Generalized boundary conditions for the Aharonov-Bohm effect combined with a homogeneous magnetic field. (English) Zbl 1059.81056

Summary: The most general admissible boundary conditions are derived for an idealized Aharonov-Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; the other two parameters of the model are the Aharonov-Bohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the Aharonov-Bohm effect with a point interaction. Spectral properties of the derived Hamiltonians are studied in detail.


81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35Q40 PDEs in connection with quantum mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI arXiv


[1] Aharonov, Phys. Rev. 115 pp 485– (1959)
[2] Ruijsenaars, Ann. Phys. (Leipzig) 146 pp 1– (1983)
[3] Dąbrowski, J. Math. Phys. 39 pp 47– (1998)
[4] Adami, Lett. Math. Phys. 43 pp 43– (1998)
[5] Albeverio, J. Reine Angew. Math. 380 pp 87– (1987)
[6] Thienel, Ann. Phys. (Leipzig) 280 pp 140– (2000)
[7] Cavalcanti
[8] Hirokawa, J. Math. Phys. 42 pp 3334– (2001)
[9] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables(Dover, New York, 1965). · Zbl 0171.38503
[10] A. P. Prudnikov, J. A. Bryčkov, and O. I. Maričev:IntěgralyiRjady(Nauka, Moskva, 1986).
[11] N. I. Akhiezer and I. M. Glazman,Theory of Linear Operators in Hilbert Spaces II(Pitman, Boston, 1981). · Zbl 0467.47001
[12] M. Reed and B. Simon,Methods of Modern Mathematical Physics IV(Academic, New York, 1975). · Zbl 0308.47002
[13] J. Weidmann,Linear Operators in Hilbert Spaces(Springer, New York, 1980). · Zbl 0434.47001
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.