Eigenfunction Expansions and Sturm-Liouville Theory

Core Idea

Sturm-Liouville theory generalizes Fourier series by showing that a wide class of second-order boundary value problems -(p(x)y')' + q(x)y = λw(x)y produce an orthogonal basis of eigenfunctions for an appropriate function space. Any sufficiently well-behaved function can be expanded in this eigenbasis, just as functions can be expanded in sines and cosines. This framework unifies the separation of variables technique: when a PDE is separated, the spatial part typically yields a Sturm-Liouville problem whose eigenfunctions provide the building blocks for the solution.

Explainer

Sturm-Liouville theory provides the spectral framework that underlies the separation of variables method for PDEs. When we separate variables in the heat equation on a finite interval, the spatial part satisfies X'' + λX = 0 with boundary conditions—the simplest Sturm-Liouville problem. Its eigenfunctions sin(nπx/L) form an orthogonal basis, and the solution is a Fourier sine series with time-dependent coefficients that decay exponentially. Sturm-Liouville theory shows this is not a coincidence but a general phenomenon.

A regular Sturm-Liouville problem has the form -(p(x)y')' + q(x)y = λw(x)y on [a,b] with separated boundary conditions, where p > 0, w > 0, and all coefficients are continuous. The operator L[y] = -(py')' + qy is self-adjoint with respect to the weighted inner product ⟨f,g⟩ = ∫f(x)g(x)w(x)dx. Self-adjointness guarantees that all eigenvalues are real, eigenfunctions for distinct eigenvalues are orthogonal, and the eigenvalues form an unbounded increasing sequence. The eigenfunctions form a complete orthonormal basis for L²([a,b], w).

The completeness of the eigenfunctions means that any square-integrable function f can be expanded as f(x) = Σ c_n φ_n(x), where the coefficients are c_n = ⟨f, φ_n⟩/⟨φ_n, φ_n⟩. This expansion converges in the L² sense, and under additional smoothness assumptions on f, it converges uniformly. This is the generalized Fourier series, and it reduces to ordinary Fourier series when p = w = 1 and q = 0. Different choices of p, q, and w yield expansions in Legendre polynomials (spherical geometry), Bessel functions (cylindrical geometry), and other classical families.

Singular Sturm-Liouville problems arise when p vanishes at an endpoint, the interval is infinite, or the coefficients are unbounded. These require more delicate analysis but remain tractable. The Bessel equation and Legendre equation are singular Sturm-Liouville problems, and their eigenfunction expansions (Fourier-Bessel series and Legendre series) are essential for solving PDEs in cylindrical and spherical coordinates. The general spectral theorem for unbounded self-adjoint operators in Hilbert spaces provides the rigorous foundation for these expansions, connecting Sturm-Liouville theory to functional analysis.