Usually spectral structure of the ordinary periodic Schr"{o}dinger 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 Schr"{o}dinger 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 Schr"{o}dinger 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 Schr"{o}dinger 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. |

__Keywords__

Periodic Schr"{o}dinger operator, Floquet-Bloch solutions, Dirichlet-to-Neumann map

__Math Review Classification__

Primary 81Q10
; Secondary 34L40

__Last Updated__

21/06/04

__Length__

26 pages

__Availability__

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