Magnetic field influence on the discrete spectrum of locally deformed leaky wires

Abstract

We consider magnetic Schr\"odinger operator $H=(i \nabla +A)^2-\alpha \delta_\Gamma$ with an attractive singular interaction supported by a piecewise smooth curve $\Gamma$ being a local deformation of a straight line. The magnetic field $B$ is supposed to be nonzero and local. We show that the essential spectrum is $[-\frac14\alpha^2,\infty)$, as for the non-magnetic operator with a straight $\Gamma$, and demonstrate a sufficient condition for the discrete spectrum of $H$ to be empty.