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

Consider the spectral problem ($0<x<1$)
$$ -y''(x)=\lambda\rho (x)y(x);\quad y(0)=y(1)=0;\quad \rho(x)>0;\quad \rho(x)\in C_{[0,1]}. $$

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
$$ \varlimsup\lambda_n^{-1/4+\varepsilon}(\rho_0)|u_n(1/2,\rho_0)|=0\qquad(n\to\infty). $$
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