Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrödinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann {For the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.submitted 11/05/04, accepted 21/09/04