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Spectral analysis for adjacency operators on graphs. (English) Zbl 1137.81013
Let \(X\) be a graph without multiple edges and loops and \(\deg(X) = \sup_{x \in X} \deg(x)\) finite, where \(\deg(x)\) is the degree of the vertex \(x.\) The paper is concerned with the spectral properties of the adjacency operator \((Hf)(x) := \sum_{y \backsim x}f(y)\) on \(\mathcal{H}= \ell^2(X),\) where \(y \backsim x\) indicates that the vertices \(y\) and \(x\) are connected. It is known that \(H\) is bounded and self-adjoint on \(\mathcal{H}\) with norm bounded by \(\deg(X).\) Few examples are known of adjacency operators on graphs with purely absolutely continuous spectrum. A non-empty singular spectrum is possibly and even a dense point spectrum. A particular case of the main result proved in this paper is that for a family of graphs called admissible, the spectrum of \(H\) is purely absolutely continuous, except at the origin where it may have an eigenvalue. The approach uses the so-called method of the weakly conjugate operator which is a commutator method claimed to be easier to use than that of Mourre in this case. It is shown that the graph associated with the one-dimensional \(XY\) Hamiltonian of solid-state physics is admissible.

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A10 Spectrum, resolvent
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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