### Abstract

Solutions to the equation ∂_{t}ε(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.

Original language | English (US) |
---|---|

Pages (from-to) | 741-758 |

Number of pages | 18 |

Journal | Communications In Mathematical Physics |

Volume | 264 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Communications In Mathematical Physics*, vol. 264, no. 3, pp. 741-758. https://doi.org/10.1007/s00220-006-1553-4

**Propagation effects on the breakdown of a linear amplifier model : Complex-mass Sehrödinger equation driven by the square of a Gaussian field.** / Mounaix, Philippe; Collet, Pierre; Lebowitz, Joel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Propagation effects on the breakdown of a linear amplifier model

T2 - Complex-mass Sehrödinger equation driven by the square of a Gaussian field

AU - Mounaix, Philippe

AU - Collet, Pierre

AU - Lebowitz, Joel

PY - 2006/6/1

Y1 - 2006/6/1

N2 - 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 qth 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.

AB - 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 qth 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.

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U2 - https://doi.org/10.1007/s00220-006-1553-4

DO - https://doi.org/10.1007/s00220-006-1553-4

M3 - Article

VL - 264

SP - 741

EP - 758

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -