Operator kernel estimates for functions of generalized Schrödinger operators

Germinet, François; Klein, Abel

doi:10.1090/S0002-9939-02-06578-4

Operator kernel estimates for functions of generalized Schrödinger operators
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by François Germinet and Abel Klein PDF

Proc. Amer. Math. Soc. 131 (2003), 911-920 Request permission

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

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
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, DOI 10.1006/jfan.1994.1103
J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251–270. MR 391792
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, 1989. MR 990239, DOI 10.1017/CBO9780511566158
E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825, DOI 10.1017/CBO9780511623721
Alexander Figotin and Abel Klein, Localization of classical waves. I. Acoustic waves, Comm. Math. Phys. 180 (1996), no. 2, 439–482. MR 1405959
Alexander Figotin and Abel Klein, Localization of classical waves. I. Acoustic waves, Comm. Math. Phys. 180 (1996), no. 2, 439–482. MR 1405959
Germinet, F., Klein, A.: Bootstrap Multiscale Analysis and Localization in random media. Commun. Math. Phys. 222, 415-448 (2001)
Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Preprint.
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
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
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)
Klein, A., Koines, A.: A general framework for localization of classical waves: II. Random media. In preparation.
Klein, A., Koines, A., Seifert, M.: Generalized eigenfunctions for waves in inhomogeneous media. J. Funct. Anal. 190, 255–291 (2002)
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