We will study the operator HT , called the Thermal Hamiltonian, originally pro- posed by Luttinger to study the e↵ects of a thermal gradient in the matter. We will start by rigurously defining the initially formally self-adjoint operator HT , as well as some unitarily equivalent operators. Then we will study their spectral prop- erties, and compute their unperturbed time evolution, as their Green functions and resolvent family. We will conclude that section by presenting a convolution poten- tials family for which the scattering conditions are satisfied. Finally we will study the dynamics defined by the classical case.