Exercise 20, § 4

Si $a$ and $b$ are real numbers with $a < b$, we denote by $\mathscr{C}^2([a, b])$ the vector space of continuous functions on $[a, b]$ whose restriction to$]a, b[$ is of class $C^2$. We denote by $\mu$ the Lebesgue measure.

a) Let $q_1$ and $q_2$ be real-valued functions in $\mathscr{C}^2([a, b])$ such that $q_1\geqslant q_2$. For $i\in \{1,2\}$, let $f_i\in \mathscr{C}^2([a, b])$ be a solution of the equation

Suppose that $f_2(a) =f_2(b) = 0$ and that $f_2\not = 0$. If the function $f_1$ does not vanish in$]a, b[$, then $q_1=q_2$ and there exists $c\in \mathbf{C}$ such that $f_1=cf_2$. (Reduce to the case where $f_1$ and $f_2$ are strictly positive on$]a, b[$; consider the wronskian $w=f_1f'_2-f'_1f_2$ and prove that $w(a) =w(b)$, then that $w= 0.$)

b) Let $q\in \mathscr{C}^2([0,1])$ satisfying $q\geqslant 0$. We denote by $\widetilde{S}L_q$ the partial differential operator on $L^2([0,1], \mu)$ with domain

such that

(“Sturm–Liouville operator”). The operator $\widetilde{S}L_q$ is symmetric and essentially self-adjoint. We denote by SL$_q$ the closure of $\widetilde{S}L_q$.

c) Let $\lambda \in$ Sp(SL$_q$) be an eigenvalue of SL$_q$. The proper subspace associated with $\lambda$ is of dimension 1 and generated by a real-valued function belonging to D.

d) Every eigenvalue $\lambda$ of SL$_q$ satisfies $\lambda \geqslant \pi^2$. (First prove that $\lambda$ is positive by using question a) with $q_1=q$ and $q_2= 0.)$

e) For $s >0$, there exists a unique function $f_s\in \mathscr{C}^2([0,1])$ which is a solution of the linear differential equation

and satisfies $f_s(0) = 0$ and $f'_s(0) =s$. Put $f(x, s) =f_s(x)$; the mapping $f$ is a function of class $C^2$ on$]0,1[\times \mathbf{R}^*_+$. The eigenvalues of SL$_q$ are of the form $\lambda =s^2$ where $s$ satisfies $f(1, s) = 0$.

f) For $t\in [0,1]$ and $s >0$, let $N(t, s)\in \overline{\mathbf{N}}$ be the number of $x\in [0, t]$ such that $f(x, s) = 0$. One has $N(t, s)\in \mathbf{N}$ and the function $s\mapsto N(t, s)$ is increasing for every fixed $t$. (Use question a).)

g) Let $t\in [0,1]$ and $s_0>0$. Let $m= N(t, s_0)\in \mathbf{N}$. Let

be the zeros of $f_{s_0}$ in $[0, t]$. For $s\geqslant 0$, let

be the first $m$ zeros of $f_s$ in $[0, t]$. The functions $s\mapsto x_j(s)$ are of class $C^1$ and one has

h) The set of eigenvalues of SL$_q$ is the set of the values of an increasing sequence of strictly positive real numbers

tending to $+\infty$. Moreover, the eigenspace corresponding to $\lambda_n$ is of dimension 1; if $\varphi_n$ is a basis of it, one has $\varphi_n\in \mathscr{C}^2([0,1])$ and the function $\varphi_n$ vanishes at exactly $n$ distinct points of $[0,1]$. (To prove that there exist infinitely many eigenvalues of SL$_q$, use question a) with $q_1= M-s^2$, where $M\in \mathbf{R}$ is suitably chosen, to verify that $N(1, s)$ tends to $+\infty$ when $s\rightarrow +\infty .)$