A regular analogue of the Smilansky model: spectral properties

TL;DR

This paper studies the spectral properties of a specific differential operator, revealing how its spectrum changes depending on the sign of the spectrum of an associated Schrödinger operator, and provides bounds on eigenvalues.

Contribution

It establishes the spectral nature of the operator in critical and subcritical cases, extending understanding of spectral transitions in analogous models.

Findings

Spectrum is purely essential and covers [0,∞) when infσ(L)=0.

In the case infσ(L)>0, the spectrum starts at ω with discrete eigenvalues in [0,ω).

Provides bounds on eigenvalue moments for the operator.

Abstract

We analyze spectral properties of the operator $H=\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} +\omega^2y^2-\lambda y^2V(x y)$ in $L^2(\mathbb{R}^2)$, where $\omega\ne 0$ and $V\ge 0$ is a compactly supported and sufficiently regular potential. It is known that the spectrum of $H$ depends on the one-dimensional Schr\"odinger operator $L=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\omega^2-\lambda V(x)$ and it changes substantially as $\inf\sigma(L)$ switches sign. We prove that in the critical case, $\inf\sigma(L)=0$, the spectrum of $H$ is purely essential and covers the interval $[0,\infty)$. In the subcritical case, $\inf\sigma(L)>0$, the essential spectrum starts from $\omega$ and there is a non-void discrete spectrum in the interval $[0,\omega)$. We also derive a bound on the corresponding eigenvalue moments.