Essential self-adjointness for combinatorial Schrödinger operators

We introduce the weighted graph Laplacian ∆ω,c and the notion of Schrödinger operator of the form ∆1,a +W on a locally finite graph G. Concerning essential self-adjointness, we extend Wojciechowski’s and Dodz-iuk’s results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the Laplacian ∆ω,c is essentially self-adjoint and the same holds for Schrödinger operators provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us to write any Schrödinger operator ∆1,a +W as a Laplacian ∆ω,c modulo a unitary transform.