Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells

Abstract

In this article, we consider the semiclassical Schrödinger operator $P=-h^2\Delta +V$ in ${\mathbb{R}}^d$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues ${\lambda }_k(P)$ as $h\to 0$. First, we give a necessary and sufficient criterion upon $V^{-1}(0)$ for ${\lambda }_1(P)h^{-2}$ to be bounded. When $d=1$ and $V^{-1}(0)=\{0\}$, we are able to control the eigenvalues ${\lambda }_k(P)$ for monotonous potentials by a quantity linked to an interval $I_h$, 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 give the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on $I_h$. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.

10 pages