A magnetic version of the Smilansky-Solomyak model

TL;DR

This paper studies how magnetic fields influence the spectral properties of certain Schrödinger operators, revealing sharp transitions depending on associated one-dimensional operators' positivity.

Contribution

It introduces a magnetic version of the Smilansky-Solomyak model and analyzes how spectral properties depend on related one-dimensional Schrödinger operators.

Findings

Spectral properties depend crucially on the positivity of associated 1D operators.

Sharp spectral transitions occur based on the positivity or negativity of these operators.

Magnetic fields significantly alter the spectral behavior of the studied operators.

Abstract

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.