Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 07, pp. 1-37.
Title: Distributional asymptotic expansions of spectral functions
and of the associated Green kernels
Authors: R. Estrada (Univ. of Costa Rica)
S. A. Fulling (Texas A\&M Univ., College Station, Texas, USA)
Abstract:
Asymptotic expansions of Green functions and spectral densities
associated with partial differential operators are widely applied
in quantum field theory and elsewhere.
The mathematical properties of these expansions can be clarified
and more precisely determined by means of tools from distribution
theory and summability theory.
(These are the same, insofar as recently the classic
Cesaro--Riesz theory of summability of series and integrals has
been given a distributional interpretation.)
When applied to the spectral analysis of Green functions
(which are then to be expanded as series in a parameter, usually
the time),these methods show:
(1) The ``local'' or ``global'' dependence of
the expansion coefficients on the background geometry, etc.,
is determined by the regularity of the asymptotic expansion of the
integrand at the origin (in ``frequency space'');
this marks the difference between a heat kernel and a Wightman
two-point function, for instance.
(2) The behavior of the integrand at infinity determines whether
the expansion of the Green function is genuinely asymptotic
in the literal, pointwise sense, or is merely valid in a
distributional (Cesaro-averaged) sense;
this is the difference between the heat kernel and the
Schrodinger kernel.
(3) The high-frequency expansion of the spectral density itself is
local in a distributional sense (but not pointwise).
These observations make rigorous sense out of calculations in the
physics literature that are sometimes dismissed as merely formal.
Submitted April 29, 1998. Published March 1, 1999.
Math Subject Classification: 35P20, 40G05, 81Q10.
Key Words: Riesz means; spectral asymptotics; heat kernel; distributions.
An addendum was attached on July 22, 2005. See last page of this article.