Sturm-Liouville Theory

Fourier Analysis · Axiom Academy

Introduction to Sturm-Liouville theory and eigenfunction expansions 1. The Sturm-Liouville Equation Form A Sturm-Liouville problem consists of a differential equation of the form: where p(x) , q(x) , and w(x) are given functions, with w(x) > 0 (the weight function), and we seek eigenvalues and corresponding eigenfunctions y(x) . This is subject to boundary conditions at the endpoints of the interval [ a , b ]. 2. Eigenvalues and Eigenfunctions For a regular Sturm-Liouville problem with appropriate boundary conditions, there exists an infinite sequence of eigenvalues: Each eigenvalue has a corresponding eigenfunction y n (x) . These eigenfunctions form a complete set, meaning any "reasonable" function can be expanded in terms of them. 3. Orthogonality of Eigenfunctions The eigenfunctions are orthogonal with respect to the weight function w(x) . This means that for distinct eigenvalues, we have: This weighted orthogonality is fundamental to eigenfunction expansions. The weight function w(x) determines the inner product structure of the function space. 4. Generalized Fourier Series Using Eigenfunctions Any function f(x) satisfying appropriate conditions can be expanded as a generalized Fourier series in terms of the eigenfunctions: The coefficients are found using the orthogonality relation: This generalizes the classical Fourier series, where the eigenfunctions were sin(nx) and cos(nx) .

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