Stochastic analysis and random Schrödinger operators

Ueki, Naomasa

doi:10.1090/suga/430

Stochastic analysis and random Schrödinger operators
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by Naomasa Ueki
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Sugaku Expositions 31 (2018), 93-115

DOI: https://doi.org/10.1090/suga/430

Published electronically: March 20, 2018

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References

Michael Aizenman, Alexander Elgart, Serguei Naboko, Jeffrey H. Schenker, and Gunter Stolz, Moment analysis for localization in random Schrödinger operators, Invent. Math. 163 (2006), no. 2, 343–413. MR 2207021, DOI 10.1007/s00222-005-0463-y
Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867
P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), 1492–1505.
Jean Bourgain and Carlos E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389–426. MR 2180453, DOI 10.1007/s00222-004-0435-7
Kurt Broderix, Dirk Hundertmark, Werner Kirsch, and Hajo Leschke, The fate of Lifshits tails in magnetic fields, J. Statist. Phys. 80 (1995), no. 1-2, 1–22. MR 1340551, DOI 10.1007/BF02178350
René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
Jean-Michel Combes, François Germinet, and Abel Klein, Generalized eigenvalue-counting estimates for the Anderson model, J. Stat. Phys. 135 (2009), no. 2, 201–216. MR 2505733, DOI 10.1007/s10955-009-9731-3
Jean-Michel Combes, François Germinet, and Abel Klein, Poisson statistics for eigenvalues of continuum random Schrödinger operators, Anal. PDE 3 (2010), no. 1, 49–80. MR 2663411, DOI 10.2140/apde.2010.3.49
R. del Rio, S. Jitomirskaya, Y. Last and B. Simon, What is localization? Phys. Rev. Lett., 75 (1995), 117–119.
R. del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one pertubations, and localization, J. Anal. Math., 69 (1996), 153–200.
Giulia Di Nunno, Bernt Øksendal, and Frank Proske, Malliavin calculus for Lévy processes with applications to finance, Universitext, Springer-Verlag, Berlin, 2009. MR 2460554, DOI 10.1007/978-3-540-78572-9
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. I, II, III, IV, Communications on Pure and Applied Mathematics, 28 (1975), 1–47, 28 (1975), 279–301, 29 (1976), 389–461, 36 (1983), 183–212.
M. D. Donsker and S. R. S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), no. 4, 525–565. MR 397901, DOI 10.1002/cpa.3160280406
László Erdős, Rayleigh-type isoperimetric inequality with a homogeneous magnetic field, Calc. Var. Partial Differential Equations 4 (1996), no. 3, 283–292. MR 1386738, DOI 10.1007/BF01254348
László Erdős, Lifschitz tail in a magnetic field: the nonclassical regime, Probab. Theory Related Fields 112 (1998), no. 3, 321–371. MR 1660914, DOI 10.1007/s004400050193
László Erdős, Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case, Probab. Theory Related Fields 121 (2001), no. 2, 219–236. MR 1865486, DOI 10.1007/PL00008803
László Erdős and David Hasler, Wegner estimate and Anderson localization for random magnetic fields, Comm. Math. Phys. 309 (2012), no. 2, 507–542. MR 2864802, DOI 10.1007/s00220-011-1373-z
László Erdős and David Hasler, Wegner estimate for random magnetic Laplacian on $\Bbb Z^2$, Ann. Henri Poincaré 13 (2012), no. 8, 1719–1731. MR 2994757, DOI 10.1007/s00023-012-0177-9
László Erdős and David Hasler, Anderson localization at band edges for random magnetic fields, J. Stat. Phys. 146 (2012), no. 5, 900–923. MR 2902447, DOI 10.1007/s10955-012-0445-6
Jürg Fröhlich and Thomas Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151–184. MR 696803
Ryoki Fukushima, Brownian survival and Lifshitz tail in perturbed lattice disorder, J. Funct. Anal. 256 (2009), no. 9, 2867–2893. MR 2502426, DOI 10.1016/j.jfa.2009.01.030
