A second-order linear Hermitian operator is an operator L^~ that satisfies integral_a^b v^_ L^~ u d x = integral_a^b uL^~ v^_ d x. where z^_ denotes a complex conjugate. As shown in Sturm-Liouville theory, if L^~ is self-adjoint and satisfies the boundary conditions v^_ p u' |_(x = a) = v^_ p u' |_(x = b), then it is automatically Hermitian. Hermitian operators have real eigenvalues, orthogonal eigenfunctions, and the corresponding eigenfunctions form a complete biorthogonal system when L^~ is second-order and linear.

adjoint | Hermitian matrix | self-adjoint