40	Scattering theory of linear operators
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P25	Scattering theory for PDEs

Keywords:

spectral and scattering theory; Aharonov-Bohm model; wave and scattering operators; boundary triple

Citations:

Zbl 0099.43102; Zbl 1197.35006

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