# Homework 3
$\renewcommand{\R}{\mathbb{R}}$
1. Recall the Volterra operator:
$$(Vf)(x):=\int_0^x f(y)dy$$
on $L^2[0,1]$.
Compute the singular value decomposition of $V$. What is its norm? Is it trace-class? Does it have a bounded inverse?
2. Show that a general second order ODE
$$ a_2(x)u''(x)+a_1(x)u'(x)+a_0(x)u(x)=f(x)$$
with $a_2(x)\neq 0, u\in C^2[a,b]$
has the same solution space as one of the form
$$ -(p(x)u'(x))' + q(x)u(x) = F(x)$$
This is called *Sturm-Liouville Form*. How does this transformation affect the boundary conditions?
3. Show that a general Sturm-Liouville eigenvalue equation
$$ -(p(x)u'(x))' +q(x)u(x) = \lambda u(x),$$
$p(x)\neq 0$, can be converted to *Schrodinger form* (which we studied in the previous lecture)
$$ -U''(s)+Q(s)U(s)=\lambda U(s)$$
via the change of variables (i.e., diffeomorphism):
$$\frac{dx}{ds}=\sqrt{p(x)},\quad f(s):=(p(x(s)))^{1/4},\quad Q(s)=q(x(s))+\frac{f''(s)}{f(s)}.$$
How does this affect the boundary conditions and domain of definition of U?
Put the Kimura equation
$$-x(1-x)u''(x)=\lambda u(x)$$
in Schrodinger form.
4. Show that $u\in C^2[a,b]$ satisfies the ODE $$-u''(x)+q(x)u(x)=\lambda u(x)$$ if and only if there are functions $\rho(x),\theta(x)$ such that the vector $(u(x),u'(x))\in\R^2$ is equal to $(\rho(x),\theta(x))$ with $u(x)=\rho(x)\sin(\theta(x))$ and $u'(x)=\rho(x)\cos(\theta(x))$, and they satisfy the system of (nonlinear) ODE:
$$\theta'(x)=\cos(\theta(x))^2 + (\lambda -q(x))\sin(\theta(x))^2, \qquad\rho'(x) = \rho(1+q(x)-\lambda)\frac{\sin(2\theta(x))}{2}$$

5. Show that the subspace $\{f\in C^2[a,b]:f(a)=f(b)=0\}$ is dense in $L^2[a,b]$.
6. Show that if $T\in L(H)$ is finite rank, then
$$ \dim\ker(I-T)=\dim\ker(I-T^*)<\infty.$$
7. Read Section 2 of http://lie.math.okstate.edu/~binegar/4233/4233-l17.pdf, which gives an example of how Sturm-Liouville problems arise naturally from physical PDE via separation of variables. Note that these are exampels of *singular* SL problems, which refers to any problem not satisfying all of the hypotheses: (1) $[a,b]$ is a finite interval (2) $p(x)$ is continuous and bounded on $(a,b)$ (3) $p(x)\neq 0$ on $[a,b]$. Similar conclusions can be reached for such problems, but it is much more complicated and will be discussed later.