Eigenvalue curves of asymmetric tridiagonal random matrices

Electronic Journal of Probability

Volumn 5, Issue , 2000, Pages 1-28

  Goldsheid, Ilya Ya ^a   Khoruzhenko, Boris A ^a  

^a QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

Complex eigenvalue; Eigenvalue distribution; Lyapunov exponent; Random matrix; Schr dinger operator

Indexed keywords

EID: 0002451703     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v5-72     Document Type: Article

References (25)

1. Avron, J. and Simon, B. (1983), Almost Periodic Schrodinger Operators II., Duke Math. Journ. 50, 369-391.
2. Bougerol, P. and Lacroix, J. (1995), Products of Random Matrices with Applications to Random Schrodinger Operators. Birkhauser, Boston.
3. Brezin, E. and Zee, A. (1998), Non-Hermitian localization: Multiple scattering and bounds. Nucl. Phys. B 509[FS], 599-614.
4. Brouwer P. W., Silvestrov, P. G., and Beenakker, C. W. J. (1997), Theory of directed localization in one dimension. Phys. Rev. B56, R4333-R4336.
5. Carmona, R. and Lacroix, J. (1990), Spectral Theory of Random Schrodinger Operators. Birkhauser, Boston.
6. Craig, W. and Simon, B. (1983), Subharmonicity of the Lyapunov Index. Duke Math. Journ. 50 (1983), 551-560.
7. Davies, E. B., Spectral properties of non-self-adjoint matrices and operators. To appear in Proc. Royal Soc. A.
8. Davies, E. B., Spectral theory of pseudo-ergodic operators. To appear in Commun. Math. Phys.
9. Hatano N. and Nelson, D. R. (1996), Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570-573.
10. Hatano N. and Nelson D. R. (1997), Vortex pinning and non-Hermitian quantum mechanics. Phys. Rev. B56, 8651-8673.
11. Hirshman Jr., I. I. (1967), The spectra of certain Toeplitz matrices. Illinois J. Math. 11, 145-159.
12. Hormander, L. (1983), The analysis of linear partial differential equations, Vol. I. Springer, New York.
13. Hormander, L. (1994) Notions of Convexity. Birkhauser, Boston.
14. Goldsheid, I. Ya. (1975), Asymptotic behaviour of a product of random matrices that depend on a parameter. Dokl. Akad. Nauk. SSSR 224, 1248-1251.
15. Goldsheid, I. Ya. (1980), Asymptotic properties of the product of random matrices depending on a parameter. In Adv. in Prob. 8, pp. 239-283, Ya. G. Sinai and R. Dobrushin, eds. Dekker, New York.
16. Goldsheid I. Ya. and Khoruzhenko, B. A. Distribution of eigenvalues in non-Hermitian Anderson models. Phys. Rev. Lett. 80, 2897-2900 (1998).
17. Naiman P. B. (1964), On the spectral theory of non-symmetric periodic Jacobi matrices (in Russian). Zap. Meh.-Mat. Fak. Har’ kov. Gos. Univ. i Har’kov. Mat. Obsc., (4), 30, 138-151.
18. Pastur, L. A. and Figotin, A. L. (1992), Spectra of Random and Almost-Periodic Operators. Springer, Berlin.
19. Reichel, L. and Trefethen, L. N. (1992), Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. Lin. Alg. Appl. 162-4, 153-185.
20. Schmidt, P. and Spitzer, F. (1960), The Toeplitz matrices of an arbitrary Laurent polynomial. Math. Scand. 8, 15-38.
21. Trefethen, L. N. (1991) Pseudospectra of matrices. In D. F. Numerical Analysis 1991, pp. 336-360, Griffiths and G. A. Watson, eds. Longman Scientific and Technical, Harlow, Essex, UK, 1992.
22. Trefethen, L. N., Contendini, M., and Embree, M., Spectra, pseudospectra, and localization for random bidiagonal matrices. To appear in Commun. Pure and Appl. Math.
23. Trefethen, L. N., Trefethen, A. E., Reddy, S. C., and Driscoll, T. A. (1993), Hydrodynamic stability without eigenvalues. Science 261, 578-584.
24. Widom, H. (1994), Eigenvalue distribution for nonselfadjoint Toeplitz matrices. In Operator Theory: Advances and Applications Vol. 71, pp. 1-8.
25. Widom, H. (1990), Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of the Toeplitz determinants in the case of nonvanishing index. In Operator Theory: Advances and Applications Vol. 48, pp. 387-421.