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

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

2003-12-10

arxiv.org/abs/math/0312200v1

Mathematics

Spectral Theory

43 pages

Scientific paper

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.

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