Magnetic Schr\"{o}dinger operator with the potential supported in a curved two-dimensional strip

We consider the magnetic Schr\"odinger operator $H=(i \nabla +A)^2- V$ with a non-negative potential $V$ supported over a strip which is a local deformation of a straight one, and the magnetic field $B:=\mathrm{rot}(A)$ is assumed to be nonzero and local. We show that the magnetic field does not change the essential spectrum of this system, and investigate a sufficient condition for the discrete spectrum of $H$ to be empty.