Loss of a spectral gap under strongly degenerate radial diffusion (${\inf }_{R\ne 0}\frac{{\int }_0^1{r}^{n+1}{R}^{\prime 2}dr}{{\int }_0^{1}r{R}^{2}dr}=0(n>2)$)

Choose a nonzero smooth bounded profile $\varphi (s)$ supported in $0\le s<1$, constant near zero, and set $R_{\epsilon }(r)=\varphi (r/\epsilon )$. Its radial diffusion Rayleigh quotient isFor $n>2$ 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 $R^{\prime }=0$ 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.