Projection methods for discrete Schrodinger operators
| dc.creator | Boulton, Lyonell S. | |
| dc.date | 2002-01-23 | |
| dc.date | 2003-02-13 | |
| dc.date.accessioned | 2026-07-07T04:46:04Z | |
| dc.date.available | 2026-07-07T04:46:04Z | |
| dc.description | Let $H$ be the discrete Schrödinger operator $Hu(n):=u(n-1)+u(n+1)+v(n)u(n)$, $u(0)=0$ acting on $l^2({\bf Z}^+)$ where the potential $v$ is real-valued and $v(n)\to 0$ as $n\to \infty$. Let $P$ be the orthogonal projection onto a closed linear subspace $L \subset l^2({\bf Z}^+)$. In a recent paper E.B. Davies defines the second order spectrum ${\rm Spec}_2(H,L)$ of $H$ relative to $L$ as the set of $z \in {\bf C}$ 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 ${\rm Spec}_2(H,L)$ 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 ${\rm Spec}_2(H,L)$ as $L$ increases towards $l^2({\bf Z}^+)$. | |
| dc.description | 24 pages, 5 figures, the version 2 contains some corrections in section 4 | |
| dc.identifier | https://arxiv.org/abs/math/0201227 | |
| dc.identifier | http://arxiv.org/abs/math/0201227 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63189 | |
| dc.subject | Spectral Theory | |
| dc.subject | 47B36; 47B39; 81-08 | |
| dc.title | Projection methods for discrete Schrodinger operators | |
| dc.type | text |