Band Structure and Accumulation Point in the Spectrum of Quantum   Collision Operator in a One-Dimensional Molecular Chain
  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. 
B. A. Tay, Kazuki Kanki, Satoshi Tanaka and Tomio Petrosky
cond-mat.stat-mech math-ph math.MP