Spectral properties of combinatorial Schrödinger operators on infinite weighted graphs

Article type: Research Article

Authors: Akkouche, Sofiane

Affiliations: Institut de Mathématiques (IMB) 351, Université Bordeaux 1, Cours de la Libération, 33405 Talence, France. E-mail: [email protected]

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.

Keywords: Schrödinger operators, spectrum, bounds, spectral functions, weighted graphs

DOI: 10.3233/ASY-2011-1041

Journal: Asymptotic Analysis, vol. 74, no. 1-2, pp. 1-31, 2011