Un método de proyección espectral para operadores de Jacobi

Let H be the Jacobi operator Hu(n) = u(n - 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l²(Z⁺) where the potential v is real and v(n] → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace ℒ ⊂ l²(Z⁺), In a recent paper E.B. Davies defines the second order spectrum Spec₂ (H, ℒ) of H relative to ℒ as the set of z ∈ C such that the restriction to ℒ of the operator P(H - z] ²P is not invertible within the space ℒ. The present paper is devoted to study properties of Spec₂ (H, ℒ} when ℒ is large but finite dimensional. Our particular interest is the connection between this set and the spectrum of H. In our main result we provide sharp bounds in terms of the potential v for the asymptotic behaviour of Spec₂(H, ℒ) as ℒ increases towards l²(Z ⁺).