Almost everywhere convergence of spectral sums for self-adjoint operators

Abstract

Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let $L={\int }_0^{\infty }\lambda d{E}_L(\lambda )$ be the spectral resolution of $L$ and $S_R(L)f={\int }_0^{R}d{E}_L(\lambda )f$ denote the spherical partial sums in terms of the resolution of $L$. In this article we give a sufficient condition on $L$ such that $$\underset{R\to \infty }{\lim }{S}_R(L)f(x)=f(x),a.e.$$ for any $f$ such that $\log (2+L)f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.