Letters in Mathematical Physics

, Volume 72, Issue 1, pp 51–64

# Fractional Moment Estimates for Random Unitary Operators

• Alain Joye
Article

## Abstract

We consider unitary analogs of d-dimensional Anderson models on l2( \$\$\mathbb(z)\$\$ d ) defined by the product Uω=Dω S where S is a deterministic unitary and Dω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman–Molchanov to get exponential estimates on fractional moments of the matrix elements of Uω(Uωz)−1, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of S. Such estimates imply almost sure localization for Uω.

## Keywords

fractional moment method unitary operators localization.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Aizenman, M., Molchanov, S. 1993Localization at large disorder and at extreme energies: an elementary derivationCommun. Math. Phys.157245278Google Scholar
2. 2.
Aizenman, M., Graf, G.-M. 1998Localization bounds for an electron gasJ. Phys. A.3167836806Google Scholar
3. 3.
Aizenman, M., Schenker, J., Friedrich, R., Hundertmark, D. 2001Finite-volume fractional-moment criteria for Anderson localizationCommun. Math. Phys224219253
4. 4.
Blatter, G., Browne, D. 1988Zener tunneling and localization in small conducting rings.Phys Rev. B.3738563880
5. 5.
Bourget, O., Howland, J.S., Joye, A. 2003Special analysis of unitary band matricesCommon. Math3738563880Google Scholar
6. 6.
Combescure, M. 1990Spectral Properties of a periodically kicked quantum Hamiltonian.J. Stat Phys.59679690
7. 7.
Cantero, M.J., Moral, L., Velázquez, L. 2003Five-diagonal matrices and zeros of orthogonal polynomials on the unit circleLinear Algebra Appl.3262956
8. 8.
Geronimo, J.S., Teplyaev, A. 1994A Difference equation arising from the trigonometric moment problem having random reflection coefficients – an operator theoretic approachJ. Func. Anal.1231245
9. 9.
Joye, A. 2004Density of states and Thouless formula for random unitary band matricesAnn. Henri Poincaré.5347379
10. 10.
Simon, B. (2004). Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory; Part~2: Spectral Theory, AMS Colloquium Series, Vol 54, American Mathematical Society, Providence RI.Google Scholar
11. 11.
Simon, B. Aizenman’s theorem for orthogonal polynomials on the unit circle, eprint, math.SP/0411388 and mp-arc:04-386.Google Scholar
12. 12.
Simon, B., Wolff, T. 1986Singular continuous spectrum under rank one perturbations and localization for random HamiltoniansCommun. Pure Appl. Math.397590Google Scholar
13. 13.
Stoiciu, M.: The statistical distribution of the zeros of random paraorthonormal polynomials on the unit circle, eprint math-ph/0412025 and mp-arc:04-405.Google Scholar
14. 14.
Teplyaev, A.V. 1992The pure point spectrum of random polynomials orthogonal on the circleSoviet. Math. Dokl.44407411Google Scholar
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9853560924530029, "perplexity": 5309.193422266366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221211316.39/warc/CC-MAIN-20180816230727-20180817010727-00405.warc.gz"}