Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells

Article type: Research Article

Authors: Bony, Jean-François | Popoff, Nicolas^{; *}

Affiliations: IMB, Université de Bordeaux, UMR 5251, 33405 Talence, France. E-mails: [email protected], [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this article, we consider the semiclassical Schrödinger operator Ph=−h2Δ+V in Rd with a confining non-negative potential V which vanishes, and study its low-lying eigenvalues λk(Ph) as h→0. First, we state a necessary and sufficient criterion upon V−1(0) for λ1(Ph)h−2 to be bounded. When d=1 and V−1(0)={0}, we show that the size of the eigenvalues λk(Ph) for potentials monotonous on both sides of 0 is given by the length of an interval Ih, 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 Ih. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.

Keywords: Semiclassical Schrödinger operator, eigenvalues asymptotic, degenerate potentials

DOI: 10.3233/ASY-181493

Journal: Asymptotic Analysis, vol. 112, no. 1-2, pp. 23-36, 2019