On the discrete spectrum of a window-coupled quantum waveguide with mixed boundary conditions

In this paper, the discrete spectrum of a straight quantum waveguide with Dirichlet and Robin boundary conditions imposed on different parts of the boundary and a Neumann window is investigated. We prove results on the existence of the discrete spectrum when the Robin parameter is nonnegative. In addition, a lower bound to the distance between the bottom of the essential spectrum and the spectral threshold in terms of the length of the Neumann window is derived.