Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-25T19:45:15.911Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic expansion of low-energy excitations for weakly interacting bosons

Part of: General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  26 March 2021

Lea Boßmann Open the ORCID record for Lea Boßmann [Opens in a new window] ,
Sören Petrat Open the ORCID record for Sören Petrat [Opens in a new window]  and
Robert Seiringer Open the ORCID record for Robert Seiringer [Opens in a new window]
Lea Boßmann
Affiliation:
Institute of Science and Technology Austria, Am Campus 1, 3400Klosterneuburg, Austria; E-mail: lea.bossmann@ist.ac.at.
Sören Petrat
Affiliation:
Department of Mathematics and Logistics, Jacobs University Bremen, Campus Ring 1, 28759Bremen, Germany; E-mail: s.petrat@jacobs-university.de. University of Bremen, Department 3 – Mathematics, Bibliothekstr. 5, 28359Bremen, Germany.
Robert Seiringer
Affiliation:
Institute of Science and Technology Austria, Am Campus 1, 3400Klosterneuburg, Austria; E-mail: robert.seiringer@ist.ac.at.

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in $1/N$.

MSC classification

Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e28
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Adhikari, A., Brennecke, C. and Schlein, B., ‘Bose–Einstein condensation beyond the Gross–Pitaevskii regime’, Ann. Henri Poincaré (NN) (2020).Google Scholar
Beliaev, S., ‘Application of the methods of quantum field theory to a system of bosons’, Sov. Phys. J. Exper. Theoret. Phys., 34(2) (1958), 289–299.Google Scholar
Beliaev, S., ‘Energy spectrum of a non-ideal Bose gas’, Sov. Phys. J. Exper. Theoret. Phys., 34(2) (1958), 299–307.Google Scholar
Boccato, C., Brennecke, C., Cenatiempo, S. and Schlein, B., ‘Complete Bose–Einstein condensation in the Gross–Pitaevskii regime’, Comm. Math. Phys. 359(3) (2018), 975–1026.Google Scholar
Boccato, C., Brennecke, C., Cenatiempo, S. and Schlein, B., ‘Bogoliubov theory in the Gross–Pitaevskii limit’, Acta Math. 222(2) (2019), 219–335.Google Scholar
Boccato, C., Brennecke, C., Cenatiempo, S. and Schlein, B., ‘The excitation spectrum of Bose gases interacting through singular potentials’, J. Eur. Math. Soc. (JEMS) 22(7) (2020), 2331–2403.Google Scholar
Boccato, C., Brennecke, C., Cenatiempo, S. and Schlein, B., ‘Optimal rate for Bose–Einstein condensation in the Gross-Pitaevskii regime’, Comm. Math. Phys. 376 (2020), 1311–1395.Google Scholar
Boccato, C., Cenatiempo, S. and Schlein, B., ‘Quantum many-body fluctuations around nonlinear Schrödinger dynamics’, Ann. Henri Poincaré 18(1) (2017), 113–191.Google Scholar
Bogoliubov, N. N., ‘On the theory of superfluidity’, Izv. Akad. Nauk Ser. Fiz. 11 (1947), 23–32.Google Scholar
Boßmann, L., Pavlović, N., Pickl, P. and Soffer, A., ‘Higher order corrections to the mean-field description of the dynamics of interacting bosons’, J. Stat. Phys. 178(6) (2020), 1362–1396.Google Scholar
Boßmann, L., Petrat, S., Pickl, P. and Soffer, A., ‘Beyond Bogoliubov dynamics’, Preprint, 2019, arXiv:1912.11004.Google Scholar
Braaten, E., Hammer, H.-W. and Hermans, S., ‘Nonuniversal effects in the homogeneous Bose gas’, Phys. Rev. A 63(6) (2001), 063609.Google Scholar
Braaten, E. and Nieto, A., ‘Quantum corrections to the energy density of a homogeneous Bose gas’, Euro. Phys. J. B 11(1) (1999), 143–159.Google Scholar
Brennecke, C., Nam, P. T., Napiórkowski, M. and Schlein, B., ‘Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation’, Ann. Inst. H. Poincaré Anal. Non Linéaire 36(5) (2019), 1201–1235.Google Scholar
Brietzke, B., Fournais, S. and Solovej, J. P., ‘A simple 2nd order lower bound to the energy of dilute Bose gases’, Comm. Math. Phys. 376 (2020), 323–351.Google Scholar
Brietzke, B. and Solovej, J. P., ‘The second-order correction to the ground state energy of the dilute Bose gas’, Ann. Henri Poincaré 21 (2020), 571–626.Google Scholar
Brueckner, K. and Sawada, K., ‘Bose–Einstein gas with repulsive interactions: General theory’, Phys. Rev. 106(6) (1957), 1117–1127.Google Scholar
