Estimates of periodic potentials in terms of gap lengths (Q1296232)

Let \(\gamma_n, n\geq 1,\) be the gap length of the Hill operator \[ Ty:=-y''(x)+q(x)y \] in \(L^2(-\infty,\infty),\) where \(q\) is a real function such that \(q(x+1)=q(x), q(x)\in L^2(0,1), \int_0^1 q(x)dx=0.\) It is proved that \(\int_0^1 | q(x)| ^2dx\leq 2\| \gamma\| (1+\| \gamma\| ^{1/3}),\) with \(\| \gamma\| ^2=\sum_{n\geq 1} | \gamma_n| ^2.\)