Eigenvalue Curves of Asymmetric Tridiagonal Matrices
Ilya Ya Goldsheid, University of London
Boris A Khoruzhenko, University of London
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
l2( Z) is a two dimensional set which is not approximated
by the spectra of the finite-interval operators.
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