Sturm–Liouville theory

Sturm–Liouville theory studies second-order linear differential operators whose eigenvalue problems use boundary conditions and a weight function. Under suitable conditions, their eigenvalues are real and their eigenfunctions are orthogonal with respect to the weight.

Sturm–Liouville operator: A differential operator of the form −(p(x)y′)′ + q(x)y, often paired with a weight function. Its differential expression supplies the structure whose eigenvalues and eigenfunctions the theory studies.
Orthogonal functions: Functions whose inner product is zero under a specified integration measure. Distinct eigenfunctions are orthogonal using the theory’s weight when its conditions hold.
Fourier–Bessel series: An expansion in Bessel functions whose orders or zeros are selected by boundary conditions. Radial boundary problems often produce Bessel eigenfunctions and their weighted expansions.
Jacques Charles François Sturm: A French mathematician known for foundational work on differential equations and oscillation. His work on zeros of solutions helped establish the theory’s oscillation results.
Boundary value problem: A differential equation problem whose solution must satisfy conditions at specified points or boundaries. Endpoint conditions help determine the spectrum and make the operator self-adjoint.
Spectral theory: The study of spectra and eigenvectors or eigenfunctions of linear operators. Sturm–Liouville problems are a foundational setting for spectral theory of differential operators.
Legendre polynomials: A sequence of orthogonal polynomials solving Legendre’s differential equation. They arise as eigenfunctions for a Sturm–Liouville problem on an interval.
Joseph Liouville: A French mathematician whose work ranged across analysis, number theory, and differential equations. His contributions to differential equations and eigenfunction expansions helped shape the theory.
Self-adjoint operator: An operator equal to its adjoint on a specified function space and domain. Self-adjointness under suitable conditions ensures real eigenvalues and orthogonal eigenfunctions.
Second-order linear differential equation: A linear differential equation involving a function and its derivatives up to order two. Sturm–Liouville equations form a structured class of these equations.

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