  In 1964 J. M. Luttinger introduced a model for the quantum thermal transport.
In this paper we study the spectral theory of the Hamiltonian operator
associated to the 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.
