Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition

TL;DR

This paper investigates how the spectral properties of a specific class of two-dimensional Schrödinger operators change abruptly at a critical coupling constant, revealing a transition from discrete to continuous spectrum.

Contribution

It provides a detailed spectral analysis of parameter-dependent Schrödinger operators, identifying the critical point and characterizing the spectrum in different regimes.

Findings

Spectrum is the entire real line in the supercritical case.

Below critical coupling, the spectrum is purely discrete with bounds on moments.

At the critical point, the essential spectrum covers positive reals, negative spectrum is discrete, and a ground state exists.

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 $\lambda\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $\lambda$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $\lambda$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.