On the spectrum of Schrödinger operators with quasiperiodic algebrogeometric KdV potentials
Abstract
We characterize the spectrum of onedimensional Schrödinger operators H=d^2/dx^2+V with quasiperiodic complexvalued algebrogeometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Kortewegde Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The corresponding problem appears to have been open since the midseventies. 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 inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semiinfinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2003
 arXiv:
 arXiv:math/0312200
 Bibcode:
 2003math.....12200B
 Keywords:

 Mathematics  Spectral Theory;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 34L05;
 35Q53;
 58F07;
 34L40;
 35Q51
 EPrint:
 43 pages