×

Localization for random Schrödinger operators with Poisson potential. (English) Zbl 0843.60058

Let \(\mu = (\mu_\omega)\) be a Poisson random measure with intensity \(\alpha > 0\) defined on a probability space \((\Omega, {\mathcal F}, {\mathcal P})\). Then there exists a sequence of random variables \(\{X_i(\omega)\}\) such that \(Y_{\pm 1} = X_{\pm 1}\), \(Y_n = X_n - X_{n - 1}\) and \(Y_{-n} = X_{-(n-1)} - X_{-n}\) are independent and identically distributed random variables with exponential distribution of parameter \(\alpha\) and \(\mu_\omega(B) = \#\{i : X_i(\omega) \in B\}\). Taking a single site potential \(f \in L^2(R)\) of non-negative and compact support function, define the Poisson potential \(V_\omega\) by \[ V_\omega(x) = \int f(x - y)d\mu_\omega(y) = \sum_i f(x - X_i(\omega)). \] Then \(H_\omega = -{d^2\over dx^2} + V_\omega\) is almost surely self-adjoint and metrically transitive. Hence there exist subsets \(\Sigma_{ac}\), \(\Sigma_{sc}\) and \(\Sigma_{pp}\) of \(R\) such that \(\Sigma_{ac} (H_\omega) = \Sigma_{ac}\), \(\Sigma_{sc} (H_\omega) = \Sigma_{sc}\) and \(\Sigma_{pp} (H_\omega) = \Sigma_{pp}\). The main result of this paper is to show that \(\Sigma_{ac} = \Sigma_{sc} = \emptyset\) and the eigenfunctions decay exponentially at the rate of Lyapunov exponent. The proof has been done along the Kotani’s method but, in the present case, the potential is not bounded. To overcome this, it is used the result that the operator \(H_a = -{d^2 \over dx^2} + V_a\), \(V_a = W_1(x - a) + W_2(x + a)\) satisfies the spectral averaging at positive energies, where \(W_1, W_2 \in L^1_{\text{loc}}(R)\), \(W_1 = 0\) in \((-\infty, 0)\) [resp. \(W_2 = 0\) in \((0,\infty)]\) and \(-{d^2\over dx^2} + W_1\) [resp. \(-{d^2\over dx^2} + W_2\)] is of limit point type at \(+\infty\) [resp. \(-\infty\)].

