= Radial Dirichlet spectrum in three dimensions
{title2=$\lambda_n=(n\pi/R)^2,\quad u_n(r)=\sin(n\pi r/R)/r$}
For the radial <Laplace operator> $L=-r^{-2}(r^2u')'$ on $0<r<R$, require a finite value at the origin and $u(R)=0$. The map $u\mapsto v=ru$ preserves the <norm> from $L^2((0,R),r^2dr)$ to $L^2(0,R)$ and sends $L$ to $-d^2/dr^2$. The regular endpoint condition becomes $v(0)=0$. The <self-adjoint> realization therefore corresponds to the ordinary <Dirichlet boundary conditions> for the one-dimensional second <derivative>. Its <eigenvalues> and <eigenfunctions> are displayed above, with $u_n(0)=n\pi/R$ understood by a <limit>. Their squared weighted <norm> is $R/2$. Completeness and <orthogonality> follow from the <Fourier sine series> on $(0,R)$.
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