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International Mathematics Research Papers (2005) 2005:511-563, doi:10.1155/IMRP.2005.511
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Copyright © 2005 Hindawi Publishing Corporation. All rights reserved.

On the spectrum of Jacobi operators with quasiperiodic algebro-geometric coefficients

Vladimir Batchenko and Fritz Gesztesy

We characterize the spectrum of one-dimensional Jacobi operators Formula with quasiperiodiccomplex-valued algebro-geometric coefficients (which satisfy one(and hence infinitely many) equation(s) of the stationary Todahierarchy) associated with nonsingular hyperelliptic curves. Thespectrum of H coincides with the conditional stability set ofH and can explicitly be described in terms of the mean value ofthe 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.


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