Tunneling for a semi-classical magnetic Schr\"odinger operator with symmetries

Abstract

We are interested in decay estimates of the ground state (or the low energy eigenstates), outside the potential wells, for a semi-classical Magnetic Schr\"odinger operator with smooth coefficients $P_A(x,hD_x)=(hD_x-\mu A(x))^2+V(x)$ on $L^2({\bf R}^d)$. We shall essentially consider the case where $\mu$ is large. This kind of estimates, in case of Schr\"odinger operator without a magnetic field, have been studied by Agmon, also in the case of a Riemannian manifold $M$. Agmon estimates hold true for any $h$, but are particularly useful in the limit $h\to0$ when studying tunneling.