Spectral properties of combinatorial Schrödinger operators on infinite weighted graphs

Abstract

Let H(λ)=−Δ+λb be the combinatorial Schrödinger operator on an infinite connected graph G with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)⊂[0,2]. When b≢0, let s(−Δ+λb):=inf σ(−Δ+λb) and M(−Δ+λb):=sup σ(−Δ+λb) be the bounds of the spectrum of the Schrödinger operator.

One of the aims of this paper is to study the influence of the potential b on the bounds of the spectrum of −Δ. More precisely, we give a condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar result for the top of the spectrum. We also prove that for a recurrent bipartite weighted graph, the only Schrödinger operator H=−Δ+b such that σ(H)⊂[0,2] is the Laplacian −Δ.

We study independently the asymptotic behaviour of the function λ↦s(−Δ+λb) for non-negative potentials b. We prove that s(∞):=lim _λ→+∞s(−Δ+λb)<+∞ if and only if inf _x∈Vb(x)=0 and we characterize this limit in the general case. As an application, we give an exact value of this limit for certain Jacobi matrices.