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Solutions to the equation $\partial_t{\cal E}(x,t)-\frac{i}{2m}\Delta {\calE}(x,t)=\lambda| S(x,t)|^2{\cal E}(x,t)$ are investigated, where $S(x,t)$ is acomplex Gaussian field with zero mean and specified covariance, and $m\ne 0$ isa complex mass with ${\rm Im}(m)\ge 0$. For real $m$ this equation describesthe backscattering of a smoothed laser beam by an optically active medium.Assuming that $S(x,t)$ is the sum of a finite number of independent complexGaussian random variables, we obtain an expression for the value of $\lambda$at which the $q$-th moment of $| {\cal E}(x,t)|$ w.r.t. the Gaussian field $S$diverges. This value is found to be less or equal for all $m\ne 0$, ${\rmIm}(m)\ge 0$ and $| m| <+\infty$ than for $| m| =+\infty$, i.e. when the$\Delta {\cal E}$ term is absent. Our solution is based on a distributionalformulation of the Feynman path-integral and the Paley-Wiener theorem.Comment: REVTeX file, 21 pages, updated references and corrected typo

Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrödinger Equation Driven by the Square of a Gaussian Field