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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
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.

References [Enhancements On Off] (What's this?)

  • [B] Bourgain, J.: On random Schrödinger operators on $ \mathbb{Z}^2$. Discrete and Continuous Dynamical Systems 8, 1 (2002). MR 1877824 (2003f:47063)
  • [BBP] Bentosela, F., Briet, Ph., Pastur, L.: On the spectral and wave propagation properties of the surface Maryland model. J. Math. Phys. 44 (2003), no. 1, 1-35.MR 1946689 (2003k:81049)
  • [CK] Christ, M., Kiselev, A.: Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials. Geom. Funct. Anal. 12 (2002), no. 6, 1174-1234. MR 1952927 (2003m:47019)
  • [CS] Chahrour, A., Sahbani, J.: On the spectral and scattering theory of the Schrödinger operator with surface potential. Rev. Math. Phys. 12, 561 (2000).MR 1763841 (2001b:81026)
  • [HK] Hundertmark, D., Kirsch, W.: Spectral theory of sparse potentials. In Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), CMS Conf. Proc. 28, 213 (2000). MR 1803388 (2002b:81029)
  • [JL1] Jakšic, V., Last, Y.: Corrugated surfaces and a.c. spectrum. Rev. Math. Phys. 12, 1465 (2000). MR 1809458 (2001m:47143)
  • [JL2] Jakšic, V., Last, Y.: Surface states and spectra. Commun. Math. Phys. 218, 459 (2001). MR 1828849 (2002g:81030)
  • [JM] Jakšic, V., Molchanov, S.: Wave operators for the surface Maryland model. J. Math. Phys. 41, 4452 (2000). MR 1765613 (2002a:81052)
  • [JMP] Jakšic, V., Molchanov, S., Pastur, L.: On the propagation properties of surface waves. In Wave Propagation in Complex Media, IMA Vol. Math. Appl. 96, 143 (1998). MR 1489748
  • [JP] Jakšic, V., Poulin, P.: In preparation.
  • [K] Krishna, M.: Anderson model with decaying randomness-extended states. Proc. Indian Acad. Sci. (MathSci.) 100, 285 (1990).MR 1081712 (91m:82069)
  • [MV1] Molchanov, S., Vainberg, B.: Scattering on the system of the sparse bumps: multidimensional case. Appl. Anal. 71, 167 (1999). MR 1690097 (2000a:47071)
  • [MV2] Molchanov, S., Vainberg, B.: Spectrum of multidimensional Schrödinger operators with sparse potentials. Analytic and computational methods in scattering and applied mathematics Chapman Hall/CRC Res. Notes Math. 417, 231 (2000).MR 1756700 (2001b:47054)
  • [P] Poulin, P.: Ph.D. Thesis, McGill University. In preparation.
  • [RS] Rodnianski, I., Schlag, W.: Classical and quantum scattering for a class of long range random potentials. Int. Math. Res. Not. 2003, no. 5, 243-300.MR 1941087 (2003k:81248)
  • [S] Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton 1993.MR 1232192 (95c:42002)
  • [SV] Shaban, W., Vainberg, B.: Radiation conditions for the difference Schrödinger operators. Appl. Anal. 80 (2001), no. 3-4, 525-556.MR 1914696 (2003d:47049)
  • [V] Vainberg, B.: Asymptotic Methods in Equations of Mathematical Physics. Gordon and Breach Science Publishers, New York 1982. MR 1054376 (91h:35081)

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Additional Information

Philippe Poulin
Affiliation: Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada H3A-2K6

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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