Asymptotic expansion of the spectral function of the Hill operator (Q580566)

Let e(\(\lambda\),x,y) be the spectral function of the Hill operator \(H(y)\equiv -y''+q(x)y,\) where \(q\in C^{\infty}(R)\) is an a-periodic function. The authors investigate an asymptotic expansion of e(\(\lambda\),x,y) for \(\lambda\to \infty\). From the main theorem then follows corollary 1: If \(\lambda\to \infty\), then \(e(\lambda,x,x)\sim \pi^{-1}\sqrt{\lambda}\sum^{\infty}_{k=1}h_ k(x)\lambda^{-k+}\) uniformly on R, where \(h_ k\in C^{\infty}(R)\) are a-periodic functions.