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

TL;DR

This paper investigates the spectral behavior of a family of quartic oscillators, revealing that for large eigenvalues, the eigenvalue functions have unique, nondegenerate minima at critical points, with detailed analysis for the first few eigenvalues.

Contribution

The paper establishes the existence and uniqueness of critical points for eigenvalue functions of quartic oscillators, including a numerically supported result for the second eigenvalue.

Findings

Eigenvalue functions for large indices have a unique, nondegenerate minimum.

The first eigenvalue function has a similar critical point property.

Numerical evidence suggests the second eigenvalue also has a unique critical point.

Abstract

We discuss spectral properties of the family of quartic oscillators $\mathfrak h_{\mathcal M}(\alpha) =-\frac{d^2}{dt^2} +\Big(\frac{1}{2} t^{2} -\alpha\Big)^2$ 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.