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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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