Diana Barseghyan, Pavel Exner
A regular analogue of the Smilansky model: spectral properties
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ABSTRACT. We analyze spectral properties of the operator
$H=rac{\partial^2}{\partial x^2} -rac{\partial^2}{\partial y^2}
+\omega^2y^2-\lambda y^2V(x y)$ in $L^2(\mathbb{R}^2)$, where
$\omega
e 0$ 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
$L=-rac{\mathrm{d}^2}{\mathrm{d}x^2}+\omega^2-\lambda V(x)$ 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.