Diana Barseghyan and Pavel Exner
A magnetic version of the Smilansky-Solomyak model
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ABSTRACT. We analyze spectral properties of two mutually related families of
magnetic Schr\"{o}dinger operators, $H_{\mathrm{Sm}}(A)=(i
abla
+A)^2+\omega^2 y^2+\lambda y \delta(x)$ and $H(A)=(i
abla
+A)^2+\omega^2 y^2+ \lambda y^2 V(x y)$ in $L^2(\R^2)$, with the
parameters $\omega>0$ and $\lambda<0$, where $A$ is a vector
potential corresponding to a homogeneous magnetic field
perpendicular to the plane and $V$ is a regular nonnegative and
compactly supported potential. We show that the spectral properties
of the operators depend crucially on the one-dimensional
Schr\"{o}dinger operators $L= -rac{\mathrm{d}^2}{\mathrm{d}x^2}
+\omega^2 +\lambda \delta (x)$ and $L (V)= -
rac{\mathrm{d}^2}{\mathrm{d}x^2} +\omega^2 +\lambda V(x)$,
respectively. Depending on whether the operators $L$ and $L(V)$ are
positive or not, the spectrum of $H_{\mathrm{Sm}}(A)$ and $H(V)$
exhibits a sharp transition.