Sturm–Liouville theory

Summary

Sturm–Liouville theory studies the eigenvalue problems $-(p y')' + q y = \lambda w y$ on an interval, with boundary conditions at the ends. For a regular problem with separated boundary conditions, the eigenvalues are real and simple and form a sequence $\lambda_1 < \lambda_2 < \cdots$ that tends to $+\infty$. The $n$-th eigenfunction has exactly $n - 1$ zeros inside the interval, and the eigenfunctions form an orthogonal basis of the weighted space $L^2((a, b), w)$. The theory explains Fourier series and many classical special functions, and it is the model case of the spectral theory of self-adjoint operators.