The spectral bounds of the discrete Schrödinger operator

Let H(λ)=− +λb be a discrete Schrödinger operator on 2(Zd ) with a potential b and a non-negative coupling constant λ. When b ≡ 0, it is well known that σ(− ) = [0, 4d]. When b ≡ 0, let s(− + λb) := infσ(− + λb) and M(− + λb) := supσ(− + 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 0 and 4d of the spectrum of − . More precisely, we give a necessary and sufficient condition on the potential b such that s(− + λb) is strictly positive for λ small enough. We obtain a similar necessary and sufficient condition on the potential b such that M(− + λb) is lower than 4d for λ small enough. In dimensions d = 1 and d = 2, the situation is more precise. The following result was proved by Killip and Simon (2003) (for d = 1) in [5], then by Damanik et al. (2003) (for d = 1 and d = 2) in [3]: If σ(− + b) ⊂ [0, 4d], then b ≡ 0. Our study on the bounds of the spectrum of (− + b) allows us to give a different and easy proof to this result. © 2010 Elsevier Inc. All rights reserved.