Dynamics of sound waves in an interacting Bose gas

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• 作者：D-A. Deckert^a ; ^{deckert@math.lmu.de" class="auth_mail" title="E-mail the corresponding author} ; J. Frö ; hlich^b ; ^{juerg@phys.ethz.ch" class="auth_mail" title="E-mail the corresponding author} ; P. Pickl^a ; ^{pickl@math.lmu.de" class="auth_mail" title="E-mail the corresponding author} ; A. Pizzo^c ; ^{pizzo@mat.uniroma2.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Interacting Bose gas ; Mean-field and large volume limit ; Effective many-body dynamics ; Effective dynamics for excitations
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：30 April 2016
• 年：2016
• 卷：293
• 期：Complete
• 页码：275-323
• 全文大小：710 K

文摘

We consider a non-relativistic quantum gas of N   bosonic atoms confined to a box of volume Λ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, width="46" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000438-si1.gif">$\rho =\frac{N}{\mathrm{\Lambda }}$, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume Λ and small ratio width="12" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000438-si2.gif">$\frac{\mathrm{\Lambda }}{\rho }$. The initial state of the gas is assumed to be close to a product state   of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of Λ. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio width="12" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000438-si2.gif">$\frac{\mathrm{\Lambda }}{\rho }$. We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.