边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

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）.

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

This paper establishes a uniqueness theorem for a kind of Sturm-Liouville problem. For a fixed index $n\ (n\in\bf{\mathbb Z})$, it is shown that the $n$-th eigenvalue $\lambda_{n}(q, a)$ of the Sturm-Liouville problem is strictly monotonous in $a$. If the spectral set $\{\lambda_{n}(q,a_k)\}_{k=1}^{+\infty}$ can be measured for distinct $a_k$, then the potential $q(x)$ on the interval $0,\pi]$ can be uniquely determined by the spectral set $\{\lambda_{n}(q,a_k)\}_{k=1}^{+\infty}$.