On the spectrum of Jacobi operators with quasiperiodic algebrogeometric coefficients
Abstract
We characterize the spectrum of onedimensional Jacobi operators H=aS^{+}+a^{}S^{}+b in l^{2}(\Z) with quasiperiodic complexvalued algebrogeometric 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2005
 DOI:
 10.48550/arXiv.math/0506138
 arXiv:
 arXiv:math/0506138
 Bibcode:
 2005math......6138B
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Mathematical Physics;
 Mathematical Physics;
 34L05;
 47B36;
 35Q58;
 35Q51
 EPrint:
 38 pages