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 PA (x, hD x ) = (hD x − µA(x))2 + V (x) on L 2(R d ). We shall essentially consider the case where µ is large. This kind of estimates, in case of Schrödinger operator without a magnetic field, have been studied by Agmon [1], also in the case of a Riemannian manifold M. Agmon estimates hold true for any h, but are particularly useful in the limit h → 0 when studying tunneling.