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 TMF, 2003, Volume 134, Number 1, Pages 55–73 (Mi tmf140)

This article is cited in 4 scientific papers (total in 4 papers)

Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$

R. K. Bullougha, N. M. Bogolyubovb, V. S. Kapitonovc, K. L. Malyshevb, I. Timonend, A. V. Rybind, G. G. Varzugine, M. Lindbergf

a University of Manchester, Department of Mathematics
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
c State Technological Institute of St. Petersburg
d University of Jyväskylä
e V. A. Fock Institute of Physics, Saint-Petersburg State University
f Åbo Akademi University

Abstract: We evaluate finite-temperature equilibrium correlators êîððåëÿòîðû $\langle T_\tau \hat{\psi}({\bold r}_1) \hat{\psi}^\dagger({\bold r}_2)\rangle$ for thermal time $\tau$ ordered Bose fields ïîëåé $\hat{\psi}$, $\hat{\psi}^\dagger$ to good approximations by new methods of functional integration in $d=1,2,3$ dimensions and with the trap potentials $V({\bold r})\not\equiv0$. As in the translationally invariant cases, asymptotic behaviors fall as $R^{-1}\equiv|{\bold r}_1-{\bold r}_2|^{-1}$ to longer-range condensate values for and only for $d=3$ in agreement with experimental observations; but there are generally significant corrections also depending on ${\bold S}\equiv({\bold r}_1+{\bold r}_2)/2$ due to the presence of the traps. For $d=1$, we regain the exact translationally invariant results as the trap frequencies $\Omega\rightarrow0$. In analyzing the attractive cases, we investigate the time-dependent $c$-number Gross–Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity $-2c\psi|\psi|^{2n}$ and $c<0$. For $n=1$, the stationary form of the GP equation appears in the steepest-descent approximation of the functional integrals. We show that collapse in the sense of Zakharov can occur for $c<0$ and $nd\geq2$ and a functional $E_{NLS}[\psi]\leq0$ even when $V({\bold r})\not\equiv0$. The singularities typically arise as $\delta$-functions centered on the trap origin ${\bold r}={\bold 0}$.

Keywords: Bose–Einstein condensation, functional integral method, quantum model of nonlinear Schrödinger equation, finite-temperature theory, magnetic traps, two-point correlations, coherence functions

DOI: https://doi.org/10.4213/tmf140

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English version:
Theoretical and Mathematical Physics, 2003, 134:1, 47–61

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Citation: R. K. Bullough, N. M. Bogolyubov, V. S. Kapitonov, K. L. Malyshev, I. Timonen, A. V. Rybin, G. G. Varzugin, M. Lindberg, “Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$”, TMF, 134:1 (2003), 55–73; Theoret. and Math. Phys., 134:1 (2003), 47–61

Citation in format AMSBIB
\Bibitem{BulBogKap03} \by R.~K.~Bullough, N.~M.~Bogolyubov, V.~S.~Kapitonov, K.~L.~Malyshev, I.~Timonen, A.~V.~Rybin, G.~G.~Varzugin, M.~Lindberg \paper Quantum Integrable and Nonintegrable Nonlinear Schr\"odinger Models for Realizable Bose--Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$ \jour TMF \yr 2003 \vol 134 \issue 1 \pages 55--73 \mathnet{http://mi.mathnet.ru/tmf140} \crossref{https://doi.org/10.4213/tmf140} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2021730} \zmath{https://zbmath.org/?q=an:1078.81534} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 134 \issue 1 \pages 47--61 \crossref{https://doi.org/10.1023/A:1021815606105} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000181042100005} 

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This publication is cited in the following articles:
1. Bogoliubov NM, Malyshev C, Bullough RK, et al, “Finite-temperature correlations in the one-dimensional trapped and untrapped Bose gases”, Physical Review A, 69:2 (2004), 023619
2. N. M. Bogolyubov, K. L. Malyshev, “Functional integration and the twopoint correlation function of the one-dimensional Bose-gas in the harmonic potential”, St. Petersburg Math. J., 17:1 (2006), 63–84
3. Xia X, Silbey RJ, “Effective Lagrangian approach to the trapped Bose gases at low temperatures”, Physical Review A, 71:6 (2005), 063604
4. Bullough R, “Goat cheese for breakfast in Istanbul or Why are certain nonlinear PDEs both widely applicable and integrable? Reminiscences of Francesco Calogero”, Journal of Nonlinear Mathematical Physics, 12 (2005), 124–137, Suppl. 1
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