Propagation effects on the breakdown of a linear amplifier model: Complex-mass Sehrödinger equation driven by the square of a Gaussian field

Solutions to the equation ∂_tε(x, t) - i/2mΔε(x, t) = λ|S(x, t)|² ε(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 at which the q^th moment of |ε(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 Δε term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.