Semiclassical eigenstates in a multidimensional well

The two-dimensional Schrödinger operator with an analytic potential, having a non-degenerated minimum (well) at the origin, is considered. Under the Diophantine condition on the frequencies, the full asymptotic series (the Plank constant h tending to zero) for eigenfunctions with given quantum numbers («1,112), concentrated at the bottom of the well , is constructed, the Gaussian-like asymptotics being valid in a neighbourhood of the origin which is independent of h. For small quantum numbers the second approximation to the eigenvalues is written in terms of the derivatives of the potential. §