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Projection methods for discrete Schrodinger operators | | Lyonell S. Boulton
; | | Date: |
23 Dec 2001 | | Subject: | Spectral Theory MSC-class: 47B36; 47B39; 81-08 | math.SP | | Abstract: | Let $H$ be the discrete Schrödinger operator $Hu(n):=u(n-1)+u(n+1)+v(n)u(n)$, $u(0)=0$ acting on $l^2({f Z}^+)$ where the potential $v$ is real-valued and $v(n) o 0$ as $n o infty$. Let $P$ be the orthogonal projection onto a closed linear subspace $L subset l^2({f Z}^+)$. In a recent paper E.B. Davies defines the second order spectrum ${
m Spec}_2(H,L)$ of $H$ relative to $L$ as the set of $z in {f C}$ such that the restriction to $L$ of the operator $P(H-z)^2P$ is not invertible within the space $L$. The purpose of this article is to investigate properties of ${
m Spec}_2(H,L)$ 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 ${
m Spec}_2(H,L)$ as $L$ increases towards $l^2({f Z}^+)$. | | Source: | arXiv, math.SP/0201227 | | Services: | Forum | Review | PDF | Favorites | |
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