Rotation numbers for Jacobi matrices with matrix entries

TL;DR

This paper extends transfer matrix techniques to define rotation numbers for eigenvalues of Jacobi matrices with matrix entries, generalizing oscillation theorems and providing a formula for the integrated density of states, including randomness effects.

Contribution

It introduces a matricial rotation number calculation for Jacobi matrices with matrix entries, generalizing classical oscillation theorems and addressing symmetry classes and randomness.

Findings

Developed a transfer matrix extension for matrix-valued Jacobi matrices.

Derived a perturbative formula for the integrated density of states with randomness.

Unified treatment of time reversal symmetry classes in the context of matrix entries.

Abstract

A selfadjoined block tridiagonal matrix with positive definite blocks on the off-diagonals is by definition a Jacobi matrix with matrix entries. Transfer matrix techniques are extended in order to develop a rotation number calculation for its eigenvalues. This is a matricial generalization of the oscillation theorem for the discrete analogues of Sturm-Liouville operators. The three universality classes of time reversal invariance are dealt with by implementing the corresponding symmetries. For Jacobi matrices with random matrix entries, this leads to a formula for the integrated density of states which can be calculated perturbatively in the coupling constant of the randomness with an optimal control on the error terms.