Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 06, pp. 1-39.
Title: Some properties of Riesz means and spectral expansions
Authors: S. A. Fulling (Texas A\&M Univ., College Station, Texas, USA)
R. A. Gustafson (Texas A\&M Univ., College Station, Texas, USA)
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.
Submitted April 29, 1998. Published March 1, 1999.
Math Subject Classification: 35P20, 40G05, 81Q10.
Key Words: Riesz means; spectral asymptotics; heat kernel.