Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps. (Q1865295)

The authors consider the Schrödinger equation on the line where the potential \(V(x)\) is real valued, periodic with period one, square integrable, and has the form \(\sum_{m=-\infty}^{\infty} v(m)\exp (2\pi i mx)\). By using techniques similar to the ones used by \textit{T. Kappeler} and the second author [Trans. Am. Math. Soc. 351, No. 2, 619--646 (1999; Zbl 0924.58074)], it is shown that the potential \(V\) belongs to the Gevrey class and satisfies \[ \sum_{m=-\infty}^{\infty} | v(m)| ^2(1+| m| )^{2s}\exp(2a| m| ^b)<\infty \] when the periodic and antiperiodic eigenvalues \(\lambda_n^-\) and \(\lambda_n^+\) satisfy \[ \sum_{n=1}^{\infty} | \lambda_n^+ -\lambda_n^-| ^2(1+n)^{2s}\exp (2an^b) <\infty \text{ with }s\geq 0 \text{ and }a>0. \]