Ansatz with Hermite polynomials for a multidimensional well

The Schrödinger operator in $\mathbb R^d$ with an analytic potential, having a nondegenerated minimum (well) at the origin, is considered. The ansatz with Hermite polynomials is used. Under a Diophantine condition on the frequencies, full asymptotic series (the Plank constant $h$ tending to zero) for eigenfunctions with given quantum numbers $n\in\mathbb N^d$ 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 $h$. The obtained asymptotic series can be prolonged on a larger domain with the help of ray methods. The way to find zero-sets of the eigenfunctions is described. Some exarnples are considered. Bibliography: 22 titles.