Tunneling for a semi-classical magnetic Schrödinger 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 Schrödinger operator with smooth coefficients $P_A(x,h{D}_x)=(h{D}_x-\mu A(x){)}^2+V(x)$ on $L^2({\mathbf{R}}^d)$. We shall essentially consider the case where $\mu$ is large. This kind of estimates, in case of Schrödinger 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 $h\to 0$ when studying tunneling.