Propagation Effects on the Breakdown of a Linear Amplifier Model:   Complex-Mass Schrodinger Equation Driven by the Square of a Gaussian Field
  Solutions to the equation $\partial_t{\cal E}(x,t)-\frac{i}{2m}\Delta {\cal E}(x,t)=\lambda| S(x,t)|^2{\cal E}(x,t)$ are investigated, where $S(x,t)$ is a complex Gaussian field with zero mean and specified covariance, and $m\ne 0$ is a complex mass with ${\rm Im}(m)\ge 0$. For real $m$ this equation describes the 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 complex Gaussian 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 $m\ne 0$, ${\rm Im}(m)\ge 0$ and $| m| <+\infty$ than for $| m| =+\infty$, i.e. when the $\Delta {\cal E}$ term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem. 
Philippe Mounaix, Pierre Collet and Joel L. Lebowitz