×

Anderson localization for time periodic Schrödinger operators. (English) Zbl 1028.81018

Time-independent random Schrödinger operator is an operator of the form \[ H_0=\Delta+rV, \text{ on } l^2({\mathbb Z}^d), \] where \(r\) is a positive parameter, the potential \(V\) is a diagonal matrix, \(V=\text{diag}(v_j), j\in {\mathbb Z}^d\), where \(\{v_j\}\) is a family of independently indentically distributed random variables with distribution \(g\).
It is known that under certain regularity conditions on \(g\), for \(r\gg 1\), and in any dimension \(d\), the spectrum of \(H_0\) is almost surely pure point with exponentially localized eigenfunctions. This is called Anderson localization, after the physicist P. W. Anderson [Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492–1505 (1958)].
In this paper, by proving that the associated quasi-energy operator has pure point spectrum, the authors show that at large disorder, Anderson localization in \({\mathbb Z}^d\) is stable under bounded localized time periodic perturbations. It is a type of quantum stability results. For other results of related interests, see e.g. J. Béllissard [Lect. Notes Phys. 257, 99–156 (1986; Zbl 0612.46022)], M. Combescure [Ann. Inst. Henri Poincaré 47, 63–83, Errata: 451–454 (1987; Zbl 0635.70018)] and P. Sarnak, [Commun. Math. Phys. 84, 377–401 (1982; Zbl 0506.35071)]. The formulation of the problem is motivated by questions of Anderson localization for nonlinear Schrödinger equations.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
60H25 Random operators and equations (aspects of stochastic analysis)
47B80 Random linear operators
47N55 Applications of operator theory in statistical physics (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aizenman M, Phys 279 pp 369– (2000)
[2] Aizenman M, Commun Math Phys pp 157– (1993)
[3] Anderson P., Phys Rev 109 pp 1492– (1958)
[4] Bellissard J., Stochastic Process in Classical and Quantum Systems (1986) · Zbl 0612.46066
[5] Combescure M., Ann Inst Henri Poincare 47 pp 63– (1987)
[6] Cycon HL, Schrödinger Operators (1987)
[7] Davies EB., Spectral theory and differential operators (1995)
[8] Devillard BJ., J Stat Phys 43 pp 423– (1986) · Zbl 0628.60086
[9] von Dreifus H, Commun Math Phys 124 pp 285– (1989) · Zbl 0698.60051
[10] Fröhlich J, Commun Math Phys 101 pp 21– (1985) · Zbl 0573.60096
[11] Fröhlich J, Commun Math Phys 88 pp 151– (1983) · Zbl 0519.60066
[12] Fröhlich J, J Stat Phys 42 pp 247– (1986) · Zbl 0629.60105
[13] Gol’dsheid Ya, Func Anal Appl 11 pp 1– (1977) · Zbl 0368.34015
[14] Helffer B, Lecture Notes in Physics pp 345– (1989)
[15] Howland JS., Indiana Univ Math J 28 pp 471– (1979) · Zbl 0444.47010
[16] Howland JS., Lect Notes Phys 43 (1992)
[17] Jauslin HR, Chaos 1 pp 114– (1991) · Zbl 0899.58059
[18] Pastur L, Spectra of Random and Almost Periodic Operators (1992)
[19] Reed M, Methods of Modern Mathematical Physics I: Functional Analysis (1980) · Zbl 0459.46001
[20] Sarnak P., Commun Math Phys 84 pp 377– (1982) · Zbl 0506.35074
[21] Simon B., Bull Am Math Soc 7 pp 447– (1982) · Zbl 0524.35002
[22] Thomas L, J Math Phys 27 pp 71– (1986) · Zbl 0608.47018
[23] Wang WM., J of Func Anal 146 pp 1– (1997) · Zbl 0872.35137
[24] Yajima K., Commun Math Phys 87 pp 331– (1982) · Zbl 0538.47010
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.