Brueckner, K. and Sawada, K., ‘Bose–Einstein gas with repulsive interactions: Hard spheres at high density’, Phys. Rev. 106(6) (1957), 1128–1135.Google Scholar
Cenatiempo, S. and Giuliani, A., ‘Renormalization theory of a two dimensional Bose gas: Quantum critical point and quasi-condensed state’, J. Stat. Phys. 157 (2014), 755–829.Google Scholar
Chong, J., ‘Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing NLS in ${\mathbb{R}}^3$’, Preprint, 2016, arXiv:1608.01615.Google Scholar
Dereziński, J. and Napiórkowski, M., ‘Excitation spectrum of interacting bosons in the mean-field infinite-volume limit’, Ann. Henri Poincaré 15(12) (2014), 2409–2439.Google Scholar
Erdős, L., Schlein, B. and Yau, H.-T., ‘Ground-state energy of a low-density Bose gas: A second-order upper bound’, Phys. Rev. A 78 (2008).Google Scholar
Fournais, S. and Solovej, J. P., ‘The energy of dilute Bose gases’, Ann. Math. 192(3) (2020), 893–976.Google Scholar
Ginibre, J. and Velo, G., ‘The classical field limit of non-relativistic bosons. II. Asymptotic expansions for general potentials’, Ann. Inst. H. Poincaré Phys. Théor. 33(4) (1980), 363–394.Google Scholar
Ginibre, J. and Velo, G., ‘The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials’, Ann. Phys. 128(2) (1980), 243–285.Google Scholar
Giuliani, A. and Seiringer, R., ‘The ground state energy of the weakly interacting Bose gas at high density’, J. Stat. Phys. 135 (2009), 915–934.Google Scholar
Grech, P. and Seiringer, R., ‘The excitation spectrum for weakly interacting bosons in a trap’, Comm. Math. Phys. 322(2) (2013), 559–591.Google Scholar
Grillakis, M. and Machedon, M., ‘Pair excitations and the mean field approximation of interacting bosons, I’, Comm. Math. Phys. 324(2) (2013), 601–636.Google Scholar
Grillakis, M. and Machedon, M., ‘Pair excitations and the mean field approximation of interacting bosons, II’, Comm. Partial Differential Equations 42(1) (2017), 24–67.Google Scholar
Grillakis, M., Machedon, M. and Margetis, D., ‘Second-order corrections to mean field evolution of weakly interacting bosons, I’, Comm. Math. Phys. 294(1) (2010), 273–301.Google Scholar
Grillakis, M., Machedon, M. and Margetis, D., ‘Second-order corrections to mean field evolution of weakly interacting bosons, II’, Adv. Math. 228(3) (2011), 1788–1815.Google Scholar
Hugenholtz, N. and Pines, D., ‘Ground-state energy and excitation spectrum of a system of interacting bosons’, Phys. Rev. 116(3) (1959), 489–506.Google Scholar
Kuz, E., ‘Exact evolution versus mean field with second-order correction for bosons interacting via short-range two-body potential’, Differential Integral Equations 30(7/8) (2017), 587–630.Google Scholar
Lewin, M., Nam, P. T. and Rougerie, N., ‘Derivation of Hartree’s theory for generic mean-field Bose systems’, Adv. Math. 254 (2014), 570–621.Google Scholar
Lewin, M., Nam, P. T. and Schlein, B., ‘Fluctuations around Hartree states in the mean field regime’, Amer. J. Math. 137(6) (2015), 1613–1650.Google Scholar
Lewin, M., Nam, P. T., Serfaty, S. and Solovej, J. P., ‘Bogoliubov spectrum of interacting Bose gases’, Comm. Pure Appl. Math. 68(3) (2015), 413–471.Google Scholar
Lieb, E. H. and Seiringer, R., ‘Proof of Bose–Einstein condensation for dilute trapped gases’, Phys. Rev. Lett. 88(17) (2002), 170409.Google ScholarPubMed
Lieb, E. H. and Seiringer, R., ‘Derivation of the Gross–Pitaevskii equation for rotating Bose gases’, Comm. Math. Phys. 264(2) (2006), 505–537.Google Scholar
Lieb, E. H., Seiringer, R., Solovej, J. P. and Yngvason, J., The Mathematics of the Bose Gas and Its Condensation (Birkhäuser, Basel, 2005).Google Scholar
Lieb, E. H., Seiringer, R. and Yngvason, J., ‘Bosons in a trap: A rigorous derivation of the Gross–Pitaevskii energy functional’, Phys. Rev. A 61(4) (2000), 043602.Google Scholar
Lieb, E. H. and Solovej, J. P., ‘Ground state energy of the one-component charged Bose gas’, Comm. Math. Phys. 217 (2001), 127–163.Google Scholar
Lieb, E. H. and Solovej, J. P., ‘Ground state energy of the two-component charged Bose gas’, Comm. Math. Phys. 252 (2004), 485–534.Google Scholar
Lieb, E. H. and Yngvason, J., ‘Ground state energy of the low density Bose gas’, Phys. Rev. Lett. 80(12) (1998), 2504–2507.Google Scholar
