A regular analogue of the Smilansky model: spectral properties

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.