The effective potential of an M-matrix

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 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⁻¹1 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.