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

Volodymyr Batchenko; Fritz Gesztesy

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

Volodymyr Batchenko·Fritz Gesztesy

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Abstract

We characterize the spectrum of one-dimensional Schrödinger 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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BibTeX RIS

Volodymyr Batchenko, Fritz Gesztesy. 2003-12-10. On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV potentials. https://arxiv.org/abs/math/0312200

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