= Loss of a spectral gap under strongly degenerate radial diffusion
{title2=$\inf_{R\ne0}\frac{\int_0^1r^{n+1}R'^2\,dr}{\int_0^1rR^2\,dr}=0\quad(n>2)$}
Choose a nonzero smooth bounded profile $\phi(s)$ supported in $0\leq s<1$, constant near zero, and set $R_\varepsilon(r)=\phi(r/\varepsilon)$. Its <radial diffusion Rayleigh quotient> is
$$
\mathcal R[R_\varepsilon]
=\varepsilon^{n-2}\frac{\int_0^1s^{n+1}\phi'(s)^2\,ds}{\int_0^1s\phi(s)^2\,ds}.
$$
For $n>2$ this tends to zero, proving that there is no positive <spectral gap>. Zero is not an <eigenvalue> with the outer <Dirichlet boundary condition>: zero energy would force $R'=0$ on $0<r<1$, hence $R=0$. Thus the bottom is not attained, and a lowest exponentially decaying <fundamental diffusion mode> must not be assumed merely because the spatial region is bounded.
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