Sturm-Liouville Theory

Sturm-Liouville problems arise throughout mathematical physics whenever separation of variables is applied to PDEs. The theory provides a complete orthogonal basis of eigenfunctions, generalizing Fourier analysis to arbitrary domains and potentials.

The Sturm-Liouville Problem

Definition5.4Regular Sturm-Liouville Problem

A regular Sturm-Liouville problem on $[a,b]$ is: $-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + q(x)y = \lambda w(x) y$ with boundary conditions $\alpha_1 y(a) + \alpha_2 y'(a) = 0$ , $\beta_1 y(b) + \beta_2 y'(b) = 0$ , where $p(x) > 0$ , $w(x) > 0$ (weight function), and $q(x)$ is continuous on $[a,b]$ . The operator $\mathcal{L}y = -\frac{1}{w}[(py')' - qy]$ is self-adjoint with respect to the inner product $\langle f, g \rangle = \int_a^b f(x)g(x)w(x)\,dx$ .

ExampleSturm-Liouville in Quantum Mechanics

The time-independent Schrodinger equation $-\frac{\hbar^2}{2m}\psi'' + V(x)\psi = E\psi$ on $[0, L]$ with $\psi(0) = \psi(L) = 0$ is a Sturm-Liouville problem with $p = \hbar^2/(2m)$ , $q = V(x)$ , $w = 1$ , $\lambda = E$ . For the infinite square well ( $V = 0$ ): eigenvalues $E_n = n^2\pi^2\hbar^2/(2mL^2)$ , eigenfunctions $\psi_n = \sqrt{2/L}\sin(n\pi x/L)$ .

Eigenvalue Theory

Definition5.5Sturm-Liouville Eigenvalue Properties

For a regular Sturm-Liouville problem: (1) eigenvalues form an increasing sequence $\lambda_1 < \lambda_2 < \cdots \to \infty$ ; (2) eigenfunctions $\phi_n$ corresponding to distinct eigenvalues are orthogonal: $\langle\phi_m, \phi_n\rangle_w = 0$ for $m \neq n$ ; (3) $\phi_n$ has exactly $n - 1$ zeros in $(a,b)$ (Sturm oscillation theorem); (4) the eigenfunctions form a complete orthonormal basis in $L^2_w([a,b])$ .

RemarkGreen's Functions for Sturm-Liouville

The Green's function $G(x, \xi; \lambda)$ solves $\mathcal{L}G - \lambda wG = \delta(x - \xi)$ and provides the resolvent: $(\mathcal{L} - \lambda)^{-1}f(x) = \int_a^b G(x,\xi;\lambda)f(\xi)w(\xi)\,d\xi$ . The eigenfunction expansion of $G$ : $G(x,\xi;\lambda) = \sum_n \frac{\phi_n(x)\phi_n(\xi)}{\lambda_n - \lambda}$ , with poles at the eigenvalues, connecting the spectral theory to integral operators.