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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.

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
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