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 | References | Similar Articles | Additional Information

Abstract: It is well known that the Green function of the standard discrete Laplacian on ,

**[B]**Jean Bourgain,*On random Schrödinger operators on ℤ²*, Discrete Contin. Dyn. Syst.**8**(2002), no. 1, 1–15. MR**1877824**, 10.3934/dcds.2002.8.1**[BBP]**F. Bentosela, Ph. Briet, and L. Pastur,*On the spectral and wave propagation properties of the surface Maryland model*, J. Math. Phys.**44**(2003), no. 1, 1–35. MR**1946689**, 10.1063/1.1521798**[CK]**M. Christ and A. Kiselev,*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**, 10.1007/s00039-002-1174-9**[CS]**Ayham Chahrour and Jaouad Sahbani,*On the spectral and scattering theory of the Schrödinger operator with surface potential*, Rev. Math. Phys.**12**(2000), no. 4, 561–573. MR**1763841**, 10.1142/S0129055X00000186**[HK]**Dirk Hundertmark and Werner Kirsch,*Spectral theory of sparse potentials*, Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), CMS Conf. Proc., vol. 28, Amer. Math. Soc., Providence, RI, 2000, pp. 213–238. MR**1803388****[JL1]**Vojkan Jakšić and Yoram Last,*Corrugated surfaces and a.c. spectrum*, Rev. Math. Phys.**12**(2000), no. 11, 1465–1503. MR**1809458**, 10.1142/S0129055X00000563**[JL2]**Vojkan Jaksić and Yoram Last,*Surface states and spectra*, Comm. Math. Phys.**218**(2001), no. 3, 459–477. MR**1828849**, 10.1007/PL00005560**[JM]**Vojkan Jakšić and Stanislav Molchanov,*Wave operators for the surface Maryland model*, J. Math. Phys.**41**(2000), no. 7, 4452–4463. MR**1765613**, 10.1063/1.533353**[JMP]**V. Jakšić, S. Molchanov, and L. Pastur,*On the propagation properties of surface waves*, Wave propagation in complex media (Minneapolis, MN, 1994) IMA Vol. Math. Appl., vol. 96, Springer, New York, 1998, pp. 143–154. MR**1489748**, 10.1007/978-1-4612-1678-0_7**[JP]**Jakšic, V., Poulin, P.: In preparation.**[K]**M. Krishna,*Anderson model with decaying randomness existence of extended states*, Proc. Indian Acad. Sci. Math. Sci.**100**(1990), no. 3, 285–294. MR**1081712****[MV1]**S. Molchanov and B. Vainberg,*Scattering on the system of the sparse bumps: multidimensional case*, Appl. Anal.**71**(1999), no. 1-4, 167–185. MR**1690097**, 10.1080/00036819908840711**[MV2]**S. Molchanov and B. Vainberg,*Spectrum of multidimensional Schrödinger operators with sparse potentials*, Analytical and computational methods in scattering and applied mathematics (Newark, DE, 1998) Chapman & Hall/CRC Res. Notes Math., vol. 417, Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 231–254. MR**1756700****[P]**Poulin, P.: Ph.D. Thesis, McGill University. In preparation.**[RS]**Igor Rodnianski and Wilhelm Schlag,*Classical and quantum scattering for a class of long range random potentials*, Int. Math. Res. Not.**5**(2003), 243–300. MR**1941087**, 10.1155/S1073792803201100**[S]**Elias M. Stein,*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192****[SV]**W. Shaban and B. Vainberg,*Radiation conditions for the difference Schrödinger operators*, Appl. Anal.**80**(2001), no. 3-4, 525–556. MR**1914696**, 10.1080/00036810108841007**[V]**B. R. Vaĭnberg,*Asymptotic methods in equations of mathematical physics*, Gordon & Breach Science Publishers, New York, 1989. Translated from the Russian by E. Primrose. MR**1054376**

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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.