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

TL;DR

This paper clarifies a mathematical proof related to the propagation effects on a linear amplifier model, emphasizing the conditions under which certain inequalities hold despite initial assumptions about the analyticity of functions involved.

Contribution

It corrects and clarifies the conditions for the validity of an inequality in the propagation model, refining the mathematical understanding of the complex-mass Schrödinger equation driven by Gaussian fields.

Findings

The inequality holds under broader conditions than initially stated.

The proof's validity is confirmed despite limitations in the analyticity assumptions.

Clarifies the mathematical properties of the involved functions.

Abstract

The proof of the inequality $\lambda_{q}(x,t)\le (q\mu_{x,t} -0^+)^{-1}$ [p 750, below Eq. (29)] is based on the statement that ${\cal E}(x,t;s)$ is an entire function of $s\in {\mathbb C}^M$ [see below Eq. (30)]. But according to Equation (9) and Lemma 1, all we know is that ${\cal E}(x,t;s)$ is an entire function of $k(s)\in {\mathbb R}^N$. Nevertheless, the above inequality holds, hence the proposition 1.