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

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 $\mathrm{\Gamma }$ 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 $[-\frac{1}{4}{\alpha }^2,\mathrm{\infty })$, as for the non-magnetic operator with a straight $\mathrm{\Gamma }$, and demonstrate a sufficient condition for the discrete spectrum of $H$ to be empty.