Abstract
We study the divergence of the solution to a Schrödinger-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 (Mounaix et al. in Commun. Math. Phys. 264:741–758, 2006, Erratum: ibid. 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 a fortiori that the breakdown of the amplifier is not affected by the possible existence of a localized regime in the amplification free limit.
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Mounaix, Ph., Collet, P., Lebowitz, J.L.: Propagation effects on the breakdown of a linear amplifier model: complex-mass Schrödinger equation driven by the square of a Gaussian field. Commun. Math. Phys. 264, 741–758 (2006). Erratum: Commun. Math. Phys. 280, 281–283 (2008)
Thouless, D.J.: Electrons in disordered systems and the theory of localization. Phys. Rep. 13, 93–142 (1974)
Nagaoka, Y., Fukuyama, H. (eds.): Anderson Localization. Springer Series in Solid-State Sciences, vol. 39. Springer, Berlin (1982)
Lee, P.A., Ramakrishnan, T.V.: Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985)
Ramakrishnan, T.V.: Electron localization. In: Souletie, J., Vannimenus, J., Stora, R. (eds.) Chance and Matter, Les Houches, Session XLVI. Elsevier, Amsterdam (1987)
Souillard, B.: Waves and electrons in inhomogeneous media. In: Souletie, J., Vannimenus, J., Stora, R. (eds.) Chance and Matter, Les Houches, Session XLVI. Elsevier, Amsterdam (1987)
John, S.: The localization of light and other classical waves in disordered media. Comments. Condens. Matter Phys. 14, 193 (1988)
Yajima, K.: Private communication (2004)
Reed, M., Simon, B.: Fourier Analysis, Self-Adjointness. Methods of Modern Mathematical Physics, vol. 2. Academic Press, New York (1975)
Conway, J.B.: Functions of One Complex Variable I. Graduate Texts in Mathematics, vol. 11. Springer, New York (1978)
Adler, J.: The Geometry of Random Fields. Wiley, New York (1981)
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Mounaix, P., Collet, P. Wave Localization Does not Affect the Breakdown of a Schrödinger-Type Amplifier Driven by the Square of a Gaussian Field. J Stat Phys 139, 466–477 (2010). https://doi.org/10.1007/s10955-010-9946-3
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DOI: https://doi.org/10.1007/s10955-010-9946-3