Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells

Abstract

In this article, we consider the semiclassical Schrödinger operator ${\mathit{P}}_{\mathit{h}}=-{\mathit{h}}^2\mathrm{\Delta }+\mathit{V}$ in ${\mathbb{R}}^{\mathit{d}}$ with a confining non-negative potential V which vanishes, and study its low-lying eigenvalues ${\mathit{\lambda }}_{\mathit{k}}\mathrm{(}{\mathit{P}}_{\mathit{h}}\mathrm{)}$ as $\mathit{h}\to 0$ . First, we state a necessary and sufficient criterion upon ${\mathit{V}}^{-1}\mathrm{(}0\mathrm{)}$ for ${\mathit{\lambda }}_1\mathrm{(}{\mathit{P}}_{\mathit{h}}\mathrm{)}{\mathit{h}}^{-2}$ to be bounded. When $\mathit{d}=1$ and ${\mathit{V}}^{-1}\mathrm{(}0\mathrm{)}=\{0\}$ , we show that the size of the eigenvalues ${\mathit{\lambda }}_{\mathit{k}}\mathrm{(}{\mathit{P}}_{\mathit{h}}\mathrm{)}$ for potentials monotonous on both sides of 0 is given by the length of an interval ${\mathit{I}}_{\mathit{h}}$ , 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 provide the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on ${\mathit{I}}_{\mathit{h}}$ . Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.