Smoothness of Schrِdinger Operator Potential in the Case of Gevrey Type Asymptotics of the Gaps

Consider the Schrödinger equation −y″+V(x)y=λy with a periodic real-valued L2-potential V of period 1, V(x)=∑m=−∞∞v(m) exp(2πimx). Let {λn−,λn+} be its zones of instability, i.e. {λn−,λn+} are pairs of periodic and antiperiodic eigenvalues, and b∈(0,1) determines Gevrey classes. M.If∑n=1∞∣λn+−λn−∣2(1+n)2se2anb<∞s⩾0, a0 , then V(x) is a Gevrey function, and moreover∑m=−∞∞∣v(m)∣2(1+∣m∣)2se2a∣m∣b<∞.