Spectral theory of the thermal Hamiltonian: 1D case (Q2078937)

Summary: In [``Theory of thermal transport coefficients'', Phys. Rev. 135, No. 135, No. 6A, A1505--A1514 (1964; \url{doi:10.1103/PhysRev.135.A1505})] \textit{J. M. Luttinger} introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated with Luttinger's model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. Moreover, the scattering by convolution-type potentials is analyzed. Finally, also the associated classical problem is completely solved, thus providing a comparison between classical and quantum behavior. This article aims to be a first contribution in the construction of a complete theory for the thermal Hamiltonian.