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

Solutions to the equation ∂tE(x, t) − i 2m E(x, t) = λ|S(x, t)|2 E(x, t) are investigated, where S(x, t) is a complex Gaussian field with zero mean and specified covariance, and m = 0 is a complex mass with Im(m) ≥ 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 λ at which the q moment of |E(x, t)| w.r.t. the Gaussian field S diverges. This value is found to be less or equal for all m = 0, Im(m) ≥ 0 and |m| < +∞ than for |m| = +∞, i.e. when the E term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.