On the spectral theory of operators on trees [Elektronische Ressource] / Matthias Keller. Gutachter: Daniel Lenz ; Simone Warzel ; Richards Froese

On the spectral theory ofoperators on treesDissertationzur Erlangung des akademischen Gradesdoctor rerum naturaliumvorgelegt dem Ratder Fakult at fur Mathematik und Informatikder Friedrich-Schiller-Universit at Jenavon Dipl. Math. Matthias Kellergeboren am 31.12.1980 in Karl-Marx-Stadt, jetzt Chemnitz1. Gutachter: Prof. Dr. Daniel Lenz, Friedrich Schiller Universit at Jena2. Gutachter: Prof. Dr. Simone Warzel, Technische Universit at Munc hen3. Gutachter: Prof. Dr. Richard Froese, University of British Columbia VancouverTag der o en tlichen Verteidigung: 17.12.2010AbstractWe study a class of rooted trees with a substitution type structure. These treesare not necessarily regular, but exhibit a lot of symmetries. We consider nearestneighbor operators which re ect the symmetries of the trees. The spectrum of suchoperators is proven to be purely absolutely continuous and to consist of nitely manyintervals. We further investigate stability of the absolutely continuous spectrumunder perturbations by su ciently small potentials. On the one hand, we look at aclass of deterministic potentials which include radial symmetric ones. The absolutelycontinuous spectrum is stable under su ciently small perturbations of this type ifand only if the tree is not regular. On the other hand, we study random potentials.