ar X iv : m at h / 02 01 22 7 v 2 [ m at h . SP ] 1 3 Fe b 20 03 PROJECTION METHODS FOR DISCRETE SCHRÖDINGER OPERATORS
نویسنده
چکیده
Let H be the discrete Schrödinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l(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 L ⊂ l(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)P is not invertible within the space L. The purpose of this article is to investigate properties of Spec2(H,L) when L 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,L) as L increases towards l(Z).
منابع مشابه
ar X iv : m at h / 02 01 22 7 v 1 [ m at h . SP ] 2 3 Ja n 20 02 D - S PROJECTION METHODS FOR DISCRETE SCHRÖDINGER OPERATORS
Let H be the discrete Schrödinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l(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 L ⊂ l(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 op...
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