Exponential decay estimates for fundamental solutions of Schrödinger-type operators

Abstract

In the present paper, we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators L = − ( ∇ − i a ) T A ( ∇ − i a ) + V L=-(\nabla -i\mathbf {a})^TA(\nabla -i\mathbf {a})+V . The latter class includes, in particular, the magnetic Schrödinger operator − ( ∇ − i a ) 2 + V -\left (\nabla -i\mathbf {a}\right )^2+V and the generalized electric Schrödinger operator − div  A ∇ + V -\textrm {div }A\nabla +V . Our exponential decay bounds rest on a generalization of the Fefferman–Phong uncertainty principle to the present context and are governed by the Agmon distance associated with the corresponding maximal function. In the presence of a scale-invariant Harnack inequality—for instance, for the generalized electric Schrödinger operator with real coefficients—we establish both lower and upper estimates for fundamental solutions, thus demonstrating the sharpness of our results. The only previously known estimates of this type pertain to the classical Schrödinger operator − Δ + V -\Delta +V .