Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

TL;DR

This paper studies the asymptotic distribution of eigenvalues of banded block Toeplitz matrices, revealing new phenomena and establishing conditions for their convergence to an equilibrium measure, extending previous scalar results.

Contribution

It generalizes scalar eigenvalue distribution results to block Toeplitz matrices, introduces an equilibrium problem for algebraic curves, and identifies new phenomena unique to the block case.

Findings

Eigenvalue measures converge to an equilibrium measure under certain conditions.

The total masses of equilibrium measures follow a combinatorial rule, not a simple series.

The limiting eigenvalue distribution can include point masses and attractors.

Abstract

We consider banded block Toeplitz matrices $T_n$ with $n$ block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of $T_n$ for $n\to\infty$ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of $T_n$ and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if $T_n$ is a Hessenberg matrix.