Abstract
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length a and the radius r of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g ~ a/r 2 is derived. Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ~ r −2 above the ground state energy.
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References
Baumgartner B., Solovej J.P., Yngvason J.: Atoms in Strong Magnetic Fields: The High Field Limit at Fixed Nuclear Charge. Commun. Math. Phys. 212, 703–724 (2000)
Bergeman T., Moore M.G., Olshanii M.: Atom-Atom Scattering under Cylindrical Harmonic Confinement: Numerical and Analytic Studies of the Confinement Induced Resonance. Phys. Rev. Lett. 91, 163201 (2003)
Bloch, I., Dalibard, J., Zwerger, W.: Many-Body Physics with Ultracold Gases. http://arxiv.org/list/0704.3011, 2007
Brummelhuis, R., Duclos, P.: Effective Hamiltonians for atoms in very strong magnetic fields. J. Math. Phys. 47, 033501 (2006); On the One-Dimensional Behaviour of Atoms in Intense Homogeneous Magnetic Fields. In: Partial Differential Equations and Spectral Theory, PDE2000 Conference in Clausthal, Germany, Demuth, M., Schulze, B.-W. (eds.), Basel: Birkhäuser 2001, pp. 25–35
Dettmer S. et al.: Observation of Phase Fluctuations in Elongated Bose-Einstein Condensates. Phys. Rev. Lett. 87, 160406 (2001)
Dunjko V., Lorent V., Olshanii M.: Bosons in Cigar-Shaped Traps: Thomas-Fermi Regime, Tonks- Girardeau Regime, and In Between. Phys. Rev. Lett. 86, 5413–5416 (2001)
Esteve J. et al.: Observations of Density Fluctuations in an Elongated Bose Gas: Ideal Gas and Quasicondensate Regimes. Phys. Rev. Lett. 96, 130403 (2006)
Girardeau M.: Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension. J. Math. Phys. 1, 516–523 (1960)
Jackson A.D., Kavoulakis G.M.: Lieb Mode in a Quasi-One-Dimensional Bose-Einstein Condensate of Atoms. Phys. Rev. Lett. 89, 070403 (2002)
Kinoshita T., Wenger T., Weiss D.S.: Observation of a One-Dimensional Tonks-Girardeau Gas. Science 305, 1125–1128 (2004)
Lieb E.H.: Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum. Phys. Rev. 130, 1616–1624 (1963)
Lieb E.H., Liniger W.: Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605–1616 (1963)
Lieb, E.H., Loss, M.: Analysis. Second edition, Providene, RI: American Math Soc. 2001
Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation. Oberwolfach Seminars, Vol. 34, Basel-Boston: Birkhäuser 2005
Lieb E.H., Seiringer R., Yngvason J.: Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A. 61, 043602 (2000)
Lieb E.H., Seiringer R., Yngvason J.: One-dimensional Bosons in Three-dimensional Traps. Phys. Rev. Lett. 91, 150401 (2003)
Lieb E.H., Seiringer R., Yngvason J.: One-Dimensional Behavior of Dilute, Trapped Bose Gases. Commun. Math. Phys. 244, 347–393 (2004)
Lieb E.H., Yngvason J.: Ground State Energy of the Low Density Bose Gas. Phys. Rev. Lett. 80, 2504–2507 (1998)
Moritz H., Stöferle T., Köhl M., Esslinger T.: Exciting Collective Oscillations in a Trapped 1D Gas. Phys. Rev. Lett. 91, 250402 (2003)
Moritz H., Stöferle T., Günter K., Köhl M., Esslinger T.: Confinement Induced Molecules in a 1D Fermi Gas. Phys. Rev. Lett. 94, 210401 (2005)
Olshanii M.: Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons. Phys. Rev. Lett. 81, 938–941 (1998)
Olsahnii M., Dunjko V.: Short-Distance Correlation Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases. Phys. Rev. Lett. 91, 090401 (2003)
Petrov D.S., Shlyapnikov G.V., Walraven J.T.M.: Regimes of Quantum Degeneracy in Trapped 1D Gases. Phys. Rev. Lett. 85, 3745–3749 (2000)
Petrov D.S., Gangardt D.M., Shlyapnikov G.V.: Low-dimensional trapped gases. J. Phys. IV. 116, 5–46 (2004)
Reed M., Simon B.: Methods of Modern Mathematical Physics II Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)
Richard S. et al.: Momentum Spectroscopy of 1D Phase Fluctuations in Bose-Einstein Condensates. Phys. Rev. Lett. 91, 010405 (2003)
Temple G.: The theory of Rayleigh’s Principle as Applied to Continuous Systems. Proc. Roy. Soc. London A. 119, 276–293 (1928)
Tolra B.L. et al.: Observation of Reduced Three-Body Recombination in a Correlated 1D Degenerate Bose Gas. Phys. Rev. Lett. 92, 190401 (2004)
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Seiringer, R., Yin, J. The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions. Commun. Math. Phys. 284, 459–479 (2008). https://doi.org/10.1007/s00220-008-0521-6
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DOI: https://doi.org/10.1007/s00220-008-0521-6