Eigenvalue Curves of Asymmetric Tridiagonal Matrices

Ilya Ya Goldsheid (University of London)
Boris A Khoruzhenko (University of London)


Random Schrödinger operators with imaginary vector potentials are studied in dimension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on finite intervals of length $n$ with periodic boundary conditions and describe the limit eigenvalue distribution when $n$ goes to infinity. We prove that this limit distribution is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a "reference" symmetric eigenvalue problem. In contrast to these results, the spectrum of the limit operator in $\ell^2(Z)$ is a two dimensional set which is not approximated by the spectra of the finite-interval operators.

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Pages: 1-28

Publication Date: November 21, 2000

DOI: 10.1214/EJP.v5-72


  1. Avron, J. and Simon, B. (1983), Almost Periodic Schroedinger Operators II. Duke Math. Journ. 50 , 369 - 391. Math. Review 85i:34009a
  2. Bougerol, P. and Lacroix J. (1985) Products of Random Matrices with Applications to Random Schroedinger Operators. Birkhaeuser, Boston. Math. Review 88f:60013
  3. Brezin, E. and Zee, A (1998), Non-Hermitian localization: Multiple scattering and bounds. Nucl. Phys. B509[FS], 599 - 614. Math. Review number not available.
  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. Math. Review number not available.
  5. Carmona, R. and Lacroix, J. (1990), Spectral Theory of Random Schroedinger Operators. Birkhaeuser, Boston. Math. Review 92k:47143
  6. Craig, W. and Simon, B. (1983), Subharmonicity of the Lyapunov Index. Duke Math. Journ. 50, 551 - 560. Math. Review 85f:34086
  7. Davies, E.B., Spectral properties of non-self-adjoint matrices and operators. To appear in Proc. Royal Soc. A. Math. Review number not available.
  8. Davies, E.B., Spectral theory of pseudo-ergodic operators. To appear in Commun. Math. Phys. Math. Review number not available.
  9. Hatano, N. and Nelson, D.R. (1996), Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570 - 573. Math. Review number not available.
  10. Hatano, N. and Nelson, D.R. (1997), Vortex pinning and non-Hermitian quantum mechanics. Phys. Rev. B56, 8651 - 8673. Math. Review number not available.
  11. Hirshman Jr., I.I. (1967), The spectra of certain Toeplitz matrices. Illinois J. Math. 11, 145 - 159. Math. Review 34:4905
  12. Hormander, L. (1983) The analysis of linear partial differential equations, Vol. I. Springer, New York. Math. Review 85g:35002a
  13. Hormander, L. (1994), Notions of Convexity. Birkhaeuser, Boston. Math. Review 95k:00002
  14. Goldsheid, I. Ya. (1975), Asymptotic behaviour of a product of random matrices that depend on a parameter (Russian) , Dokl. Akad. Nauk SSSR 224, 1248 - 1251. Math. Review 55:4328
  15. Goldsheid, I. Ya. (1980) Asymptotic properties of the product of random matrices depending on a parameter. In Multicomponent Random Systems, pp. 239 -283, Sinai, Ya.G. and Dobrushin, R., eds., Adv. in Prob. 8 . Dekker, New York. Math. Review 82f:60027
  16. Goldsheid, I.Ya. and Khoruzhenko, B.A. (1998), Distribution of eigenvalues in non-Hermitian Anderson models , Phys. Rev. Lett. 80, 2897 - 2900. Math. Review number not available.
  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. Math. Review 36:1999
  18. Pastur, L.A. and Figotin, A.L. (1992), Spectra of Random and Almost-Periodic Operators. Springer, Berlin. Math. Review 94h:47068
  19. Reichel, L. and Trefethen, L.N. (1992), Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. Lin. Alg. Appl. 162-4, 153 - 185. Math. Review 92k:15028
  20. Schmidt, P and Spitzer F. (1960), The Toeplitz matrices of an arbitrary Laurent polynomial. Math. Scand. 8 , 15 - 38. Math. Review 23:A1977
  21. Trefethen, L.N. (1992), Pseudospectra of matrices. In Numerical Analysis 1991, D. F. Griffiths and G. A. Watson, eds., pp. 336 - 360. Longman Scientific and Technical, Harlow, Essex, UK. Math. Review 93e:65064
  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. Math. Review number not available.
  23. Trefethen, L.N., Trefethen, A.E., Reddy, S.C., and Driscoll, T.A. (1993), Hydrodynamic stability without eigenvalue. Science 261 , 578 - 584. Math. Review number not available.
  24. Widom, H. (1994), Eigenvalue distribution for nonselfadjoint Toeplitz matrices. In Operator Theory: Advances and Applications 71 , 1 - 8. Math. Review 95g:47043
  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 48 , 387 - 421. Math. Review 93m:47033

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