A magnetic version of the Smilansky-Solomyak model

We analyze spectral properties of two mutually related families of magnetic Schr\"{o}dinger operators, $H_{\mathrm{Sm}}(A)=(i \nabla +A)^2+\omega^2 y^2+\lambda y \delta(x)$ and $H(A)=(i \nabla +A)^2+\omega^2 y^2+ \lambda y^2 V(x y)$ 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 $L= -\frac{\mathrm{d}^2}{\mathrm{d}x^2} +\omega^2 +\lambda \delta (x)$ and $L (V)= - \frac{\mathrm{d}^2}{\mathrm{d}x^2} +\omega^2 +\lambda V(x)$, 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.