SturmLiouvilleproblem

Sturm-Liouville problems refer to a class of boundary value problems for linear second-order differential equations with a spectral parameter. A typical regular form on an interval (a,b) is

-(p(x) y')' + q(x) y = λ w(x) y,

where p, p', q, w are real-valued functions on [a,b] with p(x) > 0 and w(x) > 0 on (a,b). The problem is completed by suitable boundary conditions, commonly separated homogeneous linear conditions such as α1 y(a) + β1 p(a) y'(a) = 0 and α2 y(b) + β2 p(b) y'(b = 0, with constants not both zero.

The differential operator L defined by L[y] = -(p y')' + q y is formally self-adjoint with respect to the weight function w, and the eigenvalue problem L[y] = λ w y is studied in the Hilbert space L^2_w([a,b]). Under regularity assumptions, L is self-adjoint and the spectrum consists of a discrete sequence of real eigenvalues λ1 < λ2 < ..., with corresponding eigenfunctions y1, y2, ..., that are orthogonal in the weighted inner product ⟨y,z⟩ = ∫_a^b y z w dx.

Key results include the Sturm oscillation theorem (the n-th eigenfunction has exactly n-1 zeros in (a,b)), and the completeness of the eigenfunctions, which allows expansion of suitable functions in a convergent series of eigenfunctions (generalized Fourier series). The Rayleigh quotient characterizes eigenvalues as minima of ∫ [p y'^2 + q y^2] dx divided by ∫ w y^2 dx.

Regular problems (finite endpoints, positive weight) are contrasted with singular problems (endpoints at infinity or p vanishing). Sturm-Liouville theory underpins many physical models, including vibrating strings and quantum mechanical radial parts.