Ryoki Fukushima and Naomasa Ueki, Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice, Ann. Henri Poincaré 11 (2010), no. 6, 1053–1083. MR 2737491, DOI 10.1007/s00023-010-0051-6
Ryoki Fukushima and Naomasa Ueki, Moment asymptotics for the parabolic Anderson problem with a perturbed lattice potential, J. Funct. Anal. 260 (2011), no. 3, 724–744. MR 2737395, DOI 10.1016/j.jfa.2010.10.016
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, A comprehensive proof of localization for continuous Anderson models with singular random potentials, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 53–143. MR 2998830, DOI 10.4171/JEMS/356
François Germinet and Frédéric Klopp, Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges, Ann. Henri Poincaré 14 (2013), no. 5, 1263–1285. MR 3070753, DOI 10.1007/s00023-012-0217-5
F. Ghribi, P. D. Hislop, and F. Klopp, Localization for Schrödinger operators with random vector potentials, Adventures in mathematical physics, Contemp. Math., vol. 447, Amer. Math. Soc., Providence, RI, 2007, pp. 123–138. MR 2423576, DOI 10.1090/conm/447/08687
Peter D. Hislop and Frédéric Klopp, The integrated density of states for some random operators with nonsign definite potentials, J. Funct. Anal. 195 (2002), no. 1, 12–47. MR 1934351, DOI 10.1006/jfan.2002.3947
D. Hundertmark, W. Kirsch, and S. Warzel, Classical magnetic Lifshits tails in three space dimensions: impurity potentials with slow anisotropic decay, Markov Process. Related Fields 9 (2003), no. 4, 651–660. MR 2072247
Thomas Hupfer, Hajo Leschke, and Simone Warzel, Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshits tailing in magnetic fields, J. Statist. Phys. 97 (1999), no. 3-4, 725–750. MR 1736101, DOI 10.1023/A:1004619409967
Nobuyuki Ikeda, Van Vleck’s formula and Jacobi fields in connection with Wiener integrals, Sūgaku 52 (2000), no. 3, 225–244 (Japanese). MR 1784771
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain J. Math. 4 (1974), 511–537. Notes by Jack Macki and Reuben Hersh. MR 510113, DOI 10.1216/RMJ-1974-4-3-511
Werner Kirsch and Fabio Martinelli, On the density of states of Schrödinger operators with a random potential, J. Phys. A 15 (1982), no. 7, 2139–2156. MR 665829
Werner Kirsch and Fabio Martinelli, On the spectrum of Schrödinger operators with a random potential, Comm. Math. Phys. 85 (1982), no. 3, 329–350. MR 678150
Werner Kirsch and Bernd Metzger, The integrated density of states for random Schrödinger operators, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 649–696. MR 2307751, DOI 10.1090/pspum/076.2/2307751
Werner Kirsch and Simone Warzel, Lifshits tails caused by anisotropic decay: the emergence of a quantum-classical regime, Math. Phys. Anal. Geom. 8 (2005), no. 3, 257–285. MR 2177468, DOI 10.1007/s11040-005-0582-0
Frédéric Klopp, Localization for some continuous random Schrödinger operators, Comm. Math. Phys. 167 (1995), no. 3, 553–569 (English, with English and French summaries). MR 1316760
Frédéric Klopp, Shu Nakamura, Fumihiko Nakano, and Yuji Nomura, Anderson localization for 2D discrete Schrödinger operators with random magnetic fields, Ann. Henri Poincaré 4 (2003), no. 4, 795–811. MR 2015430, DOI 10.1007/s00023-003-0147-3
Shinichi Kotani, Problems on random potentials. II, Sūgaku 38 (1986), no. 3, 193–201 (Japanese). MR 876441
H. Leschke, P. Müller, and S. Warzel, A survey of rigorous results on random Schrödinger operators for amorphous solids, Markov Process. Related Fields 9 (2003), no. 4, 729–760. MR 2072253
I. M. Lifshitz, Energy spectrum structure and quantum states of disordered condensed systems, Soviet Physics Uspekhi 7 (1965), 549–573. MR 0181368, DOI 10.1070/pu1965v007n04abeh003634
