The Absence of Absolutely Continuous Spectra for Radial Tree Graphs
Seminar Room 1, Newton Institute
AbstractWe will introduce a family of Schrödinger operators on tree graphs with coupling conditions given by (b_n-1)^2+4 real parameters where b_n is the branching number. We will show the unitary equivalence of the Hamiltonian on the tree graph and the orthogonal sum of the Hamiltonians on the halflines. We will use this unitary equivalence to prove that for a large family of coupling conditions there is no absolutely continuous spectrum of the Hamiltonian on the sparse tree. On the other hand, we will show nontrivial examples of trees with the spectrum which is purely absolutely continuous.
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