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 $[0,\mathrm{\infty })$. 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.