Regular Sturm-Liouville Eigenvalue Problem

Definition

A regular Sturm-Liouville eigenvalue problem consists of the Sturm-Liouville differential equation $$ \frac{d}{dx}\left[p(x)\frac{d\phi}{dx}\right]+q(x)\phi+\lambda\sigma(x)\phi=0,\qquad a<x<b, $$ subject to the boundary conditions $$\begin{align} \beta_1\phi(a)+\beta_2\frac{d\phi}{dx}(a)&=0,\\ \beta_3\phi(b)+\beta_4\frac{d\phi}{dx}(b)&=0,\qquad\beta_i\in\mathbb{R}, \end{align}$$ 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 $\phi_n(x)$, and for $a<x<b$, $\phi_n$ has $n-1$ zeros.
4. The $\phi_n$ form a complete set, i.e., $$ f(x)\sim\sum_{n=1}^{\infty}a_n\phi_n, $$ 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 $\lambda_n\neq\lambda_m$,
$$ \int_a^b\phi_n\phi_m\sigma\,dx=0. $$

The following is the Rayleigh quotient: $$ \lambda=\frac{-p\phi\,d\phi/dx|_a^b+\int_a^b[p(d\phi/dx)^2-q\phi^2]\,dx}{\int_a^b\phi^2\sigma\,dx}. $$ Most of these theorems can be proved using Green's formula $$ \int_a^b[uL(v)-vL(u)]\,dx=p\left(u\frac{dv}{dx}-v\frac{du}{dx}\right)\Bigg|_a^b, $$ where $$ L\equiv\frac d{dx}\left(p\frac d{dx}\right)+q. $$ The Rayleigh quotient proves that $\lambda\geq0$ if $-p\phi\,d\phi/dx|_a^b\geq0$ and $q\leq0$.