A matrix-valued point interactions model

TL;DR

This paper investigates a matrix-valued Schrödinger operator with random point interactions, demonstrating the absence of absolutely continuous spectrum and establishing the H"older continuity of Lyapunov exponents and the integrated density of states.

Contribution

It proves the absence of absolutely continuous spectrum for the operator and establishes the H"older continuity of Lyapunov exponents and the integrated density of states using transfer matrix analysis.

Findings

Absence of absolutely continuous spectrum away from a discrete set.

H"older continuity of Lyapunov exponents.

H"older continuity of the integrated density of states.

Abstract

We study a matrix-valued Schr\"odinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the H\"older continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its H\"older continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.