The effective potential of an $M$-matrix

Abstract

In the presence of a confining potential $V$, the eigenfunctions of a continuous Schrödinger 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 Schrödinger matrix. This note shows, perhaps surprisingly, that one can replace a discrete Schrödinger 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.

References in corpus (3)

• Many-Body Localization Landscape
• The exponential decay of eigenfunctions for tight binding Hamiltonians via landscape and dual landscape functions
• Localization landscape for Dirac fermions

Cited by in corpus (3)

• Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
• Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs
• Landscape approximation of the ground state eigenvalue for graphs and random hopping models