  We identify, through a change of variables, solution operators for evolution
equations with generators given by certain simple first-order differential
operators acting on Fock spaces. This analysis applies, through unitary
equivalence, to a broad class of supersymmetric quadratic
multiplication-differentiation operators acting on $L^2(\Bbb{R}^n)$ which
includes the elliptic and weakly elliptic quadratic operators. We demonstrate a
variety of sharp results on boundedness, decay, and return to equilibrium for
these solution operators, connecting the short-time behavior with the range of
the symbol and the long-time behavior with the eigenvalues of their generators.
This is particularly striking when it allows for the definition of solution
operators which are compact and regularizing for large times for certain
operators whose spectrum is the entire complex plane.
