The spectrum of Schr\"odinger operators with quasi-periodic algebro-geometric KdV potentials

Vladimir Batchenko ; Fritz Gesztesy

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003), pp. 447-468.

Résumé

In this announcement we report on a recent characterization of the spectrum of one-dimensional Schrödinger operators $H=-d^2/dx^2+V$ in $L^2(\mathbb R;dx)$ 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 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 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. These results extend to the $L^p(\mathbb R;dx)$-setting for $p\in [1,\infty)$.