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

Write the energy form as $Q[\psi ]={\int }_{\alpha }^{\beta }(p|{\psi }^{\prime }{|}^2+q|\psi {|}^2)dx$ and the weighted squared norm as $N[\psi ]={\int }_{\alpha }^{\beta }w|\psi {|}^{2}dx$. For real coefficients and appropriate homogeneous boundary conditions, the Sturm-Liouville problem is the stationary-point equation for their Rayleigh quotient $Q/N$.

One suitable choice is homogeneous Dirichlet boundary conditions, $\psi (\alpha )=\psi (\beta )=0$, with variations vanishing there. Assume real $q$ and the stated positive $p,w$. Introduce a Lagrange multiplier $\mu$ for $N[\psi ]=1$. For real variations, integration by parts givesThe boundary term vanishes, so the fundamental lemma of the calculus of variations yieldsMultiplying this equation by $\psi$, integrating, and using the boundary conditions gives $Q[\psi ]=\mu N[\psi ]=\mu$. Thus the multiplier, the eigenvalue, and the constrained stationary value coincide. Conversely every normalized eigenfunction makes the first variation vanish for all tangent variations satisfying $\delta N=0$.

Homogeneous Neumann boundary conditions, $p{\psi }^{\prime }=0$ at both endpoints, or mixed Dirichlet and Neumann conditions also work with this energy form. General Robin boundary conditions require the corresponding boundary terms in the form; they cannot simply be inserted into the displayed unmodified integral.

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.