EXPONENTIAL DECAY ESTIMATES FOR FUNDAMENTAL SOLUTIONS OF SCHRODINGER-TYPE OPERATORS

摘要

In the present paper, we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators L = -(del - ia)(T)A(del - ia) + V. The latter class includes, in particular, the magnetic Schrodinger operator - (del - ia)(2) + V and the generalized electric Schrodinger operator -div del + 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 Schrodinger 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 Schrodinger operator -Delta + V.