Identities and exponential bounds for transfer matrices

TL;DR

This paper investigates the analytic properties of transfer matrices from block-tridiagonal matrices, revealing exponential bounds on their singular values linked to the spectrum and localization phenomena in physics and numerical analysis.

Contribution

It establishes exponential bounds on the singular values of transfer matrices derived from general block-tridiagonal matrices, connecting spectral properties to matrix decay behavior.

Findings

Half of the transfer matrix's singular values are exponentially large.

The other half are exponentially small.

These bounds are linked to the invertibility and spectral properties of the underlying matrices.

Abstract

This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity, If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of inverse of banded matrices.