Projection methods for discrete Schrödinger operators

Let H be the discrete Schrödinger operator H u ( n ) : = u ( n − 1 ) + u ( n + 1 ) + v ( n ) u ( n ) , u ( 0 ) = 0 acting on l2 Z+, where the potential v is real-valued and v(n) → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace Λ l2 Z+). In a recent paper E. B. Davies defines the second order spectrum Spec2(H, Λ) of H relative to Λ as the set of z ∈ C such that the restriction to Λ of the operator P(H - z)2P is not invertible within the space Λ. The purpose of this article is to investigate properties of Spec2(H, Λ) when Λ is large but finite dimensional. We explore in particular the connection between this set and the spectrum of H. Our main result provides sharp bounds in terms of the potential v for the asymptotic behaviour of Spec2(H, Λ) as Λ increases towards l2 Z+). 2000 Mathematics Subject Classification 47B36 (primary), 47B39, 81-08 (secondary).