The spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV potential

Анотація

In this announcement we report on a recent characterization of the spectrum of one-dimensional Schrödinger 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 non-singular hyperelliptic curves in [1]. It turns out the spectrum of H coincides with the conditional stability set of H and that it 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 an-alytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as conffuences of spectral arcs are possible and discussed as well. These results extend to the L ^p(ℝ; dx) -setting for pÎ [1,∞).