S. A. Fulling
Some Properties of Riesz Means and Spectral Expansions
(104K, REVTeX)
ABSTRACT. 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.