Consider a second-order differential operator ℒ^~ u(x) congruent p_0 (d^2 u)/(d x^2) + p_1 (d u)/(d x) + p_2 u, where u congruent u(x) and p_i congruent p_i(x) are real functions of x on the region of interest [a, b] with 2 - i continuous derivatives and with p_0(x)!=0 on [a, b]. This means that there are no singular points in [a, b]. Then the adjoint operator (ℒ^~)^† is defined by (ℒ^~)^† u | congruent | d^2/(d x^2)(p_0 u) - d/(d x)(p_1 u) + p_2 u | = | p_0 (d^2 u)/(d x^2) + (2p_0^, - p_1)(d u)/(d x) + (p_0^, , - p_1^, + p_2) u.

adjoint | Hermitian operator