On the asymptotic behavior of the normalized eigenfunctions of the Sturm-Liouville problem on a~finite interval

M. M. Gekhtman

Sbornik. Mathematics, Tome 61 (1988) no. 1, pp. 185-199

Résumé

Consider the spectral problem ($0$) $$ -y''(x)=\lambda\rho (x)y(x);\quad y(0)=y(1)=0;\quad \rho(x)>0;\quad \rho(x)\in C_{[0,1]}. $$ Let $\lambda_n(\rho)$ and $u_n(x,\rho)$ ($n\in N$) be the eigenvalues and the corresponding eigenfunctions, normalized in $L_2(0,1;\rho)$. Theorem. 1. {\it If the weight function $\rho(x)$, continuous on $[0,1]$, is positive, then $$ \lim\lambda_n^{-1/4}(\rho)\max_{0\le x\le1}|u_n(x,\rho)|=0\qquad(n\to\infty). $$ 2. For any $\varepsilon>0$ there exists a continuous weight $\rho_0(x,\varepsilon)>0\quad(x\in[0,1])$ such that $$ \varlimsup\lambda_n^{-1/4+\varepsilon}(\rho_0)|u_n(1/2,\rho_0)|=0\qquad(n\to\infty). $$} Bibliography: 17 titles.