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] R. Carmona , One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types , J. Funct. Anal. , Vol. 51 , 1983 , pp. 229 - 258 . MR 701057 | Zbl 0516.60069 · Zbl 0516.60069
[2] R. Carmonara and J. Lacroix , Spectral theory of random Schrödinger operators , Birkhäuser , Basel - Berlin , 1990 . MR 1102675 | Zbl 0717.60074 · Zbl 0717.60074
[3] E.A. Coddington and N. Levinson , Theory of ordinary differential equations , McGraw-Hill , New York , 1955 . MR 69338 | Zbl 0064.33002 · Zbl 0064.33002
[4] J.-M. Combes and P.D. Hislop , Localization for continuous random Hamiltonians in d-dimensions , J. Funct. Anal. , Vol. 124 , 1994 , pp. 149 - 180 . MR 1284608 | Zbl 0801.60054 · Zbl 0801.60054
[5] R. Del Rio , N. Makarov and B. Simon , Operators with singular continuous spectrum, II. Rank one operators , Commun. Math. Phys. , Vol. 165 , 1994 , pp. 59 - 67 . Article | MR 1298942 | Zbl 1055.47500 · Zbl 1055.47500
[6] R. Del Rio , B. Simon and G. Stolz , Stability of spectral types for Sturm-Liouville operators , Math. Research Lett. , Vol. 1 , 1994 , pp. 437 - 450 . MR 1302387 | Zbl 0838.34090 · Zbl 0838.34090
[7] J.L. Doob , Stochastic Processes , Wiley , New York , 1953 . MR 58896 | Zbl 0053.26802 · Zbl 0053.26802
[8] A. Gordon , Pure point spectrum under 1-parameter perturbations and instability of Anderson localization , Commun. Math. Phys. , Vol. 164 , 1994 , pp. 489 - 506 . Article | MR 1291242 | Zbl 0839.47002 · Zbl 0839.47002
[9] Ph . Hartman , The number of L2-solutions of x” + q (t) x = 0 , Amer. J. Math. , Vol. 73 , 1951 , pp. 635 - 645 . MR 44695 | Zbl 0044.31202 · Zbl 0044.31202
[10] I.W. Herbst , J.S. Howland , The Stark ladder and other one-dimensional extemal field problems , Commun. Math. Phys. , Vol. 80 , 1981 , pp. 23 - 42 . Article | MR 623150 | Zbl 0473.47037 · Zbl 0473.47037
[11] P.D. Hilsop , S. Nkamura , Stark Hamiltonians with unbounded random potentials , Rev. Math. Phys. , Vol. 2 , 1990 , pp. 479 - 494 . MR 1107687 | Zbl 0727.34076 · Zbl 0727.34076
[12] O. Kallenberg , Random Measures , Akademie-Verlag , Berlin , 1975 . MR 431372 | Zbl 0345.60031 · Zbl 0345.60031
[13] J.F.C. Kingman , Poisson Processes , Clarendon Press , Oxford , 1993 . MR 1207584 | Zbl 0771.60001 · Zbl 0771.60001
[14] W. Kirsch , S. Kotani and B. Simon , Absence of absolutely continuous spectrum for some one dimensional random but deterministic potentials , Ann. Inst. Henri Poincaré , Vol. 42 , 1985 , pp. 383 - 406 . Numdam | MR 801236 | Zbl 0581.60052 · Zbl 0581.60052
[15] W. Kirsch , S.A. Molchanov and L.A. Pastur , One-dimensional Schrödinger operator with unbounded potential: The pure point spectrum , Funct. Anal. Appl. , Vol. 24 , 1990 , pp. 176 - 186 . MR 1082027 | Zbl 0747.47023 · Zbl 0747.47023
[16] W. Kirsch , S.A. Molchanov and L.A. Pastur , One-dimensional Schrödinger operator with high potential barriers , Operator Theory: Advances and Applications , Vol. 57 , pp. 163 - 170 , Birkhäuser-Verlag , 1992 . MR 1230898 | Zbl 0883.34078 · Zbl 0883.34078
[17] F. Klopp , Localization for Semiclassical Continuous Random Schrödinger Operators II: the Random Displacement Model , Helv. Phys. Acta , Vol. 66 , 1993 , pp. 810 - 841 . MR 1264047 | Zbl 0820.60043 · Zbl 0820.60043
[18] S. Kotani , Lyapunov indices determine absolute contnuous spectra of stationnary one dimensional Schrödinger operators , Proc. Taneguchi Intern. Symp. on Stochastic Analysis, Katata and Kyoto , 1982 , pp. 225 - 247 , North Holland , 1983 . MR 780760 | Zbl 0549.60058 · Zbl 0549.60058
[19] S. Kotani , Lyapunov exponents and spectra for one-dimensional Schrödinger operators , Contemp. Math. , Vol. 50 , 1986 , pp. 277 - 286 . Zbl 0587.60054 · Zbl 0587.60054
[20] S. Kotani and B. Simon , Localization in General One-Dimensional Random Systems , Commun. Math. Phys. , Vol. 112 , 1987 , pp. 103 - 119 . Article | MR 904140 | Zbl 0637.60080 · Zbl 0637.60080
[21] N. Minami , Exponential and Super-Exponential Localizations for One-Dimensional Schrödinger Operators with Lévy Noise Potentials , Tsukuba J. Math. , Vol. 13 , 1989 , pp. 225 - 282 . MR 1003604 | Zbl 0694.60058 · Zbl 0694.60058
[22] L. Pastur and A. Figotin , Spectra of Random and Almost-Periodic Operators , Springer-Verlag , 1991 . MR 1223779 | Zbl 0752.47002 · Zbl 0752.47002
[23] Th Poerschke , G. Stolz and J. Weidmann , Expansions in Generalized Eigenfunctions of Selfadjoint Operators , Math. Z. , Vol. 202 , 1989 , pp. 397 - 408 . MR 1017580 | Zbl 0661.47021 · Zbl 0661.47021
[24] B. Simon , Trace ideals and their Applications , Cambridge University Press , Cambridge , 1979 . MR 541149 | Zbl 0423.47001 · Zbl 0423.47001
[25] B. Simon , Schrödinger semigroups , Bull. Am. Math. Soc. , Vol. 7 , 1982 , pp. 447 - 526 . Article | MR 670130 | Zbl 0524.35002 · Zbl 0524.35002
[26] B. Simon , Localization in General One Dimensional Random Systems, I. Jacobi Matrices , Commun. Math. Phys. , Vol. 102 , 1985 , pp. 327 - 336 . Article | MR 820578 | Zbl 0604.60062 · Zbl 0604.60062
[27] B. Simon and T. Wolff , Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians , Commun. Pure Appl. Math. , Vol. 39 , 1986 , pp. 75 - 90 . MR 820340 | Zbl 0609.47001 · Zbl 0609.47001
[28] G. Stolz , Note to the paper by P.D. Hilsop and S. Nakamura : Stark Hamiltonian with unbounded random potentials , Rev. Math. Phys. , Vol. 5 , 1993 , pp. 453 - 456 . MR 1223529 | Zbl 0782.34088 · Zbl 0782.34088
[29] G. Stolz , Spectral theory of Schrödinger operators with potentials of infinite barriers type , Habilitationsschrift , Frankfurt , 1994 . · Zbl 0819.34049
[30] G. Stolz , Localization for the Poisson model , in ”Spectral Analysis and Partial Differential Equations”, Operator Theory: Advances and Applications, Vol. 78 , pp. 375 - 380 , Birkhäuser-Verlag , 1955 . MR 1365351 | Zbl 0841.34089 · Zbl 0841.34089
[31] J. Weidmann , Spectral Theory of Ordinary Differential Operators , Lect. Notes in Math. , Vol. 1258 , Springer/Verlag , 1987 . MR 923320 | Zbl 0647.47052 · Zbl 0647.47052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.