Accumulation Point in the Spectrum of the Collision Operator for a One-Dimensional Polaron System

Abstract

The spectrum of Hamiltonians has been well investigated in quantum mechanics, since it plays an essential role in analyzing dynamical properties of conservative systems. For example, we know there are Hamiltonians that have a discrete spectrum, or a continuous spectrum, or a band structure, and so on, each of which leads to a characteristic behavior of the system. However, our knowledge on the spectrum of the collision operators in kinetic equations in dissipative systems still remains in a poor stage in spite of the fact that their spectral properties play a fundamental role in non-equilibrium statistical physics. In this paper we give an example of quantum systems that has a rich structure of the spectrum, such as an accumulation point and band structures, in the collision operator for a momentum relaxation process. We consider a one-dimensional (1D) polaron system in which a quantum particle is weakly interacting with a thermal reservoir consisting of an acoustic phonon field. Similar systems have been studied in different contexts, see e.g. Refs. 1)–5). As a consequence of a constraint in 1D due to the resonance condition, the momenta of the particle related successively through the collision operator form a subset separated from other momenta. Momentum relaxation occurs among those momenta in such a subset independently of other momenta. In this case the collision operator is represented by a tridiagonal matrix for each such subset of momenta. Taking advantage of the tridiagonal nature of the matrix, we found a solution of the eigenvalue problem of the collision operator in terms of continued fractions. This paper is organized as follows. In §2 we introduce a model of a quantum particle weakly coupled with a phonon field, and present a kinetic equation for the