For the semiclassical magnetic Schrödinger operator, the tunnelling effect, that is the asymptotic expansion of the splitting of the first eigenvalues in case of multiple confining wells, is so far not completely understood. We present here some complete results in dimension 1, and partial results in larger dimension including first known WKB extensions and refined estimates of the splitting, following a Born-Oppenheimer strategy of dimensional reduction.

This is based on joint works with V. Bonnaillie-Noël and N. Raymond.