Low-lying eigenvalues of semiclassical Schr\"odinger operator with degenerate wells

TL;DR

This paper analyzes the behavior of low-lying eigenvalues of a semiclassical Schrödinger operator with degenerate wells, providing criteria for boundedness, asymptotic estimates, and extending results to higher dimensions.

Contribution

It introduces new criteria for eigenvalue boundedness, asymptotic formulas for degenerate wells, and extends analysis to higher dimensions.

Findings

Eigenvalues are bounded under specific conditions on the potential.

Asymptotic behavior of eigenvalues matches that of Dirichlet Laplacian on certain intervals.

Eigenvectors' asymptotics are characterized and extended to higher dimensions.

Abstract

In this article, we consider the semiclassical Schr\"odinger 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.