Microlocal WKB Expansions

In relation with microlocal tunneling, we investigate the existence of micro-local WKB expansions for the eigenfunctions of semiclassical pseudodifferential operators whose symbol admits a non degenerate minimum at some point (x0, ξ0) of R2n. More precisely, under assumptions of analyticity, we prove that the FBI (or Bargman) transform of the first eigenfunction of such an operator admits near (x0, ξ0) an expansion of the form e−Φ(x, ξ)/h ∑j⩾0 hjaj(x, ξ) as the semiclassical parameter h tends to 0. Here Φ and the ajʹs are smooth (complex) functions, and the domain in which the expansion is valid can be described in terms of deformation properties. Our result can be applied e.g. to magnetic Schrödinger operators with electric potential admitting a non degenerate well.