17-86 Diana Barseghyan and Pavel Exner

A magnetic version of the Smilansky-Solomyak model (555K, pdf) Aug 24, 17

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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.

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