Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators


Authors: Jean-Marc Bouclet, François Germinet and Abel Klein
Journal: Proc. Amer. Math. Soc. 132 (2004), 2703-2712
MSC (2000): Primary 81Q10, 47F05; Secondary 35P05
DOI: https://doi.org/10.1090/S0002-9939-04-07431-3
Published electronically: April 21, 2004
MathSciNet review: 2054797
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.


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

  • [CKM] R. Carmona, A. Klein, and F. Martinelli: Anderson localization for Bernoulli and other singular potentials, Commun. Math. Phys. 108, 41-66 (1987). MR 88f:82027
  • [CoTh] J. M. Combes and L. Thomas: Asymptotic behavior of eigenfunctions for multi-particle Schrödinger operators, Commun. Math. Phys. 34, 251-270 (1973). MR 52:12611
  • [DSS] D. Damanik, R. Sims, and G. Stolz: Localization for one-dimensional, continuum, Bernoulli-Anderson models, Duke Math. J. 114, 59-100 (2002). MR 2003h:82048
  • [Da1] E. B. Davies: Kernel estimates for functions of second order elliptic operators, Quart. J. Math. Oxford Ser. (2) 39, 37-46 (1988). MR 89c:35123
  • [Da2] E. B. Davies: Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1990. MR 92a:35035
  • [DBG] S. De Bièvre and F. Germinet: Dynamical localization for the random dimer Schrödinger operator, J. Statist. Phys. 98, 1135-1147 (2000). MR 2000m:82053
  • [DiSj] M. Dimassi and J. Sjöstrand: Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Note Series 268, Cambridge University Press, Cambridge, 1999. MR 2001b:35237
  • [ElGr] P. Elbau and G. M. Graf: Equality of bulk and edge Hall conductance revisited, Commun. Math. Phys. 229, 415-432 (2002). MR 2003h:81264
  • [GKT] F. Germinet, A. Kiselev, and S. Tcheremchantsev: Transfer matrices and transport for 1D Schrödinger operators with singular spectrum, Ann. Inst. Fourier, to appear.
  • [GK1] F. Germinet and A. Klein: Bootstrap multiscale analysis and localization in random media, Commun. Math. Phys. 222, 415-448 (2001). MR 2002m:82035
  • [GK2] F. Germinet and A. Klein: Operator kernel estimates for functions of generalized Schrödinger operators, Proc. Amer. Math. Soc. 131, 911-920 (2003). MR 2003k:47067
  • [GK3] F. Germinet and A. Klein: A characterization of the metal-insulator transport transition, submitted.
  • [GK4] F. Germinet and A. Klein: The Anderson metal-insulator transport transition, to appear in Contemp. Math.
  • [HeSj] B. Helffer and S. Sjöstrand: Équation de Schrödinger avec champ magnétique et équation de Harper, in Schrödinger operators, H. Holden and A. Jensen, eds., Lecture Notes in Physics, vol. 345, Springer-Verlag, Berlin, 1989, pp. 118-197. MR 91g:35078
  • [Ho] L. Hörmander: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Second edition. Springer Study Edition, Springer-Verlag, Berlin, 1990. MR 91m:35001b
  • [JSBS] S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz: Delocalization in random polymer models, Commun. Math. Phys. 233, 27-48 (2003).
  • [KLS] A. Klein, J. Lacroix, and A. Speis: Localization for the Anderson model on a strip with singular potentials, J. Funct. Anal. 94, 135-155 (1990). MR 92c:82060
  • [Si] B. Simon: Schrödinger semi-groups, Bull. Amer. Math. Soc. (N.S.) 7, 447-526 (1982). MR 86b:81001a
  • [Tc] S. Tcheremchantsev: Dynamical analysis of Schrödinger operators with sparse potential, Comm. Math. Phys., to appear.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 81Q10, 47F05, 35P05

Retrieve articles in all journals with MSC (2000): 81Q10, 47F05, 35P05


Additional Information

Jean-Marc Bouclet
Affiliation: UMR 8524 CNRS, UFR de Mathématiques, Université de Lille 1, F-59655 Villeneuve d’Ascq Cédex, France
Email: Jean-Marc.Bouclet@agat.univ-lille1.fr

François Germinet
Affiliation: UMR 8524 CNRS, UFR de Mathématiques, Université de Lille 1, F-59655 Villeneuve d’Ascq Cédex, France
Address at time of publication: Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Email: germinet@agat.univ-lille1.fr, germinet@math.u-cergy.fr

Abel Klein
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697-3875
Email: aklein@uci.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07431-3
Keywords: Schr\"odinger operator, acoustic operator, Maxwell operator, Combes-Thomas estimate, operator kernel, Gevrey class
Received by editor(s): February 13, 2003
Received by editor(s) in revised form: July 7, 2003
Published electronically: April 21, 2004
Additional Notes: The third author was supported in part by NSF Grant DMS-0200710.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society