Electronic Journal of Probability

Eigenvalue Curves of Asymmetric Tridiagonal Matrices

Ilya Goldsheid and Boris Khoruzhenko

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Random matrix Schrödinger operator Lyapunov exponent eigenvalue distribution complex eigenvalue

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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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