Spectral convergence of high-dimensional spheres to Gaussian spaces

We prove that the spectral structure on $N$-dimensional standard sphere of radius $(N - 1)^{1/2}$ compatible with a projection onto first $n$-coordinates converges to $n$-dimensional Gaussian space variance 1 as $N \rightarrow \infty$. also show analogue for Dirichlet eigenvalue problem ball in and halfspace space.