Spectral theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models |
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Authors: | Christian Sadel |
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Affiliation: | Facultad de Matemáitcas, Pontificia Universidad Católica de Chile, Chile |
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Abstract: | A one-channel operator is a self-adjoint operator on for some countable set with a rank 1 transition structure along the sets of a quasi-spherical partition of . Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees. |
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Keywords: | primary 47A10 secondary 60H25 82B44 Operators on graphs Anderson model Absolutely continuous spectrum Extended states |
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