Some Properties of Riesz Means and Spectral Expansions

It is well known that short-time expansions of heat kernels correlate to formal high-frequency expansions of spectral densities. It is also well known that the latter expansions are generally not literally true beyond the first term. However, the terms in the heat-kernel expansion correspond rigorously to quantities called Riesz means of the spectral expansion, which damp out oscillations in the spectral density at high frequencies by dint of performing an average over the density at all lower frequencies. In general, a change of variables leads to new Riesz means that contain different information from the old ones. In particular, for the standard second-order elliptic operators, Riesz means with respect to the square root of the spectral parameter correspond to terms in the asymptotics of elliptic and hyperbolic Green functions associated with the operator, and these quantities contain “nonlocal” information not contained in the usual Riesz means and their correlates in the heat kernel. Here the relationship between these two sets of Riesz means is worked out in detail; this involves just classical one-dimensional analysis and calculation, with no substantive input from spectral theory or quantum field theory. This work provides a general framework for calculations that are often carried out piecemeal (and without precise understanding of their rigorous meaning) in the physics literature. 1 Physical motivation Let H be a positive, self-adjoint, elliptic, second-order partial differential operator. For temporary expository simplicity, assume that H has purely discrete spectrum, with eigenvalues λn and normalized eigenfunctions φn(x). Quantum field theorists, especially those working in curved space, are accustomed to ∗1991 Mathematics Subject Classifications: 35P20, 40G05, 81Q10.