On the derivation of the Gross-Pitaevskii equation

Riccardo Adami

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 2, page 359-368
  • ISSN: 0392-4041

Abstract

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This article reflects in its content the talk the author gave at the XVII Congresso dell’Unione Matematica Italiana, held in Milano, 8-13 September 2003. We review about some recent results on the problem of deriving the Gross-Pitaevskii equation in dimension one from the dynamics of a quantum system with a large number of identical bosons. We explain the motivations for some peculiar choices (shape of the interaction potential, scaling, initial datum). Open problems are pointed out and difficulties and hindrances in replicating the strategy in higher dimension are put in evidence.

How to cite

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Adami, Riccardo. "On the derivation of the Gross-Pitaevskii equation." Bollettino dell'Unione Matematica Italiana 8-B.2 (2005): 359-368. <http://eudml.org/doc/195880>.

@article{Adami2005,
abstract = {This article reflects in its content the talk the author gave at the XVII Congresso dell’Unione Matematica Italiana, held in Milano, 8-13 September 2003. We review about some recent results on the problem of deriving the Gross-Pitaevskii equation in dimension one from the dynamics of a quantum system with a large number of identical bosons. We explain the motivations for some peculiar choices (shape of the interaction potential, scaling, initial datum). Open problems are pointed out and difficulties and hindrances in replicating the strategy in higher dimension are put in evidence.},
author = {Adami, Riccardo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {359-368},
publisher = {Unione Matematica Italiana},
title = {On the derivation of the Gross-Pitaevskii equation},
url = {http://eudml.org/doc/195880},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Adami, Riccardo
TI - On the derivation of the Gross-Pitaevskii equation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/6//
PB - Unione Matematica Italiana
VL - 8-B
IS - 2
SP - 359
EP - 368
AB - This article reflects in its content the talk the author gave at the XVII Congresso dell’Unione Matematica Italiana, held in Milano, 8-13 September 2003. We review about some recent results on the problem of deriving the Gross-Pitaevskii equation in dimension one from the dynamics of a quantum system with a large number of identical bosons. We explain the motivations for some peculiar choices (shape of the interaction potential, scaling, initial datum). Open problems are pointed out and difficulties and hindrances in replicating the strategy in higher dimension are put in evidence.
LA - eng
UR - http://eudml.org/doc/195880
ER -

References

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  1. ADAMI, R. - BARDOS, C. - GOLSE, F. - TETA, A., Towards a rigorous derivation of the cubic NLSE in dimension one, preprint, Mathematical Physics Preprint Archive n. 03-347, 2003. To appear in «Asymptotic Analysis». Zbl1069.35082MR2104130
  2. BOSE, S. N., Z. Phys., 26 (1924), 178. 
  3. BARDOS, C. - ERDÖS, L. - GOLSE, F. - MAUSER, N. - -T. YAU, H., Derivation of the Schrödinger-Poisson equation from the quantum N-body problem, C. R. Math. Acad. Sci. Paris, 334, no. 6 (2002), 515-520. Zbl1018.81009MR1890644
  4. BARDOS, C. - GOLSE, F. - MAUSER, N., Weak coupling limit of the N particles Schrödinger equation, Methods Appl. Anal., 7, n. 2 (2000), 275-293. Zbl1003.81027MR1869286
  5. DELL'ANTONIO, G. - FIGARI, R. - TETA, A., Hamiltonians for systems of N particles interacting through point interactions, Ann. Inst. H. Poincaré Phys. Théor., 60, 3 (1994), 253-290. Zbl0808.35113MR1281647
  6. EINSTEIN, A., Sitzber. Kgl. Preuss. Akad. Wiss., 261, 1924. 
  7. ERDÖS, L. - YAU, H.-T., Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys., 5, no. 6 (2001), 1169-1205. Zbl1014.81063MR1926667
  8. GROSS, E. P., Structure of a quantized vortex in boson systems, Nuovo Cimento (10) 20 (1961), 454-477. Zbl0100.42403MR128907
  9. JONA-LASINIO, G. - PRESILLA, C. - SJÖSTRAND, J., On Schrödinger equations with concentrated nonlinearities, Ann. Physics, vol. 240, no. 1 (1985), 1-21. Zbl0820.34050MR1329589
  10. LIEB, E. H. - LINIGER, W., Exact analysis of an interacting Bose gas. I. The general solution and the ground state and Lieb, E. H., Exact analysis of an interacting Bose gas. II. The excitation spectrum, Physical Review, vol. 130, n. 1 (1963), 1605-1624. Zbl0138.23001MR156630
  11. LIEB, E. H. - SEIRINGER, R. - YNGVASON, J., Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Dedicated to Joel L. Lebowitz, Comm. Math. Phys., 224, no. 1 (2001), 17-31. Zbl0996.82010MR1868990
  12. MINLOS, R. A. - FADDEEV, I. D., On the point interaction for a three-particle system in quantum mechanics, Dokl. Akad. Nauk SSSR vol. 141, 1335-1338 (Russian); translated as Soviet Physics Dokl., 6 (1962), 1072-1074. MR147136
  13. PITAEVSKII, L. P., Zh. Eksp. Theor. Fiz., 40 (1961), 646. 
  14. SPOHN, H., Kinetic equations from hamiltonian dynamics, Rev. Mod. Phys., 52, n.3 (1980), 600-640. Zbl0399.60082MR578142
  15. SPOHN, H., On the Vlasov hierarchy, Math. Methods Appl. Sci., 3, n. 4 (1981), 445-455. Zbl0492.35067MR657065

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