SturmLiouville

Sturm–Liouville theory concerns a class of linear second-order differential operators and the eigenvalue problems they generate. The standard form is to find nontrivial functions y(x) and constants λ (eigenvalues) such that

-(p(x) y')' + q(x) y = λ w(x) y

on an interval [a, b], together with specified boundary conditions at a and b. Here p, p' and q are real-valued functions, and w(x) > 0 is a weight function. The operator L[y] = -(p y')' + q y is self-adjoint with respect to the inner product ⟨f, g⟩ = ∫_a^b f(x) g(x) w(x) dx, which leads to many of the theory’s key results.

Regular and singular problems differentiate the theory’s setting. Regular problems have p, p', q, w continuous on [a, b], with p > 0 and w > 0. Singular problems occur when endpoints are infinite or coefficients are not well-behaved, yet a robust spectral theory often remains, with potential discrete and continuous spectra.

Key consequences include real eigenvalues and complete orthogonal sets of eigenfunctions. Eigenfunctions corresponding to distinct eigenvalues are orthogonal under the weight w: ∫_a^b y_m(x) y_n(x) w(x) dx = 0 for m ≠ n. The eigenvalues form a discrete, increasing sequence under regular conditions, and their eigenfunctions exhibit the Sturm separation and oscillation properties, including interlacing of zeros.

Applications of Sturm–Liouville theory are widespread in solving linear partial differential equations via separation of variables, especially in problems with spherical, cylindrical, or planar symmetry. Classical eigenfunctions arise as special cases, including Legendre, Laguerre, Hermite, and Bessel functions. Historical development is attributed to Sturm and Liouville in the 1830s.