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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.

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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