Spectral expansion of the Riemannian heat kernel
= Spectral expansion of the Riemannian heat kernel
{title2=$H(t,x,y)=\sum_j e^{-t\lambda_j}\phi_j(x)\overline{\phi_j(y)}$}
On a <closed manifold> with a <Riemannian metric>, an <orthonormal eigenbasis> for the <positive Laplace-Beltrami operator> gives this formula, with <eigenvalues> counted with multiplicity. Polynomial elliptic bounds and exponential time decay give convergence of every derivative away from time zero. The conjugate is necessary for a complex basis.