Exercise 21, § 4

We retain the notation of the preceding exercise. We denote by $\varphi_n$ the unique eigenfunction of SL$_q$ corresponding to the eigenvalue $\lambda_n$ such that $\varphi '_n(0) =\lambda_n$.

a) Let M = sup$\{q(x)\}$. For every $n\in \mathbf{N}$, one has

and moreover

when $n\rightarrow +\infty$.

b) Let $s >0$. Let $f\in \mathscr{C}^2([0,1])$ be such that

Then one has $(1-s^{-1}u_s)f=g_s$, where $g_s(x) =$ sin($sx$) and $u_s$ is the integral operator on $\mathscr{C}([0,1])$ given by

$u_s(f)(x) =\int_0^x$ sin($s(x-y)$)$q(y)f(y)d\mu(y)$.

c) For $s$ sufficiently large, one has $s \notin$ Sp($u_s$).

d) For every fixed $x$, one has

$\varphi_n(x) =$ sin(($n+ 1$)$x$) $+ O(n^{-1})$

when $n\rightarrow +\infty$.