Almost everywhere convergence of spectral sums for self-adjoint operators

Let L be a non-negative self-adjoint operator acting on the space L²(X), where X is a positive measure space. Let L = ∫₀^∞ λdE_L(λ) be the spectral resolution of L and S_R(L)f = ∫₀^R dE_L(λ)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→∞ S_R(L)f(x) = f(x), a.e.  for any f such that log(2 + L)f ∈ L²(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.