| 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
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