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.Comment: 34 pages, REVTeX; very minor correction
