The effective potential of an <i>M</i>-matrix

In the presence of a confining potential V, eigenfunctions continuous Schrödinger operator ?? + V decay exponentially with rate governed by part which is above corresponding eigenvalue; this can be quantified method Agmon. Analogous localization properties also established for eigenvectors discrete matrix. This note shows, perhaps surprisingly, that one replace matrix any real symmetric Z-matrix and still obtain eigenvector estimates. case non-singular M-matrix A (which situation arises in several contexts, including random theory statistical physics), landscape function u = A?11 plays role an effective localization. Starting from potential, create Agmon-type distance governing exponential away “wells” typical eigenfunction being localized to single such well.