The Molchanov–Vainberg Laplacian
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- by Philippe Poulin PDF
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Abstract:
It is well known that the Green function of the standard discrete Laplacian on $l^2(\mathbb {Z}^d)$, \[ \Delta _{st} \psi (n)=(2d)^{-1}\sum _{|n-m|=1}\psi (m), \] exhibits a pathological behavior in dimension $d \geq 3$. In particular, the estimate \begin{equation*} \langle \delta _0|(\Delta _{st}-E- \mathrm {i} 0)^{-1}\delta _n\rangle = O(|n|^{-\frac {d-1}{2}}) \end{equation*} 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, \begin{equation*} \Delta \psi (n) = 2^{-d} \sum _{|n-m|=\sqrt d}\psi (m), \end{equation*} and conjectured that the estimate \begin{equation*} \langle \delta _0|(\Delta -E-\mathrm {i} 0)^{-1}\delta _n \rangle = O(|n|^{-\frac {d-1}{2}}) \end{equation*} holds for all $0<|E|<1$. In this paper we prove this conjecture.References
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Additional Information
- Philippe Poulin
- Affiliation: Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada H3A-2K6
- Email: ppoulin@math.mcgill.ca
- Received by editor(s): April 20, 2005
- Received by editor(s) in revised form: July 12, 2005
- Published electronically: June 13, 2006
- Additional Notes: This research was supported in part by NSERC
- Communicated by: Mikhail Shubin
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 77-85
- MSC (2000): Primary 47B39, 34L40
- DOI: https://doi.org/10.1090/S0002-9939-06-08431-0
- MathSciNet review: 2280177