Diana Barseghyan and Pavel Exner
A magnetic version of the Smilansky-Solomyak model
(555K, pdf)

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.