# Sturm-Liouville eigenvalues and eigenvectors

Hi i was wondering if someone could give me the following definitions.

The First part of the question is to explain the terms eigenvalue and eigenfunctions in relation to the strum liouville problem :$$\frac{d}{dx}(p(x)\frac{dy}{dx})+\lambda r(x)y=0,a\leq 0 \leq b$$ subject to $y(a)=0,y(b)=0$ this part is worth (3) marks on my past exam paper that i'm using for revision, the next part was to determine the eigenvalues and eigenfunctions for a specific problem which was (17) marks which i know how to do, i just don't know what is expected for the (3) marks, any help would be highly appreciated.

• See e.g. en.wikipedia.org/wiki/… – Winther Apr 18 '17 at 23:11
• I dont see what i would need to write on there how much detail do you think they are asking for? – user395952 Apr 19 '17 at 4:43
• You could start with the definition: if $Ly = \lambda y$ for some linear operator $L$ and scalar $\lambda$ then we say that $y$ is an eigenvector of the operator $L$ with eigenvalue $\lambda$. Then show, as in the answer below, that the SL equation can be written on this form where $L = \ldots$ is a linear differential operator. – Winther Apr 19 '17 at 10:26

Let $L^2_r[a,b]$ denote the weighted inner product space with inner product $$(f,g)_r = \int_{a}^{b}f(x)\overline{g(x)}r(x)dx.$$ You can recast the problem in terms of the operator $$Lf = \frac{1}{r(x)}\left[-\frac{d}{dx}\left(p(x)\frac{df}{dx}\right)\right]$$ on the domain $\mathcal{D}(L)$ consisting of twice continuously differentiable functions (or twice absolutely continuous functions) that vanish at $x=a,b$. In this context, the eigenvalues $\lambda_n$ of the operator $L$ are those $\lambda$ for which $(L-\lambda I)f=0$ has a non-trivial solution $f\in\mathcal{D}(L)$. That is $Lf=\lambda f$ for some non-trivial $f\in\mathcal{D}(L)$ and some $\lambda\in\mathbb{C}$. For standard, non-singular Sturm-Liouville problems where $r$, $p$ are real and positive on $[a,b]$, the eigenvalues are real.