We compute the Gabor matrix for Schr\"odinger-type evolution operators.
Precisely, we analyze the Heat Equation, already presented in
\cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix
which leads to better numerical evaluations. Then, using asymptotic integration
techniques, we obtain an upper bound for the Gabor matrix in one-dimension for
the generalized Heat Equation, new in the literature. Using Maple software, we
show numeric representations of the coefficients' decay. Finally, we show the
super-exponential decay of the coefficients of the Gabor matrix for the
Harmonic Repulsor, together with some numerical evaluations. This work is the
natural prosecution of the ideas presented in \cite{2012arXiv1209.0945C} and
\cite{MR2502369}.Comment: 29 pages, 7 figure
