A magnetic version of the Smilansky-Solomyak model

Abstract

We analyze spectral properties of two mutually related families of magnetic Schrödinger operators, $H_{\mathrm{Sm}}(A)=(i\nabla +A{)}^2+{\omega }^2{y}^2+\lambda y\delta (x)$ and $H(A)=(i\nabla +A{)}^2+{\omega }^2{y}^2+\lambda y^{2}V(xy)$ 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 Schrödinger 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.

References in corpus (1)

• Spectral and resonance properties of Smilansky Hamiltonian