Hill方程的特征函数的一个迹公式

设Hill算子L=-α~2+u(x)具有周期有限带位势u(x)。众所周知,与谱带左端点E_(2j)相应的特征函数ψ_j(x)满足著名的McKean-Trubowitz迹恒等式:sum from j=0 to N ψ_j~2(x)=1. 本文证明,谱带右端点E_(2j-1)相应的特征函数φ_j(x)满足另一个迹恒等式:u(x)=-2 sum from j=1 to N φ_j(x)+σ,其中σ=E_0+ sum from j=1 to N(E_(2j)-E_(2j-1). ψ_j与φ_j满足的Hill方程组分别被此二个迹公式非线性化为两个Liouville意义下的完全可积系统:Neumann系统与Bargmamm系统。

A Trace Formula for the Eigenfunctions of the Hill's Equation

Let the Hill's operator L=-(?)2 + u(x) possess a periodic and finite-band potential u(x). It is a well-known fact that the eigenfunctions (?)j(x), corresponding to the left end-point of the spectral bands, satisfy the famous McKean-Trubowitz's identity:In the present paper we have proved that the eigenfunctions (?)j(x), corresponding to the right end-point of the spectral band, satisfy another trace identity:whereThe two systems of the Hill's equations satisfied by (?)j's and (?)j's are nonlinearized by these two trace formulas respectively to be two completely integrable systems in Liou-ville sense: the Neumann system and the Bargmann system.