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

C. Fox, V. Oleinik and B. Pavlov


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.

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

Math Review Classification
Primary 81Q10 ; Secondary 34L40

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26 pages

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