Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 10

Let $R[y]=F[y]/G[y]$, where $G[y]>0$ for a nonzero admissible real function. Writing $\lambda =R[y]$, its first variation isChoose boundary conditions making the boundary term vanish, for example homogeneous Dirichlet boundary conditions, homogeneous Neumann boundary conditions, or a corresponding mixed pair. Arbitrary interior variations then give the Sturm-Liouville problemMultiplying this equation by $y$ and integrating by parts, with $[py{y}^{\prime }{]}_{\alpha }^{\beta }=0$, yieldsThus eigenfunctions are stationary directions of the quotient. The lowest eigenvalue is its global minimum; higher eigenfunctions need not be unrestricted maxima or minima. General Robin boundary conditions require including their boundary energy in the quotient unless the displayed endpoint products also vanish.

The Rayleigh-Ritz method chooses admissible trial functions depending on finitely many coefficients and minimizes this quotient over them. Expansion in a complete orthonormal basis with weight $w$ shows $R[y]\ge {\lambda }_0$, so the trial minimum is an upper bound on the true lowest eigenvalue. In a finite basis, the minimization becomes a generalized eigenvalue problem for the stiffness and weighted mass matrices. Enlarging the trial space can only lower or preserve this upper bound.

For a spherically symmetric function, the Laplacian in spherical coordinates reduces toMultiplying the Helmholtz equation by $r^2$ givesThe endpoint at $r=0$ is singular in Sturm-Liouville terminology; bounded regular radial fields select the physical solution and eliminate the $1/r$ branch. At $r=a$ the field has the imposed zero Dirichlet value. For regular solutions and admissible trial functions, the endpoint term $r^2\psi {\psi }^{\prime }$ vanishes at the origin and at the wall.

The radial Rayleigh quotient isFor the given trial profile, the numerator is $4a^{-4}{\int }_0^a{r}^{4}dr=4a/5$. The denominator isThereforeThe profile is regular at the origin and vanishes at the wall, so it belongs to the admissible trial space. This is the variational estimate for the radial Dirichlet modes of a spherical cavity.

Set $\psi (r)=u(r)/r$ for $r>0$. Direct differentiation gives $(r^2{\psi }^{\prime }{)}^{\prime }=r{u}^{\prime \prime }$, so the radial equation reduces toBoundedness of $\psi$ requires $u(0)=0$, and the wall condition requires $u(a)=0$. The nontrivial solutions are consequentlyTheir origin values are the removable limits $C_{j}j\pi /a$. At $k=0$, the only bounded radial solution is constant and the wall forces it to vanish, so there is no zero mode. The lowest eigenvalue isIt is below $10.5/a^2$, as required by the Rayleigh-Ritz variational principle. The trial value overestimates $k_0^2$ by approximately $\boxed{6.39\%}$.