Spectrum of the Magnetic Schrodinger Operator in a Waveguide with   Combined Boundary Conditions
  We consider the magnetic Schrodinger 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 boundary condition (see Section 2, Eq. (2.2) for the definition of this boundary condition}. 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. 
Denis Borisov, Tomas Ekholm, and Hynek Kovarik