Band Structure and Accumulation Point in the Spectrum of Quantum Collision Operator in a One-Dimensional Molecular Chain

TL;DR

This paper analyzes the spectrum of a quantum collision operator in a one-dimensional molecular chain, revealing a band structure and an accumulation point, and constructs its eigenvectors with proven completeness and orthogonality.

Contribution

It introduces a novel analysis of the quantum collision operator's spectrum, including band structure and accumulation points, using the continued fraction method.

Findings

Spectrum has a band structure with an accumulation point.

Eigenvectors form a complete and orthogonal set in each subspace.

Spectrum is non-negative with a well-defined eigenstructure.

Abstract

We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.