The molchanov-vainberg laplacian

Philippe Poulin

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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 languageEnglish
Pages (from-to)77-85
Number of pages9
JournalProceedings of the American Mathematical Society
Issue number1
Publication statusPublished - Jan 2007
Externally publishedYes


  • Discrete Laplacian
  • Green function

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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