Given a sourceful Sturm-Liouville-type differential equation, these relations may be found in general for the adjoint source moment of the forward field or the forward source moment of the adjoint field, which are equal, and work by generating a second-order approximate value from a first-order approximation to an eigenfunction, using the completeness of an operator's unknown eigenspectrum.  
The Rousopolos variety of these equations suffers the limitation that it does not implicitly cancel normalizations, so care must be taken that the substituted shape fields have the correct amplitude.  Given a sourceful Sturm-Liouville-type differential equation, these relations may be found in general for the adjoint source moment of the forward field or the forward source moment of the adjoint field, which are equal, and work by generating a second-order approximate value from a first-order approximation to an eigenfunction, using the completeness of an operator's unknown eigenspectrum.  FTP, identify these widely applicable principles, best known in the Schwinger type, which utilize the namesake type of calculus to generate approximate eigenvalues.
The Rousopolos variety of these equations suffers the limitation that it does not implicitly cancel normalizations, so care must be taken that the substituted shape fields have the correct amplitude.  
variational principles