Paper 1, Section II, C

Methods | Part IB, 2012

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

(Ly)(x)λω(x)y(x)=0,axb(\mathcal{L} y)(x)-\lambda \omega(x) y(x)=0, \quad a \leqslant x \leqslant b

where

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

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

{A1y(a)+A2y(a)=0B1y(b)+B2y(b)=0\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, L\mathcal{L} is self-adjoint. By considering yLyy \mathcal{L} y, or otherwise, show that the eigenvalue λ\lambda can be written as

λ=ab(py2+qy2)dx[pyy]ababωy2dx\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=0a=0 and b=b=\ell, that p(x)=1,q(x)0p(x)=1, q(x) \geqslant 0 and ω(x)=1\omega(x)=1 for all x[0,]x \in[0, \ell], and that A1=1,A2=0,B1=kR+A_{1}=1, A_{2}=0, B_{1}=k \in \mathbb{R}^{+}and B2=1B_{2}=1. Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that q(x)=0q(x)=0, solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues λ1<λ2<<λn<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}<\cdots. Describe the behaviour of λn\lambda_{n} as nn \rightarrow \infty.

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