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

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

2005-05-23

arxiv.org/abs/math-ph/0505060v2

Commun. Math. Phys. 264, 741-758 (2006)

Physics

Mathematical Physics

REVTeX file, 21 pages, updated references and corrected typos

Scientific paper

10.1007/s00220-006-1553-4

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.

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