Spectral Properties of Non-Unitary Band Matrices
We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several deterministic and almost sure results about the location and nature of the spectrum of such non-normal operators as a function of their parameters. We relate these results to the analysis of certain random quantum walks, the dynamics of which can be studied by means of iterates of such random non-unitary contraction operators.
We generalize the algebraic multiplicity of the eigenvalues of nonlinear eigenvalue problems (NEPs) to the rational form and give the extension of the argument principle. In addition, we propose a novel numerical method to determine the algebraic multiplicity of the eigenvalues of the NEPs in a given region by the contour integral method. Finally, some numerical experiments are reported to illustrate the effectiveness of our method.