边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

A Uniqueness Theorem for Sturm-Liouville Operators with Eigenparameter in Boundary Conditions

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n(q,a)关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n(q,a_k)}_(k=1)^(+∞),则谱集合{λ_n(q,a_k)}_(k=1)^(+∞)能够唯一确定[0,π]上的势函数q(x).

This paper establishes a uniqueness theorem for a kind of Sturm-Liouville prob lem. For a fixed index n （n ∈ Z）, it is shown that the n-th eigenvalue An（q, a） of the ＋∞ Sturm-Liouville problem is strictly monotonous in a. If the spectral set {λn（q,ak）}^＋∞ k=1 can be measured for distinct ak, then the potential q（x） on the interval [0, π] can be uniquely determined by the spectral set {λ（q,ak）}^＋∞ k=1.