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

Abstract

Solutions to the equation ${\partial }_t\mathcal{E}(x,t)-\frac{i}{2m}\Delta \mathcal{E}(x,t)=\lambda |S(x,t){|}^2\mathcal{E}(x,t)$ 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 $\mathrm{Im}(m)\ge 0$. 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 $|\mathcal{E}(x,t)|$ w.r.t. the Gaussian field $S$ diverges. This value is found to be less or equal for all $m\ne 0$, $\mathrm{Im}(m)\ge 0$ and $|m|<+\infty$ than for $|m|=+\infty$, i.e. when the $\Delta \mathcal{E}$ term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.

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