D-S projection methods for discrete Schrodinger operators

Lyonell Boulton Universidad Simon Bolivar, Caracas, Venezuela

Let H be the discrete Schrodinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l^{2}(Z^{+}) where the potential v is real and v(n) → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace Λ ⊂ l^{2}(Z^{+}). In a recent paper E.B. Davies defines the second order spectrum Spec_{2}(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)^{2}P is not invertible within the space Λ. We study properties of Spec_{2}(H, Λ) when Λ is large but finite dimensional. Our particular interest is the conection between this set, the spectrum and the “resonances” of H. In our main result we provide sharp bounds in terms of the potential v for the asymptotic behaviour of Spec_{2}(H, Λ) as Λ increases towards l^{2}(Z^{+}).