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)

 

 

Operator kernel estimates for functions of generalized Schrödinger operators


Authors: François Germinet and Abel Klein
Journal: Proc. Amer. Math. Soc. 131 (2003), 911-920
MSC (1991): Primary 81Q10, 47F05; Secondary 35P05
Published electronically: July 17, 2002
MathSciNet review: 1937430
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, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.


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

  • 1. J. M. Barbaroux, J. M. Combes, and P. D. Hislop, Localization near band edges for random Schrödinger operators, Helv. Phys. Acta 70 (1997), no. 1-2, 16–43. Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II (Zürich, 1995). MR 1441595
  • 2. J.-M. Combes and P. D. Hislop, Localization for some continuous, random Hamiltonians in 𝑑-dimensions, J. Funct. Anal. 124 (1994), no. 1, 149–180. MR 1284608, 10.1006/jfan.1994.1103
  • 3. J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251–270. MR 0391792
  • 4. E. B. Davies, Kernel estimates for functions of second order elliptic operators, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 37–46. MR 929793, 10.1093/qmath/39.1.37
  • 5. E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239
  • 6. E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825
  • 7. Alexander Figotin and Abel Klein, Localization of classical waves. I. Acoustic waves, Comm. Math. Phys. 180 (1996), no. 2, 439–482. MR 1405959
  • 8. Alexander Figotin and Abel Klein, Localization of classical waves. II. Electromagnetic waves, Comm. Math. Phys. 184 (1997), no. 2, 411–441. MR 1462752, 10.1007/s002200050066
  • 9. Germinet, F., Klein, A.: Bootstrap Multiscale Analysis and Localization in random media. Commun. Math. Phys. 222, 415-448 (2001)
  • 10. Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Preprint.
  • 11. B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 118–197 (French). MR 1037319, 10.1007/3-540-51783-9_19
  • 12. Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
  • 13. Klein, A., Koines, A.: A general framework for localization of classical waves: I. Inhomogeneous media and defect eigenmodes. Math. Phys. Anal. Geom. 4, 97-130 (2001)
  • 14. Klein, A., Koines, A.: A general framework for localization of classical waves: II. Random media. In preparation.
  • 15. Klein, A., Koines, A., Seifert, M.: Generalized eigenfunctions for waves in inhomogeneous media. J. Funct. Anal. 190, 255-291 (2002)
  • 16. Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, 10.1090/S0273-0979-1982-15041-8

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 81Q10, 47F05, 35P05

Retrieve articles in all journals with MSC (1991): 81Q10, 47F05, 35P05


Additional Information

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

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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06578-4
Keywords: Schr\"odinger operator, magnetic Schr\"odinger operator, classical wave operator, acoustic operator, Maxwell operator, Combes-Thomas estimate
Received by editor(s): July 18, 2001
Received by editor(s) in revised form: October 22, 2001
Published electronically: July 17, 2002
Additional Notes: The second author was supported in part by NSF Grants DMS-9800883 and DMS-9800860
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society