 Radial Dirichlet modes of a spherical cavity

A bounded spherically symmetric solution of the Helmholtz equation in a ball with zero value at radius $a$ satisfies $-(r^2{\psi }^{\prime }{)}^{\prime }=k^2{r}^2\psi$. The substitution $u=r\psi$ gives $u^{\prime \prime }+k^{2}u=0$ with $u(0)=u(a)=0$, so the displayed modes have $n\ge 1$; the apparent singularity at zero is removable. Their Rayleigh quotient is ${\int }_0^a{r}^2{\psi }^{\prime 2}dr/{\int }_0^a{r}^2{\psi }^{2}dr$. The trial profile $1-r^2/a^2$ gives the upper bound $21/(2a^2)$, compared with the exact lowest value ${\pi }^2/a^2$. Boundedness at zero eliminates the $1/r$ solution and makes the endpoint term in the Sturm-Liouville problem vanish.