On the discrete spectrum of a quantum waveguide with Neumann windows in presence of exterior field

The discrete spectrum of the Hamiltonian describing a quantum particle living in three dimensional straight layer of width $d$ in the presence of a constant electric field of strength $F$ is studied. The Neumann boundary conditions are imposed on a finite set of bounded domains (windows) posed at one of the boundary planes and the Dirichlet boundary conditions on the remaining part of the boundary (it is a reduced problem for two identical coupled layers with symmetric electric field). It is proved that such system has eigenvalues below the lower bound of the essential spectrum for any $F\ge0$. Then we closer examine a dependence of bound state energies on $F$ and window's parameters, using numerical methods.