Eigenvalue curves of asymmetric tridiagonal random matrices

TL;DR

This paper investigates the eigenvalue distribution of large asymmetric tridiagonal random matrices, revealing that their spectra concentrate on specific curves in the complex plane and deriving equations for these curves.

Contribution

It provides a detailed analysis of the limit eigenvalue distribution for non-Hermitian operators, linking it to Lyapunov exponents and density of states, and distinguishes finite-interval spectra from the infinite operator spectrum.

Findings

Eigenvalues concentrate on specific curves in the complex plane.

Derived equations for eigenvalue curves and density in terms of Lyapunov exponent.

Finite-interval spectra do not approximate the spectrum of the infinite operator.

Abstract

Random Schroedinger operators with imaginary vector potentials are studied in dimension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on finite intervals of length n with periodic boundary conditions and describe the limit eigenvalue distribution when n goes to infinity. We prove that this limit distribution is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a "reference" symmetric eigenvalue problem. In contrast to these results, the spectrum of the limit operator in l^2(Z) is a two dimensional set which is not approximated by the spectra of the finite-interval operators.