# 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.