On the spectrum of schrodinger operators with quasi-periodic algebro-geometric KdV potentials

We characterize the spectrum of one-dimensional Schrodinger operators H = -d(2)/dx(2) + V in L-2(R; dx) 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 spectrum of H coincides with the conditional stability set of H and can be described explicitly 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 are discussed as well. These results extend to the L-P(R; dx)-setting for p is an element of [1, infinity).