一类边界条件含参数的Sturm-Liouville算子的逆问题

An Inverse Problem for Sturm-Liouville Operators with Boundary Conditon Dependent on the Spectral Parameter

本文讨论了一类两个边界条件中都含有参数的Sturm-Liouville算子的逆问题,运用Hochstadt-Lieberman的方法以及整函数的性质得到两方面的结论:一是对固定的非负整数n,证明了该Sturm-Liouville问题的第n个特征值λn(q,bk)关于bk是严格单调的;二是如果测得一组不同参数边界条件下该问题的第n个特征值的无穷集合,则该谱集合能惟一确定区间[0,π]上的势函数q(x).

This paper discussed the inverse problem of Sturm-Liouville operators.Two boundary conditions depend on spectral parameter.Applying the Hochstadt-Lieberman’s method and properties of entire functions,two conclusions are proved.One is the n-th eigenvalueλn(q,bk)of the Sturm-Liouville problem is strictly monotonous in bk for a fixed index n(n∈N 0).Another is the potential q(x)on the interval[0,π]can be uniquely determined by the spectral set{λn(q,bk)}=∞n=1,if the spectral set{λn(q,bk)}=∞n=1 can be measured for distinct bk.