Deckert, D.A.; Fröhlich, J.; Pickl, P.; Pizzo, A.
(2016):
Dynamics of sound waves in an interacting Bose gas.
In: Advances in Mathematics, Vol. 293: pp. 275323

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Abstract
We consider a nonrelativistic quantum gas of N bosonic atoms confined to a box of volume Lambda in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, rho = N/Lambda of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume Lambda and small ratio Lambda/rho. The initial state of the gas is assumed to be close to a product state of oneparticle wave functions that are approximately constant throughout the box. The initial oneparticle wave function of an excitation is assumed to have a compact support independent of Lambda. We derive an effective nonlinear equation for the time evolution of the oneparticle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective nonlinear dynamics in terms of the ratio Lambda/rho. We conclude with a discussion of the dispersion law of lowenergy excitations, recovering Bogolyubov's wellknown formula for the speed of sound in the gas, and a dynamical instability for attractive twobody potentials. (C) 2016 ELSEVIER. All rights reserved.