Spectral estimates for a class of Schr\"odinger operators with infinite phase space and potential unbounded from below

TL;DR

This paper studies two-dimensional Schrödinger operators with specific unbounded potentials, identifying a critical parameter value that determines whether the spectrum is bounded or unbounded, and providing bounds for eigenvalue sums.

Contribution

It introduces spectral estimates for a class of Schrödinger operators with unbounded potentials and characterizes the spectral transition at a critical parameter value.

Findings

Existence of a critical lambda separating bounded and unbounded spectra.

Spectral bounds established for the subcritical case.

Spectral transition characterized by potential parameters.

Abstract

We analyze two-dimensional Schr\"odinger operators with the potential $|xy|^p - \lambda (x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $\lambda\ge 0$. We show that there is a critical value of $\lambda$ such that the spectrum for $\lambda<\lambda_\mathrm{crit}$ is below bounded and purely discrete, while for $\lambda>\lambda_\mathrm{crit}$ it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.