2012-50.md

course	course_year	question_number	tags	title	year
Methods	IB	50	IB 2012 Methods	Paper 1, Section II, C	2012

Consider the regular Sturm-Liouville (S-L) system

$$(\mathcal{L}y)(x)-\lambda \omega (x)y(x)=0,a⩽x⩽b$$

where

$$(\mathcal{L}y)(x):=-{[p(x)y^{\prime }(x)]}^{\prime }+q(x)y(x)$$

with $\omega (x)>0$ and $p(x)>0$ for all $x$ in $[a,b]$, and the boundary conditions on $y$ are

Missing or unrecognized delimiter for \left

$$\left{\begin{array}{l}
A_{1} y(a)+A_{2} y^{\prime}(a)=0 \\
B_{1} y(b)+B_{2} y^{\prime}(b)=0
\end{array}\right.$$

Show that with these boundary conditions, $\mathcal{L}$ is self-adjoint. By considering $y\mathcal{L}y$, or otherwise, show that the eigenvalue $\lambda$ can be written as

$$\lambda=\frac{\int_{a}^{b}\left(p y^{\prime 2}+q y^{2}\right) d x-\left[p y y^{\prime}\right]{a}^{b}}{\int{a}^{b} \omega y^{2} d x}$$

Now suppose that $a=0$ and $b=\ell$, that $p(x)=1,q(x)⩾0$ and $\omega (x)=1$ for all $x\in [0,\ell ]$, and that $A_{1}=1, A_{2}=0, B_{1}=k \in \mathbb{R}^{+}$and $B_2=1$. Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that $q(x)=0$, solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues ${\lambda }_1<{\lambda }_2<\cdots <{\lambda }_n<\cdots$. Describe the behaviour of ${\lambda }_n$ as $n\to \mathrm{\infty }$.