×

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

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)].

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Abramowitz M., National Bureau of Standards Applied Mathematics Series 55, in: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1964) · Zbl 0171.38503
[2] DOI: 10.1023/A:1007330512611 · Zbl 0907.47058
[3] DOI: 10.1103/PhysRev.115.485 · Zbl 0099.43102
[4] Albeverio S., Solvable Models in Quantum Mechanics (2005)
[5] DOI: 10.1088/0305-4470/38/22/010 · Zbl 1071.47003
[6] Amrein W. O., Lecture Notes and Supplements in Physics 16, in: Scattering Theory in Quantum Mechanics (1977)
[7] DOI: 10.1007/s00220-008-0579-1 · Zbl 1228.81262
[8] Ballesteros M., J. Math. Phys. 50 pp 54–
[9] DOI: 10.1063/1.3480092
[10] DOI: 10.1063/1.1534893 · Zbl 1061.58025
[11] DOI: 10.1142/S0129055X08003249 · Zbl 1163.81007
[12] DOI: 10.1063/1.532307 · Zbl 0916.47053
[13] DOI: 10.1016/0022-1236(91)90024-Y · Zbl 0748.47004
[14] DOI: 10.1063/1.1463712 · Zbl 1059.81056
[15] DOI: 10.1134/S0001434608070110 · Zbl 1219.34109
[16] DOI: 10.1088/0305-4470/33/50/305 · Zbl 0983.53051
[17] DOI: 10.1007/BF01237049 · Zbl 0483.47031
[18] Kato T., Classics in Mathematics, in: Perturbation Theory for Linear Operators (1995)
[19] DOI: 10.1080/10652460802321485 · Zbl 1163.33313
[20] Lisovyy O., J. Math. Phys. 48 pp 17–
[21] DOI: 10.1007/978-3-642-65975-1
[22] DOI: 10.1016/S0034-4877(06)80048-0 · Zbl 1143.47017
[23] Posilicano A., Oper. Matrices 2 pp 483–
[24] DOI: 10.1090/conm/500/09828
[25] DOI: 10.1016/0003-4916(83)90051-9 · Zbl 0554.47003
[26] Tamura H., Nagoya Math. J. 155 pp 95– · Zbl 0934.35140
[27] Tamura H., Rev. Math. Phys. 13 pp 465–
[28] Watson G. N., A Treatise on the Theory of Bessel Functions (1966)
[29] DOI: 10.1088/0266-5611/18/4/307 · Zbl 1024.81053
[30] Yafaev D. R., Translations of Mathematical Monographs 105, in: Mathematical Scattering Theory. General Theory (1992) · Zbl 0761.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.