ON THE EIGENVALUES AND THE NODAL POINTS OF THE EIGENFUNCTIONS OF SOME EIGENVALUE PROBLEMS WITH EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

CHAN, CHI-HUA ; HUANG, PO-CHUN 

Glasgow mathematical journal, Tome 63 (2021) no. 1, pp. 158-178

Résumé

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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DOI : 10.1017/S0017089520000087