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.
Categories: math.SP math-ph math.MP