We characterize the spectrum of one-dimensional Jacobi operators
H=aS^{+}+a^{-}S^{-}+b in l^{2}(\Z) with quasi-periodic complex-valued
algebro-geometric coefficients (which satisfy one (and hence infinitely many)
equation(s) of the stationary Toda hierarchy) associated with nonsingular
hyperelliptic curves. The spectrum of H coincides with the conditional
stability set of H and can explicitly be described in terms of the mean value
of the Green's function of H.
As a result, the spectrum of H consists of finitely many simple analytic arcs
in the complex plane. Crossings as well as confluences of spectral arcs are
possible and discussed as well.
@article{0506138,
author = {Batchenko, Vladimir and Gesztesy, Fritz},
title = {On the spectrum of Jacobi operators with quasi-periodic
algebro-geometric coefficients},
journal = {arXiv},
volume = {2005},
number = {0},
year = {2005},
language = {en},
url = {http://dml.mathdoc.fr/item/0506138}
}
Batchenko, Vladimir; Gesztesy, Fritz. On the spectrum of Jacobi operators with quasi-periodic
algebro-geometric coefficients. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0506138/