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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.

##### 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
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