Mitrouskas, D., Derivation of Mean Field Equations and Their Next-Order Corrections: Bosons and Fermions, PhD thesis, LMU Munich, 2017.Google Scholar
Mitrouskas, D., Petrat, S. and Pickl, P., ‘Bogoliubov corrections and trace norm convergence for the Hartree dynamics’, Rev. Math. Phys. 31(8) (2019), 1–36.Google Scholar
Nam, P. T., ‘Bogoliubov theory and bosonic atoms’, Preprint, 2011, arXiv:1109.2875.Google Scholar
Nam, P. T., Contributions to the Rigorous Study of the Structure of Atoms, PhD thesis, University of Copenhagen, 2011.Google Scholar
Nam, P. T., ‘Binding energy of homogeneous Bose gases’, Lett. Math. Phys. 108(1) (2018), 141–159.Google Scholar
Nam, P. T. and Napiórkowski, M., ‘Bogoliubov correction to the mean-field dynamics of interacting bosons’, Adv. Theor. Math. Phys. 21(3) (2017), 683–738.Google Scholar
Nam, P. T. and Napiórkowski, M., ‘A note on the validity of Bogoliubov correction to mean-field dynamics’, J. Math. Pures Appl. (9) 108(5) (2017), 662–688.Google Scholar
Nam, P. T. and Napiórkowski, M., ‘Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas’, Preprint, 2020, arXiv:2010.03595.Google Scholar
Nam, P. T., Napiórkowski, M., Ricaud, J. and Triay, A., ‘Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime’, Preprint, 2020, arXiv:2001.04364.Google Scholar
Nam, P. T., Napiórkowski, M. and Solovej, J. P., ‘Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations’, J. Funct. Anal. 270(11) (2016), 4340–4368.Google Scholar
Nam, P. T., Rougerie, N. and Seiringer, R., ‘Ground states of large bosonic systems: The Gross–Pitaevskii limit revisited’, Anal. PDE 9(2) (2016), 459–485.Google Scholar
Nam, P. T. and Seiringer, R., ‘Collective excitations of Bose gases in the mean-field regime’, Arch. Ration. Mech. Anal. 215 (2015), 381–417.Google Scholar
Napiórkowski, M., ‘Recent advances in the theory of Bogoliubov Hamiltonians’ in Workshop on Macroscopic Limits of Quantum Systems (Springer, Heidelberg, 2017), 101–121.Google Scholar
Paul, T. and Pulvirenti, M., ‘Asymptotic expansion of the mean-field approximation’, Discrete Contin. Dyn. Syst. 39(4) (2019), 1891–1921.Google Scholar
Petrat, S., Pickl, P. and Soffer, A., ‘Derivation of the Bogoliubov time evolution for a large volume mean-field limit’, Ann. Henri Poincaré, 21(2) (2020), 461–498.Google Scholar
Pizzo, A., ‘Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian’, Preprint, 2015, arXiv:1511.07022.Google Scholar
Pizzo, A., ‘Bose particles in a box II. A convergent expansion of the ground state of the Bogoliubov Hamiltonian in the mean field limiting regime’, Preprint, 2015, arXiv:1511.07025.Google Scholar
Pizzo, A., ‘Bose particles in a box III. A convergent expansion of the ground state of the Hamiltonian in the mean field limiting regime’, Preprint, 2015, arXiv:1511.07026.Google Scholar
Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators (Academic Press, London, 1978).Google Scholar
Sakurai, J. and Napolitano, J., Modern Quantum Mechanics, second edn (Person New International, San Fransico, 2014).Google Scholar
Sawada, K., ‘Ground-state energy of Bose–Einstein gas with repulsive interaction’, Phys. Rev. 116(6) (1959), 1344–1358.Google Scholar
Seiringer, R., ‘The excitation spectrum for weakly interacting bosons’, Comm. Math. Phys. 306(2) (2011), 565–578.Google Scholar
Solovej, J. P., ‘Upper bounds to the ground state energies of the one- and two-component charged Bose gases’, Comm. Math. Phys. 266 (2006), 797–818.Google Scholar
Solovej, J. P., ‘Many body quantum mechanics’, unpublished notes (2007). URL: http://www.mathematik.uni-muenchen.de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf.Google Scholar
Weiss, C. and Eckardt, A., ‘Ground state energy of a homogeneous Bose–Einstein condensate beyond Bogoliubov’, Europhys. Lett. 68(1) (2004), 8–14.Google Scholar
Wu, T. T., ‘Ground state of a Bose system of hard spheres’, Phys. Rev. 115(6) (1959), 1390–1404.Google Scholar
Yau, H.-T. and Yin, J., ‘The second order upper bound for the ground energy of a Bose gas’, J. Stat. Phys. 136 (2009), 453–503.Google Scholar