Exponential Decay Estimates for Fundamental Matrices of Generalized Schr\"odinger Systems

TL;DR

This paper establishes the existence and exponential decay estimates for fundamental matrices of generalized Schr"odinger systems with non-self-adjoint operators, using novel reverse H"older class potentials and Agmon distances.

Contribution

It introduces sharp exponential decay bounds for fundamental matrices of non-self-adjoint Schr"odinger systems with potentials in a new reverse H"older class.

Findings

Existence of fundamental matrices for generalized Schr"odinger operators.

Sharp exponential decay estimates governed by Agmon distances.

Analysis of the relationship between different matrix potential classes.

Abstract

In this article, we investigate systems of generalized Schr\"odinger 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 Schr\"odinger 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 H\"older 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 H\"older matrix that serves as the potential function. Furthermore, we thoroughly discuss the relationship between this new reverse H\"{o}lder class of matrices, the more classical matrix $\mathcal{A}_{p,\infty}$ class, and the matrix $\mathcal{A}_{\infty}$ class introduced in [Dall15].