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
   
   $$a_2(x)u^{″}(x)+a_1(x)u^{\prime }(x)+a_0(x)u(x)=f(x)$$
   with
   $a_2(x)\ne 0,u\in C^2[a,b]$
   has the same solution space as one of the form
   
   $$-(p(x)u^{\prime }(x){)}^{\prime }+q(x)u(x)=F(x)$$
   This is called Sturm-Liouville Form. How does this transformation affect the boundary conditions?
3. Show that a general Sturm-Liouville eigenvalue equation
   
   $$-(p(x)u^{\prime }(x){)}^{\prime }+q(x)u(x)=\lambda u(x),$$
   
   $p(x)\ne 0$, 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):
   
   $$\frac{dx}{ds}=\sqrt{p(x)},f(s):=(p(x(s)){)}^{1/4},Q(s)=q(x(s))+\frac{f^{″}(s)}{f(s)}.$$
   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