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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 Luttingers 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.
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 dim
Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli-Dirac one and, in contrast to the discr
This paper is a synopsis of the recent book A. Boritchev, S. Kuksin, textit{One-Dimensional Turbulence and the Stochastic Burgers Equation}, AMS Publications, 2021 (to appear). The book is dedicated to the stochastic Burgers equation as a model for 1
An analytic definition of a $mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through $0$ alon
In this paper we consider an Anderson model with a large number of sites with zero interaction. For such models we study the spectral statistics in the region of complete localization. We show that Poisson statistics holds for such energies, by proving the Minami estimate.