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Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators. (English) Zbl 1053.81028
The paper is devoted to the improvement of the authors’ previous paper [F. Germinet and A. Klei, Proc. Am. Math. Soc. 131, No. 3, 911–920 (2003; Zbl 1013.81009)]. Let \(H\) be a generalized Schrödinger operator on \(L^2(\mathbb{R}^d, dq;\mathbb{C}^k)\). The following inequality \[ \| \chi_xf(H)\chi_y\| \leq C\exp(-c| x-y| ^\alpha), \;\;\;x,y\in\mathbb{R}^d, \] is proved for \(C^\infty\) functions \(f\) belonging to a \(L^1\)-Gervey class. Here \(\alpha\in(0,1]\) is depending on \(f\), and \(\chi_x(q)=\chi_0(q-x)\), \(x,q\in\mathbb{R}^d\), where \(\chi_0\) is the characteristic function of the cube centered at 0 with side 1.

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35P05 General topics in linear spectral theory for PDEs
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[1] René Carmona, Abel Klein, and Fabio Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41 – 66. · Zbl 0615.60098
[2] J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251 – 270. · Zbl 0271.35062
[3] David Damanik, Robert Sims, and Günter Stolz, Localization for one-dimensional, continuum, Bernoulli-Anderson models, Duke Math. J. 114 (2002), no. 1, 59 – 100. · Zbl 1107.82025 · doi:10.1215/S0012-7094-02-11414-8 · doi.org
[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. · Zbl 0652.35024 · doi:10.1093/qmath/39.1.37 · doi.org
[5] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. · Zbl 0699.35006
[6] Stephan De Bièvre and François Germinet, Dynamical localization for the random dimer Schrödinger operator, J. Statist. Phys. 98 (2000), no. 5-6, 1135 – 1148. · Zbl 1005.82028 · doi:10.1023/A:1018615728507 · doi.org
[7] 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. · Zbl 0926.35002
[8] P. Elbau and G. M. Graf, Equality of bulk and edge Hall conductance revisited, Comm. Math. Phys. 229 (2002), no. 3, 415 – 432. · Zbl 1001.81091 · doi:10.1007/s00220-002-0698-z · doi.org
[9] F. Germinet, A. Kiselev, and S. Tcheremchantsev: Transfer matrices and transport for 1D Schrödinger operators with singular spectrum, Ann. Inst. Fourier, to appear.
[10] François Germinet and Abel Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415 – 448. · Zbl 0982.82030 · doi:10.1007/s002200100518 · doi.org
[11] François Germinet and Abel Klein, Operator kernel estimates for functions of generalized Schrödinger operators, Proc. Amer. Math. Soc. 131 (2003), no. 3, 911 – 920. · Zbl 1013.81009
[12] F. Germinet and A. Klein: A characterization of the metal-insulator transport transition, submitted. · Zbl 1062.82020
[13] F. Germinet and A. Klein: The Anderson metal-insulator transport transition, to appear in Contemp. Math. · Zbl 1130.82312
[14] 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). · Zbl 0699.35189 · doi:10.1007/3-540-51783-9_19 · doi.org
[15] 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. Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Springer Study Edition, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. · Zbl 0712.35001
[16] S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz: Delocalization in random polymer models, Commun. Math. Phys. 233, 27-48 (2003). · Zbl 1013.82027
[17] 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. · Zbl 0780.60063 · doi:10.1016/0022-1236(90)90031-F · doi.org
[18] Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447 – 526. , https://doi.org/10.1090/S0273-0979-1982-15041-8 Barry Simon, Erratum: ”Schrödinger semigroups”, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 426. · doi:10.1090/S0273-0979-1984-15288-1 · doi.org
[19] S. Tcheremchantsev: Dynamical analysis of Schrödinger operators with sparse potential, Comm. Math. Phys., to appear. · Zbl 1100.47027
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