Wave localization does not affect the breakdown of a Schr\"odinger-type amplifier driven by the square of a Gaussian field

Abstract

We study the divergence of the solution to a Schr\"odinger-type amplifier driven by the square of a Gaussian noise in presence of a random potential. We follow the same approach as Mounaix, Collet, and Lebowitz (MCL) in terms of a distributional formulation of the amplified field and the use of the Paley-Wiener theorem [Commun. Math. Phys. {\bf 264}, 741-758 (2006) and {\bf 280}, 281-283 (2008)]. Our results show that the divergence is not affected by the random potential, in the sense that it occurs at exactly the same coupling constant as what was found by MCL without a potential. It follows {\it a fortiori} that the breakdown of the amplifier is not affected by the possible existence of a localized regime in the amplification free limit.