Perturbation Theory for the Thermal Hamiltonian: 1D Case

TL;DR

This paper develops a detailed perturbation theory for the one-dimensional thermal Hamiltonian, analyzing its domain, perturbations, and wave operators, extending previous work on the free model and scattering results.

Contribution

It provides a comprehensive analysis of perturbations and wave operators for the 1D thermal Hamiltonian, advancing understanding of thermal current models in solids.

Findings

Established regularity and decay properties of the unperturbed Hamiltonian

Identified classes of self-adjoint and relatively compact perturbations

Proved existence and completeness of wave operators for certain potentials

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.