Perturbation Theory for the Thermal Hamiltonian: 1D Case

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 (De Nittis and Lenz in Spectral theory of the thermal Hamiltonian, 1D case, 2020) 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 De Nittis and Lenz (2020) by providing a detailed analysis of the perturbation theory for the one-dimensional thermal Hamiltonian. In more detail, the following results 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.