Abstract
We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.
Acknowledgements
The author gratefully acknowledges stimulating discussions with Ivan Veselić and Christoph Schumacher.
References
[1] M. Aizenman, Localization at weak disorder: Some elementary bounds, Rev. Math. Phys. 6 (1994), no. 5A, 1163–1182. 10.1142/S0129055X94000419Search in Google Scholar
[2] M. Aizenman and G. M. Graf, Localization bounds for an electron gas, J. Phys. A 31 (1998), no. 32, 6783–6806. 10.1088/0305-4470/31/32/004Search in Google Scholar
[3] M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. 10.1007/BF02099760Search in Google Scholar
[4] M. Aizenman, J. H. Schenker, R. M. Friedrich and D. Hundertmark, Finite-volume fractional-moment criteria for Anderson localization, Comm. Math. Phys. 224 (2001), no. 1, 219–253. 10.1007/s002200100441Search in Google Scholar
[5] 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. 10.1006/jfan.1994.1103Search in Google Scholar
[6] A. Elgart, M. Shamis and S. Sodin, Localisation for non-monotone Schrödinger operators, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 909–924. 10.4171/JEMS/451Search in Google Scholar
[7]
A. Elgart, M. Tautenhahn and I. Veselić,
Localization via fractional moments for models on
[8] A. Elgart, M. Tautenhahn and I. Veselić, Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method, Ann. Henri Poincaré 12 (2011), no. 8, 1571–1599. 10.1007/s00023-011-0112-5Search in Google Scholar
[9] W. Fischer, T. Hupfer, H. Leschke and P. Müller, Existence of the density of states for multi-dimensional continuum Schrödinger operators with Gaussian random potentials, Comm. Math. Phys. 190 (1997), no. 1, 133–141. 10.1007/s002200050236Search in Google Scholar
[10] J. Fröhlich, F. Martinelli, E. Scoppola and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), no. 1, 21–46. 10.1007/BF01212355Search in Google Scholar
[11] J. Fröhlich and T. 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. 10.1007/BF01209475Search in Google Scholar
[12] F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415–448. 10.1007/s002200100518Search in Google Scholar
[13] F. Germinet and A. Klein, Explicit finite volume criteria for localization in continuous random media and applications, Geom. Funct. Anal. 13 (2003), no. 6, 1201–1238. 10.1007/s00039-003-0444-5Search in Google Scholar
[14] F. Germinet and A. Klein, New characterizations of the region of complete localization for random Schrödinger operators, J. Stat. Phys. 122 (2006), no. 1, 73–94. 10.1007/s10955-005-8068-9Search in Google Scholar
[15] G. M. Graf, Anderson localization and the space-time characteristic of continuum states, J. Statist. Phys. 75 (1994), no. 1–2, 337–346. 10.1007/BF02186292Search in Google Scholar
[16] P. D. Hislop and F. Klopp, The integrated density of states for some random operators with nonsign definite potentials, J. Funct. Anal. 195 (2002), no. 1, 12–47. 10.1006/jfan.2002.3947Search in Google Scholar
[17] D. Hundertmark, On the time-dependent approach to Anderson localization, Math. Nachr. 214 (2000), 25–38. 10.1002/1522-2616(200006)214:1<25::AID-MANA25>3.0.CO;2-ZSearch in Google Scholar
[18] D. Hundertmark, A short introduction to Anderson localization, Analysis and Stochastics of Growth Processes and Interface Models, Oxford University Press, Oxford (2008), 194–218. 10.1093/acprof:oso/9780199239252.003.0009Search in Google Scholar
[19] W. Kirsch, An invitation to random Schrödinger operators, Random Schrödinger Operators, Panor. Synthèses 25, Société Mathématique de France, Paris (2008), 1–119. Search in Google Scholar
[20] W. Kirsch, P. Stollmann and G. Stolz, Anderson localization for random Schrödinger operators with long range interactions, Comm. Math. Phys. 195 (1998), no. 3, 495–507. 10.1007/s002200050399Search in Google Scholar
[21] F. Klopp, Localization for some continuous random Schrödinger operators, Comm. Math. Phys. 167 (1995), no. 3, 553–569. 10.1007/BF02101535Search in Google Scholar
[22] V. Kostrykin and I. Veselić, On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials, Math. Z. 252 (2006), no. 2, 367–392. 10.1007/s00209-005-0860-3Search in Google Scholar
[23] H. Krüger, Localization for random operators with non-monotone potentials with exponentially decaying correlations, Ann. Henri Poincaré 13 (2012), no. 3, 543–598. 10.1007/s00023-011-0130-3Search in Google Scholar
[24] K. Leonhardt, N. Peyerimhoff, M. Tautenhahn and I. Veselić, Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials, Rev. Math. Phys. 27 (2015), no. 4, Article ID 1550007. 10.1142/S0129055X15500075Search in Google Scholar
[25] S. C. Port, Theoretical Probability for Applications, Wiley Ser. in Probab. Math. Statist., John Wiley & Sons, New York, 1994. Search in Google Scholar
[26] N. Ueki, Wegner estimates and localization for Gaussian random potentials, Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, 29–90. 10.2977/prims/1145475966Search in Google Scholar
[27] I. Veselić, Wegner estimate and the density of states of some indefinite alloy-type Schrödinger operators, Lett. Math. Phys. 59 (2002), no. 3, 199–214. 10.1023/A:1015580402816Search in Google Scholar
[28] I. Veselić, Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators, Lecture Notes in Math. 1917, Springer, Berlin, 2008. Search in Google Scholar
[29] I. Veselić, Wegner estimate for discrete alloy-type models, Ann. Henri Poincaré 11 (2010), no. 5, 991–1005. 10.1007/s00023-010-0052-5Search in Google Scholar
[30] I. Veselić, Wegner estimates for sign-changing single site potentials, Math. Phys. Anal. Geom. 13 (2010), no. 4, 299–313. 10.1007/s11040-010-9081-zSearch in Google Scholar
[31] I. Veselić, Lipschitz-continuity of the integrated density of states for Gaussian random potentials, Lett. Math. Phys. 97 (2011), no. 1, 25–27. 10.1007/s11005-011-0465-1Search in Google Scholar
[32] H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), no. 2, 285–299. 10.1007/BF01219198Search in Google Scholar
[33] H. von Dreifus and A. Klein, Localization for random Schrödinger operators with correlated potentials, Comm. Math. Phys. 140 (1991), no. 1, 133–147. 10.1007/BF02099294Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Wegner estimate for discrete Schrödinger operators with Gaussian random potentials
- A general maximum principle for mean-field forward-backward doubly stochastic differential equations with jumps processes
- BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient
- Fast decay of eigenfunction correlators in long-range continuous random alloys
- Inverting weak random operators
- The limit G-Law for the solutions of systems of linear algebraic equations with independent random coefficients under the G-Lindeberg condition
Articles in the same Issue
- Frontmatter
- Wegner estimate for discrete Schrödinger operators with Gaussian random potentials
- A general maximum principle for mean-field forward-backward doubly stochastic differential equations with jumps processes
- BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient
- Fast decay of eigenfunction correlators in long-range continuous random alloys
- Inverting weak random operators
- The limit G-Law for the solutions of systems of linear algebraic equations with independent random coefficients under the G-Lindeberg condition