A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

doi:10.1016/S0252-9602(10)60083-5

Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (3): 841-856.doi: 10.1016/S0252-9602(10)60083-5

• Articles • Previous Articles Next Articles

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

LIU Chuang-Ye

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China

Received:2007-09-11 Revised:2008-04-07 Online:2010-05-20 Published:2010-05-20
Supported by:
This work is partially supported by NSFC (10571176)

Abstract

Abstract:

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N^-1V_a(x_i-x_j) where V_a(x)=a^-2V(x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrodinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if u_tsolves the GP equation, then the family of k-particle density matrices { _ku_t, k ≥1} solves the GP hierarchy. Denote by ψ_{N, t} the solution to the N-particle Schr\"odinger equation. Under the assumption that a =N^-ε for 0< ε<3/4, we prove that as N → ∞ the limit points of the k-particle density matrices of ψ^{_{N, t}}are solutions of the GP hierarchy with the coupling constant in the
nonlinear term of the GP equation given by ∫V(x)dx.

Key words: Gross-Pitaevskii equation, Boson system, density matrix, BBGKY hierarchy

CLC Number:

35Q40

LIU Chuang-Ye. A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS[J].Acta mathematica scientia,Series B, 2010, 30(3): 841-856.

Trendmd

References

[1] Dalfovo F, Giorgini S, Pitaevskii L P et al. Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern Phys, 1999, 71: 463--512

[2] Elgart A, Erdos L, Schlein B et al. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal, 2006, 179(2): 265--283

[3] Erdos L, Schlein B, Yau H T. Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun Pure Appl Math, 2006, 59(12): 1659--1741

[4] Erdos L, Schlein B, Yau H T. Derivation of the cubic non-linear Schr\"odinger equation from quantum dynamics of many-body systems. Invent Math, 2007, 167: 515--614

[5] Erdos L, Schlein B, Yau H T. Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. arXiv:math-ph/0606017 v3

[6] Erdos L, Yau H T. Derivation of the nonlinear Schr\"odinger equation from a many body Coulomb system. Adv Theor Math Phys, 2001, 5(6): 1169--1205

[7] Gross E P. Structure of a quantized vortex in boson systems. Nuovo Cimento, 1961, 20: 454--466

[8] Gross E P. Hydrodynamics of a superfluid condensate. J Math Phys, 1963, 4: 195--207

[9] Lieb E H, Seiringer R. Proof of Bose-Einstein condensation for dilute trapped gases. Phys Rev Lett, 2002, 88: 170409

[10] Lieb E H, Seiringer R, Yngvason J. Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys Rev A, 2000, 61: 043602

[11] Lieb E H, Seiringer R, Yngvason J. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Comm Math Phys, 2001, 224: 17--31

[12] Pitaevskii L P. Vortex lines in an imperfect Bose gas. Sov Phys JETP, 1961, 13: 451--454

[13] Rudin W. Functional Analysis. 2nd ed. Beijing: China Machine Press, 2004: 70--70

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 10