Perturbation Theory for the Thermal Hamiltonian: 1D Case

Abstract: This work continues the study of the thermal Hamiltonian, initially proposed by J. M. Luttinger in 1964 as a model for the conduction of thermal currents in solids. The previous work [DL] contains a complete study of the "free" model in one spatial dimension along with a preliminary scattering result for convolution-type perturbations. This work complements the results obtained in [DL] by providing a detailed analysis of the perturbation theory for the one-dimensional thermal Hamiltonian. In more detail the following result are established: the regularity and decay properties for elements in the domain of the unperturbed thermal Hamiltonian; the determination of a class of self-adjoint and relatively compact perturbations of the thermal Hamiltonian; the proof of the existence and completeness of wave operators for a subclass of such potentials.