The effective potential of an $M$-matrix

TL;DR

This paper extends eigenvector localization estimates from Schr"odinger operators to real symmetric $Z$-matrices, especially $M$-matrices, using the landscape function as an effective potential to describe eigenfunction decay.

Contribution

It demonstrates that eigenvector localization estimates apply to any real symmetric $Z$-matrix, generalizing previous results for Schr"odinger operators and introducing the landscape function for $M$-matrices.

Findings

Eigenvector localization estimates hold for any real symmetric $Z$-matrix.

The landscape function $u = A^{-1} 1$ acts as an effective potential for localization.

Eigenfunctions decay exponentially away from potential wells defined by the landscape function.

Abstract

In the presence of a confining potential $V$, the eigenfunctions of a continuous Schr\"odinger operator $-\Delta +V$ decay exponentially with the rate governed by the part of $V$ which is above the corresponding eigenvalue; this can be quantified by a method of Agmon. Analogous localization properties can also be established for the eigenvectors of a discrete Schr\"odinger matrix. This note shows, perhaps surprisingly, that one can replace a discrete Schr\"odinger matrix by \emph{any} real symmetric $Z$-matrix and still obtain eigenvector localization estimates. In the case of a real symmetric non-singular $M$-matrix $A$ (which is a situation that arises in several contexts, including random matrix theory and statistical physics), the \emph{landscape function} $u = A^{-1} 1$ plays the role of an effective potential of localization. Starting from this potential, one can create an Agmon-type distance function governing the exponential decay of the eigenfunctions away from the "wells" of the potential, a typical eigenfunction being localized to a single such well.