= Radial Dirichlet modes of a spherical cavity
{title2=$\psi_n(r)\propto\sin(n\pi r/a)/r,\quad k_n^2=n^2\pi^2/a^2$}
A bounded spherically symmetric solution of the <Helmholtz equation> in a ball with zero value at radius $a$ satisfies $-(r^2\psi')'=k^2r^2\psi$. The substitution $u=r\psi$ gives $u''+k^2u=0$ with $u(0)=u(a)=0$, so the displayed modes have $n\geq1$; the apparent singularity at zero is removable. Their <Rayleigh quotient> is $\int_0^ar^2\psi'^2dr/\int_0^ar^2\psi^2dr$. 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.
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