Consider the following two eigenvalue problems: (0.1) \begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases} and (0.2) \begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0, az'(\pi)+\mu z(\pi)=0, \end{cases} where $q(x)$ is real-valued and integrable on [0, $\pi$ ]. Let $\{\lambda_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ and $\{\mu_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ denote the eigenvalues of equations (0.1) and (0.2), respectively. Then \[\cdots\lt\mu_{-3}\lt\lambda_{-2}\lt\mu_{-2}\lt\lambda_{-1}\lt\mu_{-1}\lt\mu_1\lt\lambda_1\lt\mu_2\lt\lambda_2\lt\mu_3\lt\cdots.\] Moreover, the number of zeros of the eigenfunctions of (0.1) ((0.2), respectively) corresponding to $\lambda_n$ ( $\mu_n$ , respectively) in (0, $\pi$ ) is equal to $|n|-1$ .

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关于具有特征参数相关边界条件的一些特征值问题的特征值和特征函数的节点

考虑以下两个特征值问题：(0.1)\begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases}和(0.2)\begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0 , az'(\pi)+\mu z(\pi)=0, \end{cases}在哪里$q(x)$是实值且可在 [0 上积，$\pi$]。让$\{\lambda_n\}_{n\in \mathbb{Z}\setminus \{0\}}$和$\{\mu_n\}_{n\in \mathbb{Z}\setminus \{0\}}$分别表示方程 (0.1) 和 (0.2) 的特征值。然后\[\cdots\lt\mu_{-3}\lt\lambda_{-2}\lt\mu_{-2}\lt\lambda_{-1}\lt\mu_{-1}\lt\mu_1\lt \lambda_1\lt\mu_2\lt\lambda_2\lt\mu_3\lt\cdots.\]此外，（0.1）（分别为（0.2））的特征函数的零点数对应于$\lambda_n$($\亩_n$, 分别) 在 (0,$\pi$) 等于$|n|-1$.