Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells

In this article, we consider the semiclassical Schrödinger operator [Formula: see text] in [Formula: see text] with a confining non-negative potential V which vanishes, and study its low-lying eigenvalues [Formula: see text] as [Formula: see text]. First, we state a necessary and sufficient criterion upon [Formula: see text] for [Formula: see text] to be bounded. When [Formula: see text] and [Formula: see text], we show that the size of the eigenvalues [Formula: see text] for potentials monotonous on both sides of 0 is given by the length of an interval [Formula: see text], determined by an implicit relation involving V and h. Next, we consider the case where V has a flat minimum, in the sense that it vanishes to infinite order. We provide the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on [Formula: see text]. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.