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arXiv · 2101.01672

The effective potential of an $M$-matrix

Marcel Filoche·Svitlana Mayboroda·Terence Tao
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Abstract

In the presence of a confining potential $V$, the eigenfunctions of a continuous Schrödinger operator $-Δ+V$ 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.

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Marcel Filoche, Svitlana Mayboroda, Terence Tao. 2021-03-30. The effective potential of an $M$-matrix. https://doi.org/10.1063/5.0042629

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