Sturm-Liouville Problem: Homogeneous Boundary Conditions Essential Role

This lecture delves into the Sturm-Liouville eigenvalue problem, a second-order ODE with homogeneous boundary conditions over an interval. It explores the self-adjointness of the operator, the weight function, orthogonal eigenfunctions, and the orthogonal basis they provide. The lecture emphasizes the significance of boundary conditions in ensuring self-adjointness and the formation of an orthogonal basis. It discusses the spectral theorem, reality of eigenvalues, orthogonality of eigenfunctions, and the countable discrete eigenvalues in the Sturm-Liouville spectra.