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

Bouclet, Jean-Marc; Germinet, François; Klein, Abel

doi:10.1090/S0002-9939-04-07431-3

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators
HTML articles powered by AMS MathViewer

by Jean-Marc Bouclet, François Germinet and Abel Klein PDF

Proc. Amer. Math. Soc. 132 (2004), 2703-2712 Request permission

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

René Carmona, Abel Klein, and Fabio Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41–66. MR 872140, DOI 10.1007/BF01210702
J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251–270. MR 391792, DOI 10.1007/BF01646473
David Damanik, Robert Sims, and Günter Stolz, Localization for one-dimensional, continuum, Bernoulli-Anderson models, Duke Math. J. 114 (2002), no. 1, 59–100. MR 1915036, DOI 10.1215/S0012-7094-02-11414-8
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, DOI 10.1093/qmath/39.1.37
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR 1103113
Stephan De Bièvre and François Germinet, Dynamical localization for the random dimer Schrödinger operator, J. Statist. Phys. 98 (2000), no. 5-6, 1135–1148. MR 1751697, DOI 10.1023/A:1018615728507
Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR 1735654, DOI 10.1017/CBO9780511662195
P. Elbau and G. M. Graf, Equality of bulk and edge Hall conductance revisited, Comm. Math. Phys. 229 (2002), no. 3, 415–432. MR 1924362, DOI 10.1007/s00220-002-0698-z
F. Germinet, A. Kiselev, and S. Tcheremchantsev: Transfer matrices and transport for 1D Schrödinger operators with singular spectrum, Ann. Inst. Fourier, to appear.
François Germinet and Abel Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415–448. MR 1859605, DOI 10.1007/s002200100518
François Germinet and Abel Klein, Operator kernel estimates for functions of generalized Schrödinger operators, Proc. Amer. Math. Soc. 131 (2003), no. 3, 911–920. MR 1937430, DOI 10.1090/S0002-9939-02-06578-4
F. Germinet and A. Klein: A characterization of the metal-insulator transport transition, submitted.
F. Germinet and A. Klein: The Anderson metal-insulator transport transition, to appear in Contemp. Math.
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, DOI 10.1007/3-540-51783-9_{1}9
Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065993, DOI 10.1007/978-3-642-61497-2
S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz: Delocalization in random polymer models, Commun. Math. Phys. 233, 27-48 (2003).
Abel Klein, Jean Lacroix, and Athanasios Speis, Localization for the Anderson model on a strip with singular potentials, J. Funct. Anal. 94 (1990), no. 1, 135–155. MR 1077548, DOI 10.1016/0022-1236(90)90031-F
Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
S. Tcheremchantsev: Dynamical analysis of Schrödinger operators with sparse potential, Comm. Math. Phys., to appear.