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Nonlinear bound states with prescribed angular momentum

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Abstract

We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schrödinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of a non-radially symmetric spatial profile which in itself is obtained via a doubly constrained energy minimization. One of the two constraints imposed is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also allows for a new description of the set of minimizers subject to only a single mass constraint.

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References

  1. Antonelli, P., Marahrens, D., Sparber, C.: On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete Contin. Dyn. Syst. 32(3), 703–715 (2012)

    Article  MathSciNet  Google Scholar 

  2. Arbunich, J., Nenciu, I., Sparber, C.: Stability and instability properties of rotating Bose-Einstein condensates. Lett. Math. Phys. 109(6), 1415–1432 (2019)

    Article  MathSciNet  Google Scholar 

  3. Arora, A.K., Sparber, C.: Self-bound vortex states in nonlinear Schrödinger equations with LHY correction. NoDEA Nonlinear Differ. Equ. Appl. 30, 14 (2023)

    Article  Google Scholar 

  4. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet. Relat. Models 6(1), 1–135 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bao, W., Du, Q., Zhang, Y.Z.: Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation. SIAM J. Appl. Math. 66, 758–786 (2006)

    Article  MathSciNet  Google Scholar 

  6. Bru, J.B., Correggi, M., Pickl, P., Yngvason, J.: The TF limit for rapidly rotating Bose gases in anharmonic traps. Commun. Math. Phys. 280, 517–544 (2008)

    Article  MathSciNet  Google Scholar 

  7. Carles, R.: Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential. Z. Angew. Math. Phys. 66(4), 2087–2094 (2015)

    Article  MathSciNet  Google Scholar 

  8. Carles, R., Sparber, C.: Orbital stability vs. scattering in the cubic-quintic Schrödinger equation. Rev. Math. Phys. 33, 2150004 (2021)

    Article  Google Scholar 

  9. Cooper, N.R.: Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008)

    Article  Google Scholar 

  10. Correggi, M., Pinsker, F., Rougerie, N., Yngvason, J.: Giant vortex phase transition in rapidly rotating trapped Bose-Einstein condensates. Eur. Phys. J. Spec. Top. 217, 183–188 (2013)

    Article  Google Scholar 

  11. Correggi, M., Rindler-Daller, T., Yngvason, J.: Rapidly rotating Bose-Einstein condensates in homogeneous traps. J. Math. Phys. 48, 102103 (2007)

    Article  MathSciNet  Google Scholar 

  12. Fetter, A.: Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys. 81, 647 (2009)

    Article  Google Scholar 

  13. Hadj Selem, F., Hajaiej, H., Markowich, P.A., Trabelsi, S.: Variational approach to the orbital stability of standing waves of the gross-Pitaevskii equation. Milan J. Math. 82, 273–295 (2014)

    Article  MathSciNet  Google Scholar 

  14. Hajaiej, H., Stuart, C.A.: On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Adv. Nonlinear Stud. 4, 469–501 (2004)

    Article  MathSciNet  Google Scholar 

  15. Jeanjean, L., Lu, S.S.: On global minimizers for a mass constrained problem. Calc. Var. Partial Differ. Equ. 61, 214 (2022)

    Article  MathSciNet  Google Scholar 

  16. Lewin, M., Rota Nodari, S.: The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications. Calc. Var. Partial Differ. Equ. 59, 197 (2020)

    Article  Google Scholar 

  17. Lieb, E., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (1997)

    Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Messiah, A.: Quantum Mechanics. Dover Publications Inc., Mineola (1995)

    Google Scholar 

  20. Seiringer, R.: Gross-Pitaevskii theory of the rotating gas. Commun. Math. Phys. 229, 491–509 (2002)

    Article  MathSciNet  Google Scholar 

  21. Seiringer, R.: Ground state asymptotics of a dilute, rotating gas. J. Phys. A Math. Gen. 36, 9755–9778 (2003)

    Article  MathSciNet  Google Scholar 

  22. Yajima, K., Zhang, G.: Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Differ. Equ. 202, 81–110 (2004)

    Article  Google Scholar 

  23. Zhang, J.: Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials. Z. Angew. Math. Phys. 51(3), 498–503 (2000)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, M/C 249, Chicago, IL, 60607, USA

    Irina Nenciu, Xiaoan Shen & Christof Sparber

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Griviţei, Sector 1, 010702, Bucharest, Romania

    Irina Nenciu

Authors
  1. Irina Nenciu
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  2. Xiaoan Shen
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  3. Christof Sparber
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Correspondence to Xiaoan Shen.

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Communicated by A. Mondino.

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This publication is supported by the MPS Simons foundation through awards no. 851720 and no. 709025.

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Nenciu, I., Shen, X. & Sparber, C. Nonlinear bound states with prescribed angular momentum. Calc. Var. 63, 1 (2024). https://doi.org/10.1007/s00526-023-02599-z

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