×

zbMATH — the first resource for mathematics

Symmetry, duality, and anholonomy of point interactions in one dimension. (English) Zbl 1016.81026
Summary: We analyze the spectral structure of a one-dimensional quantum mechanical system with point interaction, which is known to be parametrized by the group \(U(2)\). Based on the classification of the interactions in terms of symmetries, we show, on a general basis, how the fermion-boson duality and the spectral anholonomy recently discovered can arise. A vital role is played by a hidden \(su(2)\) formed by a certain set of discrete transformations, which becomes a symmetry if the point interaction belongs to a distinguished \(U(1)\) subfamily in which all states are doubly degenerate. Within the \(U(1)\), there is a particular interaction that admits the interpretation of the system as a supersymmetric Witten model.

MSC:
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q60 Supersymmetry and quantum mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Reed, M.; Simon, B., Methods of modern mathematical physics, (1980), Academic Press New York
[2] Albeverio, S.; Gesztesy, F.; Hø egh-Krohn, R.; Holden, H., Solvable models in quantum mechanics, (1988), Springer-Verlag New York
[3] Albeverio, S.; Kurasov, P., Singular perturbations of differential operators, (2000), Cambridge Univ. Press Cambridge · Zbl 0945.47015
[4] Avron, J.E.; Exner, P.; Last, Y., Phys. rev. lett., 72, 869, (1994)
[5] Kiselev, A., J. math. anal. appl., 212, 263, (1997)
[6] Chernoff, P.R.; Hughes, R.J., J. funct. anal., 111, 97, (1993)
[7] Cheon, T.; Shigehara, T., Phys. lett. A, 243, 111, (1998)
[8] S. Albeverio, and, L. Nizhnik, Approximation of general zero-range potentials, Uni. Bonn Preprint 585, 1999. · Zbl 1037.81532
[9] R. Jackiw, in, Diverse Topics in Theoretical and Mathematical Physics, World Scientific, Singapore, 1995. · Zbl 0960.00509
[10] Cheon, T.; Shigehara, T., Phys. rev. lett., 82, 2536, (1999)
[11] Cheon, T., Phys. lett. A, 248, 285, (1998)
[12] Tsutsui, I.; Fülöp, T.; Cheon, T., J. phys. soc. jpn., 69, 3473, (2000)
[13] Albeverio, S.; Dabrowski, L.; Kurasov, P., Lett. math. phys., 45, 33, (1998)
[14] Fülöp, T.; Tsutsui, I., Phys. lett. A, 264, 366, (2000)
[15] Šeba, P., Czech. J. phys., 36, 667, (1986)
[16] Albeverio, S.; Brzeźniak, Z.; Dabrowski, L., J. funct. anal., 130, 220, (1995)
[17] Román, J.M.; Tarrach, R., J. phys. A, 29, 6073, (1996)
[18] Kurasov, P., J. math. anal. appl., 201, 297, (1996)
[19] Albeverio, S.; Dabrowski, L.; Fei, S.-M., Int. J. mod. phys. B, 14, 721, (2000)
[20] Shapere, A.; Wilczek, F., Geometric phases in physics, (1989), World Scientific Singapore · Zbl 0914.00014
[21] P. Exner, and, H. Grosse, Some properties of the one-dimensional generalized point interactions (a torso), math-ph/9910029.
[22] Junker, G., Supersymmetric methods in quantum and statistical physics, (1996), Springer-Verlag Berlin · Zbl 0867.00011
[23] Witten, E., Nucl. phys. B, 188, 513, (1988)
[24] Bertoni, A.; Bordone, P.; Brunetti, R.; Jacoboni, C.; Reggiani, S., Phys. rev. lett., 84, 5912, (2000)
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.