On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KDV potentials

We characterize the spectrum of one-dimensional Schrödinger operatorsH=?d ² /dx ² +V inL ²(?dx) with quasi-periodic complex-valued algebro-geometric potentialsV, i.e., potentialsV which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy, associated with nonsingular hyperelliptic curves. The spectrum ofH coincides with the conditional stability set ofH and can be described explicitly in terms of the mean value of the inverse of the diagonal Green’s function ofH. As a result, the spectrum ofH 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 theL ^p (?dx) forp∈1, ∞).