Below is the ascii version of the abstract for 03-532.
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Volodymyr Batchenko and Fritz Gesztesy
On the spectrum of Schr\"odinger operators with quasi-periodic
algebro-geometric KdV potentials
ABSTRACT. We characterize the spectrum of one-dimensional Schr\"odinger
operators H=-d^2/dx^2+V with quasi-periodic complex-valued
algebro-geometric potentials V (i.e., potentials V which satisfy
one (and hence infinitely many) equation(s) of the stationary
Korteweg-de Vries (KdV) hierarchy) associated with nonsingular
hyperelliptic curves. The corresponding problem appears to have
been open since the mid-seventies. 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 semi-infinite simple analytic arc in the
complex plane. Crossings as well as confluences of spectral arcs are
possible and discussed as well.