Abstract
A rigorous proof is outlined to exclude the absolutely continuous spectrum at sufficiently low energies for a quantum-mechanical particle moving in multi-dimensional Euclidean space under the influence of certain Gaussian random potentials, which are homogeneous with respect to Euclidean translations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, P. W.: Absence of diffusion in certain random lattices,Phys. Rev. 109 (1958), 1492.
Shklovskii, B. I. and Efros, A. L.,Electronic Properties of Doped Semiconductors, Springer, Berlin, 1984.
Lifshits, I. M., Gredeskul, S. A. and Pastur, L. A.:Introduction to the Theory of Disordered Systems, Wiley, New York, 1988.
Carmona, R. and Lacroix, J.:Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
Pastur, L. and Figotin, A.:Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992.
Martinelli, F. and Holden, H.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator onL 2 (ℝv),Comm. Math. Phys. 93 (1984), 197.
Combes, J.-M. and Hislop, P. D.: Localisation for some continuous, random Hamiltonians ind-dimensions,J. Funct. Anal. 124 (1994), 149.
Klopp, F.: Localisation for some continuous random Schrödinger operators,Comm. Math. Phys. 167 (1995), 553.
Kirsch, W.: Wegner estimates and Anderson localisation for alloy-type potentials,Math. Z. 221 (1996), 507.
Martinelli, F. and Scoppola, E.: Introduction to the mathematical theory of Anderson localisation,Riv. Nuovo Cimento 10(10) (1987), 1.
Fischer, W., Leschke, H. and Müller, P.: in preparation, to be submitted toJ. Stat. Phys.
Doukhan, P.:Mixing, Properties and Examples, Springer, New York, 1994.
Fernique, X. M.: Regularité des trajectoires des fonctions aléatoires Gaussiennes, in: P.-L. Hennequin (ed.),Ecole d'Eté de Probabilités de Saint-Flour IV-1974, Lecture Notes in Mathematics 480, Springer, Berlin, 1975, p. 1.
Ibragimov, I. A. and Rozanov, Y. A.:Gaussian Random Processes, Springer, New York, 1978.
Kirsch, W.: Random Schrödinger operators: a course, in: H. Holden and A. Jensen (eds),Schrödinger Operators, Lecture Notes in Physics 345, Springer, Berlin, 1989, p. 264.
Fröhlich, J. and Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Comm. Math. Phys. 88 (1983), 151.
von Dreifus, H. and Klein, A.: Localisation for random Schrödinger operators with correlated potentials,Comm. Math. Phys. 140 (1991), 133.
Wegner, F.: Bounds on the density of states in disordered systems,Z. Phys. B 44 (1981), 9.
Glimm, J. and Jaffe, A.:Quantum Physics: A Functional Integral Point of View, 2nd edn, Springer, New York, 1987.
Gihman, I. I. and Skorokhod, A. V.:The Theory of Stochastic Processes I, Springer, Berlin, 1974.
Combes, J.-M. and Hislop, P. D.: Landau Hamiltonians with random potentials: Localisation and the density of states,Comm. Math. Phys. 177 (1996), 603.
Dorlas, T. C., Macris, N. and Pulé, J. V.: Localisation in a single-band approximation to random Schrödinger operators in a magnetic field,Helv. Phys. Acta 68 (1995), 329.
Dorlas, T. C., Macris, N. and Pulé, J. V.: Localisation in single Landau bands,J. Math. Phys. 37 (1996), 1574.
Wang, W.-M.: Microlocalisation, percolation and Anderson localisation for the magnetic Schrödinger operatorswith a random potential, to appear inJ. Funct. Anal.
Matsumoto, H.: On the integrated density of states for the Schrödinger operators with certain random electromagnetic potentials,J. Math. Soc. Japan 45 (1993), 197.
Ueki, N.: On spectra of random Schrödinger operators with magnetic fields,Osaka J. Math. 31 (1994), 177.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fischer, W., Leschke, H. & Müller, P. Towards localisation by Gaussian random potentials in multi-dimensional continuous space. Lett Math Phys 38, 343–348 (1996). https://doi.org/10.1007/BF01815517
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01815517