Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn