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THE EFFECTIVE POTENTIAL OF AN M-MATRIX

Abstract : In the presence of a confining potential V, the eigenfunctions of a continuous Schrödinger operator −∆ + 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 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 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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https://hal.archives-ouvertes.fr/hal-03189307
Contributor : Marcel Filoche <>
Submitted on : Saturday, April 3, 2021 - 10:06:08 AM
Last modification on : Friday, April 9, 2021 - 3:32:12 AM

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arXiv.2101.01672.pdf
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  • HAL Id : hal-03189307, version 1

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Marcel Filoche, Svitlana Mayboroda, Terence Tao. THE EFFECTIVE POTENTIAL OF AN M-MATRIX. 2021. ⟨hal-03189307⟩

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