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The Molchanov-Vainberg Laplacian


Author: Philippe Poulin
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
Published electronically: June 13, 2006
MathSciNet review: 2280177
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Abstract: It is well known that the Green function of the standard discrete Laplacian on $ l^2(\mathbb{Z}^d)$,

$\displaystyle \Delta_{\mathit{st}} \psi(n)=(2d)^{-1}\sum_{\vert n-m\vert=1}\psi(m), $

exhibits a pathological behavior in dimension $ d \geq 3$. In particular, the estimate

$\displaystyle \langle \delta_0\vert(\Delta_{\mathit{st}}-E- \mathrm{i} 0)^{-1}\delta_n\rangle = O(\vert n\vert^{-\frac{d-1}{2}}) $

fails for $ 0 <\vert E\vert<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,

$\displaystyle \Delta \psi(n) = 2^{-d} \sum_{\vert n-m\vert=\sqrt d}\psi(m), $

and conjectured that the estimate

$\displaystyle \langle \delta_0\vert(\Delta-E-\mathrm{i} 0)^{-1}\delta_n \rangle = O(\vert n\vert^{-\frac{d-1}{2}}) $

holds for all $ 0<\vert E\vert<1$. In this paper we prove this conjecture.


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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

DOI: https://doi.org/10.1090/S0002-9939-06-08431-0
Keywords: Discrete Laplacian, Green function
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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