Many particles problems for quantum layers

In quantum computer implementation using resonance phenomena in quantum waveguides coupled through small windows is suggested. It has some interesting features, but faces some objectives. There are two variants of such devices. In the first one we have two quantum waveguides for each qubit, and the state of the qubit is determined by the position of the electron. In the second variant we have one waveguide for each qubit and the state of the qubit is determined by the electron spin projection ("up" or "down"). In each case, to perform many-qubit operations successfully, we need to solve the two-electrons problem in the area of waveguide and to synchronize the electrons in different waveguides. Also preparation of the initial state is here very important problem. In the present paper we study the Green function for the simplest cases of two noninteracting particles in two-dimensional infinite straight waveguide and three-dimensional infinite layer with Neumann boundary conditions. We use the convolution method based on known representation of the one-particle Green function