Almost everywhere convergence of spectral sums for self-adjoint operators

TL;DR

This paper establishes conditions under which spectral sums of certain self-adjoint operators converge almost everywhere, extending classical results to a broad class of operators including Dirichlet, Hermite, and Schrödinger operators.

Contribution

It provides a new sufficient condition ensuring almost everywhere convergence of spectral sums for a wide class of self-adjoint operators.

Findings

Spectral sums converge almost everywhere under the given condition.

Results apply to Dirichlet, Hermite, and Schrödinger operators.

The condition involves the logarithmic integrability of the operator applied to functions.

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 ${ L}=\int_0^{\infty} \lambda dE_{ L}({\lambda})$ 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 $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$ for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schr\"odinger operators with inverse square potentials.