Regular Sturm-Liouville Eigenvalue Problem

Definition

A regular Sturm-Liouville eigenvalue problem consists of the Sturm-Liouville differential equation $$\frac{d}{dx}[p(x)\frac{d\varphi }{dx}]+q(x)\varphi +\lambda \sigma (x)\varphi =0,a<x<b,$$ subject to the boundary conditions $$\begin{array}{rl}{\beta }_1\varphi (a)+{\beta }_2\frac{d\varphi }{dx}(a)& =0,\\ {\beta }_3\varphi (b)+{\beta }_4\frac{d\varphi }{dx}(b)& =0,{\beta }_i\in \mathbb{R},\end{array}$$ where $p$, $q$, and $\sigma$ are real and continuous, and both $p>0$ and $\sigma >0$. Moreover, only Dirichlet, Neumann and Robin boundary conditions are considered.

Theorems

The following theorems about it have been derived:

1. Each eigenvalue ${\lambda }_n\in \mathbb{R}$
2. ${\lambda }_1<{\lambda }_2<\cdots$
3. ${\lambda }_n$ has eigenfunction ${\varphi }_n(x)$, and for $a<x<b$, ${\varphi }_n$ has $n-1$ zeros.
4. The ${\varphi }_n$ form a complete set, i.e., $$f(x)\sim \sum _{n=1}^{\mathrm{\infty }}{a}_n{\varphi }_n,$$ where $f$ is piecewise smooth. Also, with properly chosen $a_n$, this converges to $[f(x^{+})+f(x^{-})]/2$ on $a<x<b$.
5. If ${\lambda }_n\ne {\lambda }_m$,
$${\int }_a^b{\varphi }_n{\varphi }_m\sigma dx=0.$$

The following is the Rayleigh quotient: $$\lambda =\frac{-p\varphi d\varphi /dx{|}_a^b+{\int }_a^b[p(d\varphi /dx{)}^2-q{\varphi }^2]dx}{{\int }_a^b{\varphi }^2\sigma dx}.$$ Most of these theorems can be proved using Green's formula $${\int }_a^b[uL(v)-vL(u)]dx=p(u\frac{dv}{dx}-v\frac{du}{dx}){|}_a^b,$$ where $$L\equiv \frac{d}{dx}\left(p\frac{d}{dx}\right)+q.$$ The Rayleigh quotient proves that $\lambda \ge 0$ if $-p\varphi d\varphi /dx{|}_a^b\ge 0$ and $q\le 0$.