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In the present paper we establish sharp exponential decay estimates foroperator and integral kernels of the (not necessarily self-adjoint) operators$L=-(\nabla-i\mathbf{a})^TA(\nabla-i\mathbf{a})+V$. The latter class includes,in particular, the magnetic Schr\"odinger operator$-\left(\nabla-i\mathbf{a}\right)^2+V$ and the generalized electricSchr\"odinger operator $-{\rm div }A\nabla+V$. 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 $-\Delta +V$.

Exponential decay estimates for fundamental solutions of Schr\"odinger-type operators