Quantization in the neighborhood of classical solutions in the $\boldsymbol N$ particle problem and superfluidity
Teoretičeskaâ i matematičeskaâ fizika, Tome 98 (1994) no. 2, pp. 266-288
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We have considered the system of $N$ similar interacting bosons in the external field.
Hamiltonian of the system is
$$
\widehat H_N=\sum_{i=1}^{N}\bigl(-\Delta_i/2+U(x_i)\bigr)+\varepsilon\sum_{1\le i\le N} V(x_i-x_j).
$$
We have found asimptotical series of eigenvalues and eigenfunctions of
$\widehat H_N$ if $N\to\infty$, $\varepsilon\to0$, $\varepsilon N\to\alpha=\text{const}$. These
series correspond with stable solutions of Hartree equation
$$
\bigl(-\Delta/2+U(x)\bigr) f(x)+\alpha\int dy\,V(x-y)\,|f(y)|^2f(x)=\Omega f(x).
$$
If $U=0$, $f(x)=\text{const}\cdot\exp(ipx)$ then out result is in agreement with
Bogolubov's work about
superfluidity. Phenomena analogous with superfluidity arises in other cases,
too.
@article{TMF_1994_98_2_a7,
author = {V. P. Maslov and O. Yu. Shvedov},
title = {Quantization in the neighborhood of classical solutions in the $\boldsymbol N$ particle problem and superfluidity},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {266--288},
publisher = {mathdoc},
volume = {98},
number = {2},
year = {1994},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1994_98_2_a7/}
}
TY - JOUR AU - V. P. Maslov AU - O. Yu. Shvedov TI - Quantization in the neighborhood of classical solutions in the $\boldsymbol N$ particle problem and superfluidity JO - Teoretičeskaâ i matematičeskaâ fizika PY - 1994 SP - 266 EP - 288 VL - 98 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_1994_98_2_a7/ LA - ru ID - TMF_1994_98_2_a7 ER -
%0 Journal Article %A V. P. Maslov %A O. Yu. Shvedov %T Quantization in the neighborhood of classical solutions in the $\boldsymbol N$ particle problem and superfluidity %J Teoretičeskaâ i matematičeskaâ fizika %D 1994 %P 266-288 %V 98 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_1994_98_2_a7/ %G ru %F TMF_1994_98_2_a7
V. P. Maslov; O. Yu. Shvedov. Quantization in the neighborhood of classical solutions in the $\boldsymbol N$ particle problem and superfluidity. Teoretičeskaâ i matematičeskaâ fizika, Tome 98 (1994) no. 2, pp. 266-288. http://geodesic.mathdoc.fr/item/TMF_1994_98_2_a7/