Exercise 22, § 4

We resume the notation of the preceding exercise.

a) There exist functions $f_1$ and $f_2$ in $\mathscr{C}^2([a, b])$ such that $-f''_i+qf_i= 0$ and

These functions are the unique solutions of these equations and the function $w=$ $f_1f'_2-f'_1f_2$ is constant.

b) Let $g\in \mathscr{C}([0,1])$. There exists a unique function $\varphi \in \mathscr{C}^2([0,1])$ such that

and one has $\varphi =c_1f_1+c_2f_2$, where

for every $x\in [0,1]$. c) The partial operator SL$_q$ is surjective; its inverse is the mapping $g\mapsto \varphi$ and there exists a function $G\in \mathscr{C}([0,1]^2)$ such that

SL$^{-1}_q(g)(x) =\int_0^1G(x, y)g(y)d\mu(y)$

for every $g\in \mathscr{C}([0,1])$ and almost every $x\in [0,1]$. One has $G(x, y) = G(y, x)$ for every $(x, y)\in [0,1]^2$.

d) The operator SL$^{-1}_q$ is compact and injective on $L^2([0,1], \mu)$.

e) The functions $(\varphi_n)$ form an orthonormal basis of $L^2([0,1])$, and the spectrum of SL$_q$ coincides with the set of its eigenvalues.