The spectral bounds of the discrete Schrödinger operator

Let H(λ)=−Δ+λbH(λ)=−Δ+λb be a discrete Schrödinger operator on ℓ2(Zd)ℓ2(Zd) with a potential b and a non-negative coupling constant λ  . When b≡0b≡0, it is well known that σ(−Δ)=[0,4d]σ(−Δ)=[0,4d]. When b≢0b≢0, let s(−Δ+λb):=infσ(−Δ+λb) and M(−Δ+λb):=supσ(−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 0 and 4d of the spectrum of −Δ. More precisely, we give a necessary and sufficient condition on the potential b   such that s(−Δ+λb)s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar necessary and sufficient condition on the potential b   such that M(−Δ+λb)M(−Δ+λb) is lower than 4d for λ   small enough. In dimensions d=1d=1 and d=2d=2, the situation is more precise. The following result was proved by Killip and Simon (2003) (for d=1d=1) in [5], then by Damanik et al. (2003) (for d=1d=1 and d=2d=2) in [3]:If σ(−Δ+b)⊂[0,4d],then b≡0. Our study on the bounds of the spectrum of (−Δ+b)(−Δ+b) allows us to give a different and easy proof to this result.