The blowup along the diagonal of the spectral function of the Laplacian (Q331864)

Let \((M,g)\) be a compact connected Riemannian manifold of dimension \(m\). Let \(\{\psi_n,\lambda_n\}\) be a spectral resolution of the scalar Laplacian on \(M\) where \(0=\lambda_0\leq\lambda_1\leq\cdots\). Let \(\mathcal{E}_\nu=\oplus_{n\leq\nu}\psi_n\otimes\psi_n\); this converges in the sense of distributions to the Dirac delta function supported on the diagonal. In this paper, the author describes a universal law which governs the behavior of \(\mathcal{E}_\nu\) as \(\nu\rightarrow\infty\) if the manifold is real analytic. The first section provides an introduction to the matter at hand. The second section deals with the ``universality conjecture'' in the real analytic setting. The third section deals with the case of a round sphere. The appendix provides the necessary sharp elliptic estimates.