  Although marginally more complicated than the traditional Laplace sum-rules,
Gaussian sum-rules have the advantage of being able to probe excited and ground
states with similar sensitivity. Gaussian sum-rule analysis techniques are
applied to the problematic scalar glueball channel to determine masses, widths
and relative resonance strengths of low-lying scalar glueball states
contributing to the hadronic spectral function. A feature of our analysis is
the inclusion of instanton contributions to the scalar gluonic correlation
function. Compared with the next-to-leading Gaussian sum-rule, the analysis of
the lowest-weighted sum-rule (which contains a large scale-independent
contribution from the low energy theorem) is shown to be unreliable because of
instability under QCD uncertainties. However, the presence of instanton effects
leads to approximately consistent mass scales in the lowest weighted and
next-lowest weighted sum-rules. The analysis of the next-to-leading sum-rule
demonstrates that a single narrow resonance model does not provide an adequate
description of the hadronic spectral function. Consequently, we consider a wide
variety of phenomenological models which distribute resonance strength over a
broad region---some of which lead to excellent agreement between the
theoretical prediction and phenomenological models. Including QCD
uncertainties, our results indicate that the hadronic contributions to the
spectral function stem from a pair of resonances with masses in the range
0.8--1.6 GeV, with the lighter of the two potentially having a large width.
