In the present paper we establish sharp exponential decay estimates foroperator and integral kernels of the (not necessarily self-adjoint) operators$L=-(abla-imathbf{a})^TA(abla-imathbf{a})+V$. The latter class includes,in particular, the magnetic Schr"odinger operator$-left(abla-imathbf{a}ight)^2+V$ and the generalized electricSchr"odinger operator $-{m div }Aabla+V$. Our exponential decay boundsrest on a generalization of the Fefferman-Phong uncertainty principle to thepresent context and are governed by the Agmon distance associated to thecorresponding maximal function. In the presence of a scale-invariant Harnackinequality, for instance, for the generalized electric Schr"odinger operatorwith real coefficients, we establish both lower and upper estimates forfundamental solutions, thus demonstrating sharpness of our results. The onlypreviously known estimates of this type pertain to the classical Schr"odingeroperator $-Delta +V$.
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机译：在本文中，我们建立了尖锐的指数衰减估计（不一定自伴）运营商$ l = - （ nabla-i mathbf {a}）^ ta（ nabla-i mathbf {a }）+ v $。后一级包括，特别是磁共氏“odinger运算符$ - left（ nabla-i mathbf {a}右）^ 2 + v $和广义电气chr ”odinger运算符$ - { rm div} a nabla + v $。我们的指数衰减界限在FEFFERMAN-PHONG不确定性原则上对PRESENT语境的概括，并且由与相应的最大函数相关的AGMM距离来治理。在存在鳞片不变的竖琴状态的情况下，例如，对于广义电气SCHR “Odinger Operatorwith真实系数，我们建立了较低和上层估计的估计，因此展示了我们的结果。唯一的知名估计这种类型的估计到古典的schr “odingeroperator $ - delta + v $。
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