Proof of Completeness of Sturm-Liouville Eigenfunctions

The completeness of Sturm-Liouville eigenfunctions guarantees that any square-integrable function can be expanded in a generalized Fourier series. This underpins the separation of variables technique throughout mathematical physics.

Statement

Theorem5.3Sturm-Liouville Completeness

The eigenfunctions $\{\phi_n\}_{n=1}^\infty$ of a regular Sturm-Liouville problem form a complete orthonormal system in $L^2_w([a,b])$ : for any $f \in L^2_w$ , $f = \sum_{n=1}^\infty c_n \phi_n$ where $c_n = \langle f, \phi_n\rangle_w = \int_a^b f(x)\phi_n(x)w(x)\,dx$ , with convergence in $L^2_w$ .

Proof

The proof proceeds via the theory of compact self-adjoint operators.

Step 1: The Green's operator is compact. The Sturm-Liouville operator $\mathcal{L}$ has a Green's function $G(x,\xi)$ (for any $\lambda$ not an eigenvalue). Define the integral operator $T f(x) = \int_a^b G(x,\xi)f(\xi)w(\xi)\,d\xi$ . Since $G$ is continuous on $[a,b]^2$ (for a regular problem), $T$ is a compact operator on $L^2_w([a,b])$ (Hilbert-Schmidt kernel).

Step 2: Self-adjointness. The Green's function satisfies $G(x,\xi) = G(\xi,x)$ (reciprocity, from the self-adjointness of $\mathcal{L}$ ). Therefore $T$ is a compact self-adjoint operator.

Step 3: Spectral theorem for compact self-adjoint operators. By the spectral theorem, $T$ has a countable set of eigenvalues $\{\mu_n\}$ with $\mu_n \to 0$ , and the eigenfunctions form a complete orthonormal system in the range of $T$ . Since $T\phi_n = \frac{1}{\lambda_n - \lambda_0}\phi_n$ (where $\lambda_0$ is the shift used in defining $G$ ), the eigenfunctions of $T$ are exactly $\{\phi_n\}$ .

Step 4: Completeness. If $f \perp \phi_n$ for all $n$ in $L^2_w$ , then $Tf \perp T\phi_n$ and... more directly: if $f \perp \phi_n$ for all $n$ , then $Tf = 0$ (since $f$ is orthogonal to all eigenfunctions of the compact self-adjoint operator $T$ , and the eigenspaces span the range). But $Tf = 0$ means $\int G(x,\xi)f(\xi)w(\xi)\,d\xi = 0$ for all $x$ . Since $G$ is the inverse of $\mathcal{L}$ , this implies $f$ is in the kernel of $T$ .

For $T$ corresponding to $(\mathcal{L} - \lambda_0)^{-1}$ with $\lambda_0$ not an eigenvalue: $Tf = 0$ implies $f = 0$ (since $T$ is injective: $Tf = 0 \Rightarrow f = (\mathcal{L} - \lambda_0) \cdot 0 = 0$ ).

Therefore $f = 0$ , proving that no nonzero function is orthogonal to all $\phi_n$ . This is completeness.

Step 5: Parseval's equality. For the orthonormal eigenfunctions, completeness implies $\|f\|_w^2 = \sum_{n=1}^\infty |c_n|^2$ and $f = \sum c_n \phi_n$ in $L^2_w$ for all $f \in L^2_w([a,b])$ . $\square$

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ExampleFourier Series as a Special Case

For $p = 1$ , $q = 0$ , $w = 1$ on $[0, \pi]$ with $y(0) = y(\pi) = 0$ : eigenfunctions are $\phi_n(x) = \sqrt{2/\pi}\sin(nx)$ , $\lambda_n = n^2$ . The completeness theorem gives the classical Fourier sine series expansion. The general Sturm-Liouville result shows that Legendre polynomials, Bessel functions, Hermite polynomials, etc., all provide complete bases for their respective weight functions.

RemarkSingular Sturm-Liouville Problems

For singular problems (e.g., $p(a) = 0$ or $[a,b] = [0, \infty)$ ), completeness still holds under appropriate boundary conditions (limit point/limit circle classification). The spectrum may include a continuous part: $f = \sum c_n \phi_n + \int c(\lambda)\phi_\lambda\, d\rho(\lambda)$ where $\rho$ is the spectral measure. The hydrogen atom (with a Coulomb potential on $[0,\infty)$ ) exhibits both discrete eigenvalues (bound states) and continuous spectrum (scattering states).