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 positive measure space. Let ${ L}=\int _0^{\infty } \lambda dE_{ L}({\lambda })$ be the spectral resolution of $L$ and $S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that \begin{equation*} \lim _{R\rightarrow \infty } S_R({ L})f(x) =f(x),\ \ \text {a.e.} \end{equation*} for any $f$ such that $\operatorname {log}(2+L) f\in L^2(X)$. 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.