Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 10 / i / b

For a regular self-adjoint problem with a lowest eigenvalue and a complete orthonormal eigenbasis in the weighted inner product, expand an admissible trial function as $\psi ={\sum }_j{c}_j{\psi }_j$. ThenThe Rayleigh-Ritz variational principle therefore provides the upper boundChoose a family satisfying the essential boundary conditions and finite energy, evaluate the quotient, and minimize over its parameters. Enlarging the trial family can only improve or preserve the bound; equality is attained if it includes an exact ground eigenfunction. For a finite linear trial space, this becomes a generalized matrix eigenvalue problem.