On the spectrum of Jacobi operators with quasi-periodic   algebro-geometric coefficients
  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. 
Vladimir Batchenko and Fritz Gesztesy