Real eigenvalues in the non-Hermitian Anderson model. (English) Zbl 06974773

Summary: The eigenvalues of the Hatano-Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.


47B80 Random linear operators
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
Full Text: DOI arXiv Euclid


[1] Aizenman, M. and Molchanov, S. (1993). Localization at large disorder and at extreme energies: An elementary derivation. Comm. Math. Phys.157 245–278. · Zbl 0782.60044
[2] Bernstein, S. (1924). Über eine modifikation der ungleichung von Tschebyscheff und über die abweichung der laplaceschen formel. Charkov Ann. Sc.1 38–49. · JFM 50.0342.01
[3] Bernstein, S. (1927). Sur l’extension du théoréme limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Ann.97 1–59.
[4] Bourgain, J. (2004). On localization for lattice Schrödinger operators involving Bernoulli variables. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.1850 77–99. Springer, Berlin.
[5] Bourgain, J. (2012). On the Furstenberg measure and density of states for the Anderson-Bernoulli model at small disorder. J. Anal. Math.117 273–295. · Zbl 1275.82006
[6] Bourgain, J. (2014). On eigenvalue spacings for the 1-D Anderson model with singular site distribution. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.2116 71–83. Springer, Cham. · Zbl 1319.35125
[7] Brézin, E. and Zee, A. (1998). Non-Hermitean delocalization: Multiple scattering and bounds. Nuclear Phys. B509 599–614. · Zbl 0953.82026
[8] Brouwer, P. W., Silvestrov, P. G. and Beenakker, C. WJ. (1997). Theory of directed localization in one dimension. Phys. Rev. B56 Article ID R4333.
[9] Carmona, R., Klein, A. and Martinelli, F. (1987). Anderson localization for Bernoulli and other singular potentials. Comm. Math. Phys.108 41–66. · Zbl 0615.60098
[10] Combes, J.-M., Germinet, F. and Klein, A. (2009). Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys.135 201–216. · Zbl 1168.82016
[11] Feinberg, J. and Zee, A. (1999). Spectral curves of non-Hermitian Hamiltonians. Nuclear Phys. B552 599–623. · Zbl 0944.82017
[12] Fröhlich, J. and Spencer, T. (1983). Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys.88 151–184. · Zbl 0519.60066
[13] Furstenberg, H. (1963). Noncommuting random products. Trans. Amer. Math. Soc.108 377–428. · Zbl 0203.19102
[14] Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Stat.31 457–469. · Zbl 0137.35501
[15] Goldsheid, I. Ya. and Khoruzhenko, B. A. (1998). Distribution of eigenvalues in non-Hermitian Anderson models. Phys. Rev. Lett.80 2897. · Zbl 1032.82004
[16] Goldsheid, I. Ja. (1980). Asymptotic properties of the product of random matrices depending on a parameter. In Multicomponent Random Systems. Adv. Probab. Related Topics6 239–283. Dekker, New York. · Zbl 0443.60100
[17] Goldsheid, I. Ya. and Khoruzhenko, B. A. (2000). Eigenvalue curves of asymmetric tridiagonal random matrices. Electron. J. Probab.5 Article ID 16. · Zbl 0983.82006
[18] Goldsheid, I. Ya. and Khoruzhenko, B. A. (2003). Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model. Comm. Math. Phys.238 505–524. · Zbl 1032.82004
[19] Graf, G. M. and Vaghi, A. (2007). A remark on the estimate of a determinant by Minami. Lett. Math. Phys.79 17–22. · Zbl 1104.82008
[20] Hatano, N. and Nelson, D. R. (1996). Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett.77 570.
[21] Hatano, N. and Nelson, D. R. (1998). Non-Hermitian delocalization and eigenfunctions. Phys. Rev. B58 8384.
[22] Kuwae, T. and Taniguchi, N. (2001). Two-dimensional non-Hermitian delocalization transition as a probe for the localization length. Phys. Rev. B64 201321.
[23] Le Page, É. (1982). Théorèmes limites pour les produits de matrices aléatoires. In Probability Measures on Groups (Oberwolfach, 1981). Lecture Notes in Math.928 258–303. Springer, Berlin.
[24] Le Page, É. (1984). Répartition d’état d’un opérateur de Schrödinger aléatoire. Distribution empirique des valeurs propres d’une matrice de Jacobi. In Probability Measures on Groups, VII (Oberwolfach, 1983). Lecture Notes in Math.1064 309–367. Springer, Berlin.
[25] Minami, N. (1996). Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Comm. Math. Phys.177 709–725. · Zbl 0851.60100
[26] Molchanov, S. A. (1982). The local structure of the spectrum of a random one-dimensional Schrödinger operator. Tr. Semin. im. I. G. Petrovskogo8 195–210.
[27] Molinari, L. G. (2009). Non-Hermitian spectra and Anderson localization. J. Phys. A42 Article ID 265204. · Zbl 1167.82317
[28] Oseledec, V. I. (1968). A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Tr. Mosk. Mat. Obŝ.19 179–210.
[29] Reddy, N. K. Lyapunov exponents and eigenvalues of products of random matrices. Preprint. Available at arXiv:1606.07704.
[30] Sert, C. Large deviation principle for random matrix products. Preprint. Available at arXiv:1704.00615. · Zbl 1373.60017
[31] Shubin, C., Vakilian, R. and Wolff, T. (1998). Some harmonic analysis questions suggested by Anderson–Bernoulli models. Geom. Funct. Anal.8 932–964. · Zbl 0920.42005
[32] Tutubalin, V. N. (1965). Limit theorems for a product of random matrices. Teor. Veroâtn. Ee Primen.10 19–32. · Zbl 0147.17105
[33] Wegner, F. (1981). Bounds on the density of states in disordered systems. Z. Phys. B44 9–15.
[34] Zee, A.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.