On critical points of eigenvalues of the Montgomery family of quartic oscillators

Abstract

We discuss spectral properties of the family of quartic oscillators ${\mathfrak{h}}_{\mathcal{M}}(\alpha )=-\frac{d^2}{d{t}^2}+(\frac{1}{2}{t}^2-\alpha {)}^2$ on the real line, where $\alpha \in \mathbb{R}$ is a parameter. This operator appears in a variety of applications coming from quantum mechanics to harmonic analysis on Lie groups, Riemannian geometry and superconductivity. We study the variations of the eigenvalues ${\lambda }_j(\alpha )$ of ${\mathfrak{h}}_{\mathcal{M}}(\alpha )$ as functions of the parameter $\alpha$.We prove that for $j$ sufficiently large, $\alpha \mapsto {\lambda }_j(\alpha )$ has a unique critical point, which is a nondegenerate minimum.We also prove that the first eigenvalue ${\lambda }_1(\alpha )$ enjoys the same property and give a numerically assisted proof that the same holds for the second eigenvalue ${\lambda }_2(\alpha )$. The proof for excited states relies on a semiclassical reformulation of the problem. In particular, we develop a method permitting to differentiate with respect to the semiclassical parameter, which may be of independent interest.