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.