Magnetic Schrödinger Operator with the Potential Supported in a Curved Two-Dimensional Strip

We consider the magnetic Schrödinger operator H=(i ∇ +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:=rot(A) is assumed to be non-zero 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.