R. Estrada and S. A. Fulling
Distributional Asymptotic Expansions of Spectral Functions and of the
Associated Green Kernels
(107K, REVTeX)
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, in so far
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 ismerely 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.