Abstract
We study the spectrum of a large system of N identical bosons interacting via a two-body potential with strength 1/N. In this mean-field regime, Bogoliubov’s theory predicts that the spectrum of the N-particle Hamiltonian can be approximated by that of an effective quadratic Hamiltonian acting on Fock space, which describes the fluctuations around a condensed state. Recently, Bogoliubov’s theory has been justified rigorously in the case that the low-energy eigenvectors of the N-particle Hamiltonian display complete condensation in the unique minimizer of the corresponding Hartree functional. In this paper, we shall justify Bogoliubov’s theory for the high-energy part of the spectrum of the N-particle Hamiltonian corresponding to (non-linear) excited states of the Hartree functional. Moreover, we shall extend the existing results on the excitation spectrum to the case of non-uniqueness and/or degeneracy of the Hartree minimizer. In particular, the latter covers the case of rotating Bose gases, when the rotation speed is large enough to break the symmetry and to produce multiple quantized vortices in the Hartree minimizer.
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References
Aftalion, A.: Vortices in Bose–Einstein Condensates. In series: Progress in nonlinear differential equations and their applications, vol. 67. Springer, Birkhäuser Basel (2006)
Aschbacher W., Fröhlich J., Graf G., Schnee K., Troyer M.: Symmetry breaking regime in the nonlinear Hartree equation. J. Math. Phys. 43, 3879–3891 (2002)
Bogoliubov N.N.: On the theory of superfluidity. J. Phys. (USSR) 11, 23 (1947)
Dereziński, J., Napiórkowski, M.: Excitation spectrum of interacting bosons in the mean-field infinite-volume limit. Ann. Henri Poincaré (to appear). arXiv:1305.3641
Fannes M., Spohn H., Verbeure A.: Equilibrium states for mean field models. J. Math. Phys. 21, 355–358 (1980)
Fetter A.: Rotating trapped Bose–Einstein condensates. Rev. Mod. Phys. 81, 647–691 (2009)
Grech P., Seiringer R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 322, 559–591 (2013)
Guo Y., Seiringer R.: On the mass concentration for Bose–Einstein condensates with attractive interactions. Lett. Math. Phys. 104, 141–156 (2014)
Hudson, R.L., Moody, G.R.: Locally normal symmetric states and an analogue of de Finetti’s theorem. Z. Wahrscheinlichkeitstheorie Verw. Geb. 33, 343–351 (1975/76)
Lewin M.: Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260, 3535–3595 (2011)
Lewin M., Nam P.T., Rougerie N.: Derivation of Hartree’s theory for generic mean-field Bose gases. Adv. Math. 254, 570–621 (2014)
Lewin, M., Nam, P.T., Serfaty, S., Solovej, J.P.: Bogoliubov spectrum of interacting Bose gases. Commun. Pure Appl. Math. (2014). doi:10.1002/cpa.21519
Petz D., Raggio G.A., Verbeure A.: Asymptotics of Varadhan-type and the Gibbs variational principle. Commun. Math. Phys. 121, 271–282 (1989)
Raggio G.A., Werner R.F.: Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta 62, 980–1003 (1989)
Reed M., Simon B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, (1975)
Reed M., Simon B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)
Seiringer R.: Gross–Pitaevskii theory of the rotating Bose gas. Commun. Math. Phys. 229, 491–509 (2002)
Seiringer R.: Ground state asymptotics of a dilute, rotating gas. J. Phys. A 36, 9755–9778 (2003)
Seiringer R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)
Størmer E.: Symmetric states of infinite tensor products of C *-algebras. J. Funct. Anal. 3, 48– (1969)
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Nam, P.T., Seiringer, R. Collective Excitations of Bose Gases in the Mean-Field Regime. Arch Rational Mech Anal 215, 381–417 (2015). https://doi.org/10.1007/s00205-014-0781-6
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DOI: https://doi.org/10.1007/s00205-014-0781-6