We consider second-quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions at zero temperature. We discuss the energy-momentum spectrum of the Bose gas and its physical significance. We review various rigorous and heuristic results as well as open conjectures about its properties. Our main aim is to convince the readers, including those with mainly mathematical background, that this subject has many interesting problems for rigorous research. In particular, we investigate the upper bound on the infimum of the energy for a fixed total momentum given by the expectation value of one-particle excitations over a squeezed states. This bound can be viewed as a rigorous version of the famous Bogoliubov method. We show that this approach seems to lead to a (nonphysical) energy gap. The variational problem involving squeezed states can serve as the preparatory step in a perturbative approach that should be useful in computing excitation spectrum. This version of a perturbative approach to the Bose gas seems (at least in principle) superior to the commonly used approach based on the -number substitution.
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June 2009
Research Article|
June 16 2009
On the infimum of the energy-momentum spectrum of a homogeneous Bose gas
H. D. Cornean;
H. D. Cornean
a)
1Department of Mathematical Sciences,
Aalborg University
, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark
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J. Dereziński;
J. Dereziński
b)
2Department of Mathematical Methods in Physics,
University of Warsaw
, Hoza 74, 00-682 Warszawa, Poland
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a)
Electronic mail: cornean@math.aau.dk.
b)
Electronic mail: jan.derezinski@fuw.edu.pl.
c)
Electronic mail: pawel.zin@fuw.edu.pl.
J. Math. Phys. 50, 062103 (2009)
Article history
Received:
December 18 2008
Accepted:
April 14 2009
Citation
H. D. Cornean, J. Dereziński, P. Ziń; On the infimum of the energy-momentum spectrum of a homogeneous Bose gas. J. Math. Phys. 1 June 2009; 50 (6): 062103. https://doi.org/10.1063/1.3129489
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