  In the present paper we establish sharp exponential decay estimates for
operator 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 electric
Schr\"odinger operator $-{\rm div }A\nabla+V$. 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 to the
corresponding maximal function. In the presence of a scale-invariant Harnack
inequality, for instance, for the generalized electric Schr\"odinger operator
with real coefficients, we establish both lower and upper estimates for
fundamental solutions, thus demonstrating sharpness of our results. The only
previously known estimates of this type pertain to the classical Schr\"odinger
operator $-\Delta +V$.
