具有转移条件的向量型Sturm-Liouville问题的特征值重数及Ambarzumyan定理

On the Multiplicity of Eigenvalues of a Vectorial Sturm-Liouville Problem with Discontinuous Conditions and Related Ambarzumyan’s Theorem

研究了定义在有限区间内具有转移条件的m维向量型Sturm-Liouville问题.主要得到了该问题特征值重数的若干结论.证明了当矩阵值势函数Q满足一定的条件时,只能有有限个重数为m的特征值.作为重数结果的应用,证明了该问题的Ambarzumyan定理.

The m-dimensional vectorial Sturm-Liouville problem with discontinuous conditions inside a finite interval is considered.We mainly obtain some conclusions about multiplicity of the eigenvalues based on the estimation of solutions.It is proved that,under certain conditions on potential matrix,the problem can only have a finite number of eigenvalues with multiplicity m.As an application,Ambarzumyan’s theorem is proved for the problem.