# Lecture 10: Group Work

Form groups of three people, ideally strangers from different departments, and solve the following problems in order.

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.

After the class, get one person from each group to (1) write down the names of the group members (2) comment on anything you thought was interesting / were confused about.

## Group Summaries

1. Dipti Jasrasaria, Paul Wrona, Sam Olivier, Nicole Farias, Sarvesh Sadana, Jack Spilecki, Gang Yang, Jonathan Liu, Mathias Palmstroem, Shizhe Liu (we all worked together) - we spent most of our time figuring out problem one
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