Abstract
It is well known that the Green function of the standard discrete Laplacian on l2(ℤd), Δstψ(n) = (2d)-1 Σ |n-m|=1 ψ(m), exhibits a pathological behavior in dimension d 7≥ 3. In particular, the estimate δd0|(Δst - E - i0) -1dnδ = O(|n|-d-1/ 2 ) fails for 0 < |E| < 1 - 2/d. This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian, Δψ(n) = 2-dΣ |n-m|=√d ψ(m), and conjectured that the estimate δ 0|(Delta; - E - i0)-1δn = O(|n|-d-1/ 2 ) holds for all 0 < |E| < 1. In this paper we prove this conjecture.
Original language | English |
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Pages (from-to) | 77-85 |
Number of pages | 9 |
Journal | Proceedings of the American Mathematical Society |
Volume | 135 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2007 |
Externally published | Yes |
Keywords
- Discrete Laplacian
- Green function
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics