On the spectrum of Jacobi operators with quasi-periodic algebro-geometric coefficients

TL;DR

This paper characterizes the spectrum of one-dimensional Jacobi operators with quasi-periodic algebro-geometric coefficients, showing it consists of finitely many analytic arcs in the complex plane and providing explicit descriptions.

Contribution

It provides a complete spectral characterization of Jacobi operators with quasi-periodic algebro-geometric coefficients linked to hyperelliptic curves, including explicit descriptions and arc structures.

Findings

Spectrum consists of finitely many simple analytic arcs.

Crossings and confluences of spectral arcs are possible.

Spectrum coincides with the conditional stability set and is described via the Green's function.

Abstract

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.