Abstract. We analyze the spectral structure of the 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 ground, 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 which admits the interpretation of the system as a supersymmetric Witten model.

The quantum dynamics of a free particle on a circle with point interaction is described by a U (2) family of self-adjoint Hamiltonians. We provide a classification of the family by introducing a number of subfamilies and thereby analyze the spectral structure in detail. We find that the spectrum depends on a subset of U (2) parameters rather than the entire U (2) needed for the Hamiltonians, and that in particular there exists a subfamily in U (2) where the spectrum becomes parameter-independent. We also show that, in some specific cases, the WKB semiclassical approximation becomes exact (modulo phases) for the system.

A linear irreversible thermodynamic framework of heat conduction in rigid conductors is introduced. The deviation from local equilibrium is characterized by a single internal variable and a current multiplier. A general constitutive evolution equation of the current density of the internal energy is derived by introducing a linear relationship between the thermodynamic forces and fluxes. The well-known Fourier, Maxwell-Cattaneo-Vernotte, Guyer-Krumhansl, Jeffreys-type, and Green-Naghdi-type equations of heat conduction are obtained as special cases. The universal character of the approach is demonstrated by two examples. Solutions illustrating the properties of the equation with jump boundary conditions are given.