Exponential decay estimates for fundamental matrices of generalized Schrödinger systems

Abstract

In this article, we investigate systems of generalized Schrödinger operators and their fundamental matrices. More specifically, we establish the existence of such fundamental matrices and then prove sharp upper and lower exponential decay estimates for them. The Schrödinger operators that we consider have leading coefficients that are bounded and uniformly elliptic, while the zeroth-order terms are assumed to be nondegenerate and belong to a reverse Hölder class of matrices. In particular, our operators need not be self-adjoint. The exponential bounds are governed by the so-called upper and lower Agmon distances associated to the reverse Hölder matrix that serves as the potential function. Furthermore, we thoroughly discuss the relationship between this new reverse Hölder class of matrices, the more classical matrix \({\mathcal {A}_{p,\infty }}\) class, and the matrix \({\mathcal {A}_\infty }\) class introduced by Dall’Ara (J Funct Anal 268(12):3649–3679, 2015).