# An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices

## TL;DR

This paper characterizes the limiting eigenvalue distribution of large banded Toeplitz matrices using an equilibrium problem, linking spectral measures to energy minimization and generalized eigenvalues.

## Contribution

It introduces a novel equilibrium framework to describe the limiting spectral measure of banded Toeplitz matrices, extending classical results.

## Key findings

- Limiting eigenvalue measure is characterized by an energy minimization problem.
- The measure is part of a vector of measures solving a unique equilibrium system.
- Other components of the vector correspond to limits of generalized eigenvalues.

## Abstract

We study the limiting eigenvalue distribution of $n\times n$ banded Toeplitz matrices as $n\to \infty$. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as $n\to\infty$. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.