Dirichlet-to-Neumann map machinery for resonance gaps and bands of periodic Networks

Usually spectral structure of the ordinary periodic Schrödinger operator is revealed based on analysis of the corresponding transfer-matrix. In this approach the quasi-momentum exponentials appear as eigenvalues of the transfer-matrix which correspond to quasi-periodic solutions of the homogeneous Schrödinger equation, and the corresponding Weyl functions are obtained as coordinates of the appropriate eigenvectors. This approach, though effective for tight-binding analysis of one-dimensional periodic Schrödinger operators, is inconvenient for spectral analysis on realistic periodic quantum networks with multi-dimensional period, where several leads are attached to each vertex, and can’t be extended to partial Schrödinger equation. We propose an alternative approach where the Dirichlet-to-Neumann map is used instead of the transfer matrix. We apply this approach to obtain, for realistic quantum networks, conditions of existence of resonance gaps or bands. PACS numbers: 73.63.Hs,73.23.Ad,85.35.-p,85.35.Be