Spectral properties of higher order anharmonic oscillators

TL;DR

This paper analyzes the spectral properties of a class of higher order anharmonic oscillators, establishing the existence, uniqueness, and behavior of the minimum ground state energy as a parameter varies, with implications for magnetic Schrödinger operators.

Contribution

It provides new results on the existence, uniqueness, and asymptotic behavior of the minimum ground state energy for a family of higher order anharmonic oscillators, extending previous spectral analysis work.

Findings

The minimum of the ground state energy over lpha is attained at a unique point.

The minimizing lpha tends to zero as the order k increases.

The minimum is proven to be non-degenerate.

Abstract

We discuss spectral properties of the self-adjoint operator \[ -d^2/dt^2 + (t^{k+1}/(k+1)-\alpha)^2 \] in $L^2(\mathbb{R})$ for odd integers $k$. We prove that the minimum over $\alpha$ of the ground state energy of this operator is attained at a unique point which tends to zero as $k$ tends to infinity. Moreover, we show that the minimum is non-degenerate. These questions arise naturally in the spectral analysis of Schr\"{o}dinger operators with magnetic field. This extends or clarifies previous results by Pan-Kwek, Helffer-Morame, Aramaki, Helffer-Kordyukov and Helffer.