Paul Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) Wiley, New York-Chichester-Brisbane, 1978, pp. 195–263. MR 536013
Nariyuki Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), no. 3, 709–725. MR 1385082
Shu Nakamura, A remark on the Dirichlet-Neumann decoupling and the integrated density of states, J. Funct. Anal. 179 (2001), no. 1, 136–152. MR 1807255, DOI 10.1006/jfan.2000.3683
Shintaro Nakao, On the spectral distribution of the Schrödinger operator with random potential, Japan. J. Math. (N.S.) 3 (1977), no. 1, 111–139. MR 651925, DOI 10.4099/math1924.3.111
David Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1344217, DOI 10.1007/978-1-4757-2437-0
Hiroyuki Ôkura, An asymptotic property of a certain Brownian motion expectation for large time, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no. 3, 155–159. MR 618081
Robert Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1–29. MR 519520, DOI 10.2307/2320297
Shin Ozawa, Spectra of random media, Sūgaku 44 (1992), no. 4, 306–319 (Japanese). MR 1197556
L. A. Pastur, The behavior of certain Wiener integrals as $t\rightarrow \infty$ and the density of states of Schrödinger equations with random potential, Teoret. Mat. Fiz. 32 (1977), no. 1, 88–95 (Russian, with English summary). MR 449356
Leonid Pastur and Alexander Figotin, Spectra of random and almost-periodic operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779, DOI 10.1007/978-3-642-74346-7
Jeffrey Rauch, The mathematical theory of crushed ice, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 370–379. MR 0427863
Jeffrey Rauch and Michael Taylor, Potential and scattering theory on wildly perturbed domains, J. Functional Analysis 18 (1975), 27–59. MR 377303, DOI 10.1016/0022-1236(75)90028-2
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Barry Simon, Functional integration and quantum physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. MR 2105995, DOI 10.1090/chel/351
Ichiro Shigekawa, Stochastic analysis, Translations of Mathematical Monographs, vol. 224, American Mathematical Society, Providence, RI, 2004. Translated from the 1998 Japanese original by the author; Iwanami Series in Modern Mathematics. MR 2060917, DOI 10.1090/mmono/224
Peter Stollmann, Caught by disorder, Progress in Mathematical Physics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 2001. Bound states in random media. MR 1935594, DOI 10.1007/978-1-4612-0169-4
Alain-Sol Sznitman, Lifschitz tail and Wiener sausage. I, II, J. Funct. Anal. 94 (1990), no. 2, 223–246, 247–272. MR 1081644, DOI 10.1016/0022-1236(90)90012-A
Alain-Sol Sznitman, Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1717054, DOI 10.1007/978-3-662-11281-6
Naomasa Ueki, Simple examples of Lifschitz tails in Gaussian random magnetic fields, Ann. Henri Poincaré 1 (2000), no. 3, 473–498 (English, with English and French summaries). MR 1777309, DOI 10.1007/s000230050004
Naomasa Ueki, Wegner estimate and localization for random magnetic fields, Osaka J. Math. 45 (2008), no. 3, 565–608. MR 2468583
N. Ueki, Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed lattice, Markov Process. Related Fields 17 (2011), no. 3, 347–368. MR 2895556
Naomasa Ueki, Quantum behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed lattice, Spectra of random operators and related topics, RIMS Kôkyûroku Bessatsu, B27, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 149–169. MR 2885261
Naomasa Ueki, Wegner estimate for Gaussian random magnetic fields, J. Funct. Anal. 263 (2012), no. 7, 2024–2050. MR 2956933, DOI 10.1016/j.jfa.2012.06.021
S. Warzel, Lifshits tails in magnetic fields, PhD Thesis, Universität Erlangen-Nürnberg, (2001).
Franz Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), no. 1-2, 9–15. MR 639135, DOI 10.1007/BF01292646