Sturm-Liouville problem

A regular Sturm-Liouville eigenvalue problem has the formon a bounded interval with self-adjoint boundary conditions and positive weight $w$.

A Dirichlet eigenvalue is an eigenvalue for an eigenfunction constrained to vanish at the endpoints of its interval or on the boundary of its domain.

For a bounded domain $\Omega$, the first Dirichlet eigenvalue of $-\Delta$ has the variational characterization

For a regular self-adjoint Sturm-Liouville problem, eigenfunctions are orthogonal in the weighted inner productand the coefficient of $y_n$ is $⟨f,y_n{⟩}_w/⟨y_n,y_n{⟩}_w$.