In this thesis, we study the spectral properties of the hierarchical Anderson model. This model is an approximation of the Anderson tight-binding model on Zd , with the usual discrete Laplacian replaced by a hierarchical long-range interaction operator. In the hierarchical Anderson model, we are given a countable set X endowed with a hierarchical structure. The free hierarchical Laplacian is a self-adjoint operator Delta acting on the Hilbert space l 2( X ). The spectrum of Delta consists of isolated infinitely degenerate eigenvalues. We look at small random perturbations of the operator Delta. The disorder is modeled by a random potential Vo, (Vopsi)(x) = o( x)psi(x) for psi ∈ l 2( X ). The numbers o(x) are identically distributed independent random variables with a bounded density. The hierarchical Anderson model is the random self-adjoint operator Ho = Delta + Vo. We prove the following two results. If the model has a spectral dimension dsp ≤ 4 then, almost surely, the spectrum of Ho is dense pure-point. The second result is on eigenvalue statistics. For dsp < 1, the energy levels for Ho are asymptotically a Poisson point process in the thermodynamic limit, after a proper rescaling.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.115890 |
| Date | January 2008 |
| Creators | Kritchevski, Evgenij. |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Doctor of Philosophy (Department of Mathematics and Statistics.) |
| Rights | All items in eScholarship@McGill are protected by copyright with all rights reserved unless otherwise indicated. |
| Relation | alephsysno: 002840015, proquestno: AAINR66676, Theses scanned by UMI/ProQuest. |
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