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Functional integration and the two-point correlation function of the one-dimensional Bose gas in a harmonic potential


Authors: N. M. Bogoliubov and C. Malyshev
Translated by: the authors
Original publication: Algebra i Analiz, tom 17 (2005), nomer 1.
Journal: St. Petersburg Math. J. 17 (2006), 63-84
MSC (2000): Primary 81S40; Secondary 42C10
DOI: https://doi.org/10.1090/S1061-0022-06-00893-4
Published electronically: January 19, 2006
MathSciNet review: 2140675
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Abstract: A quantum field-theoretical model which describes a spatially nonhomogeneous one-dimensional nonrelativistic repulsive Bose gas in an external harmonic potential is considered. The two-point thermal correlation function of the Bose gas is calculated in the framework of the functional integration approach. The calculations are done in the coordinate representation. The method of successive integration first over the ``high-energy'' functional variables and then over the ``low-energy'' ones is used. The effective action functional for the low-energy variables is calculated in one-loop approximation. The functional integral representation for the correlation function is obtained in terms of the low-energy variables and is estimated with the help of stationary phase approximation. The asymptotics of the correlation function is studied in the limit when the temperature tends to zero while the volume occupied by the nonhomogeneous Bose gas increases infinitely. It is demonstrated that the behavior of the thermal correlation function in this limit is power-like and is governed by the critical exponent that depends on the spatial and thermal arguments.


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Additional Information

N. M. Bogoliubov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: bogoliub@pdmi.ras.ru

C. Malyshev
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: malyshev@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-06-00893-4
Keywords: Bose gas, functional integration, correlation function
Received by editor(s): August 13, 2004
Published electronically: January 19, 2006
Additional Notes: Supported in part by RFBR (grant no. 04–01–00825) and by the Russian Academy of Sciences program “Mathematical Methods in Non-Linear Dynamics”.
Dedicated: Dedicated to Ludwig Dmitrievich Faddeev
Article copyright: © Copyright 2006 American Mathematical Society

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