Electronic Journal of Probability

Eigenvalue Curves of Asymmetric Tridiagonal Matrices

Ilya Goldsheid and Boris Khoruzhenko

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 5 (2000), paper no. 16, 28 pp.

Dates
Accepted: 21 November 2000
First available in Project Euclid: 7 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457376451

Digital Object Identifier
doi:10.1214/EJP.v5-72

Mathematical Reviews number (MathSciNet)
MR1800072

Zentralblatt MATH identifier
0983.82006

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 15A52 47B80: Random operators [See also 47H40, 60H25] 47B39: Difference operators [See also 39A70] 60H25: Random operators and equations [See also 47B80] 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx]

Keywords
Random matrix Schrödinger operator Lyapunov exponent eigenvalue distribution complex eigenvalue

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Goldsheid, Ilya; Khoruzhenko, Boris. Eigenvalue Curves of Asymmetric Tridiagonal Matrices. Electron. J. Probab. 5 (2000), paper no. 16, 28 pp. doi:10.1214/EJP.v5-72. https://projecteuclid.org/euclid.ejp/1457376451


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