Cheon, Taksu; Fülöp, Tamás; Tsutsui, Izumi Symmetry, duality, and anholonomy of point interactions in one dimension. (English) Zbl 1016.81026 Ann. Phys. 294, No. 1, 1-23 (2001). 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. Cited in 11 Documents 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 Keywords:spectral structure; fermion-boson duality; spectral anholonomy; hidden \(su(2)\); supersymmetric Witten model PDFBibTeX XMLCite \textit{T. Cheon} et al., Ann. Phys. 294, No. 1, 1--23 (2001; Zbl 1016.81026) Full Text: DOI arXiv References: [1] Reed, M.; Simon, B., Methods of Modern Mathematical Physics (1980), Academic Press: Academic Press New York [2] Albeverio, S.; Gesztesy, F.; Hø egh-Krohn, R.; Holden, H., Solvable Models in Quantum Mechanics (1988), Springer-Verlag: Springer-Verlag New York [3] Albeverio, S.; Kurasov, P., Singular Perturbations of Differential Operators (2000), Cambridge Univ. Press: 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.; S. Albeverio, and, L. Nizhnik, Approximation of general zero-range potentials, Uni. Bonn Preprint 585, 1999. · Zbl 1037.81532 [9] R. Jackiw, in; R. Jackiw, in · 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: 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.; 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: 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.