Geodesic
Quantization in the neighborhood of classical solutions in the $N$ particle problem and superfluidity
Teoretičeskaâ i matematičeskaâ fizika, Tome 98 (1994) no. 2, pp. 266-288 Cet article a éte moissonné depuis la source Math-Net.Ru

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We have considered the system of $N$ similar interacting bosons in the external field. Hamiltonian of the system is $$ \widehat H_N=\sum_{i=1}^{N}\bigl(-\Delta_i/2+U(x_i)\bigr)+\varepsilon\sum_{1\le i<j\le N} V(x_i-x_j). $$ 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. 
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     title = {Quantization in the neighborhood of classical solutions in the $N$ particle problem and superfluidity},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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V. P. Maslov; O. Yu. Shvedov. Quantization in the neighborhood of classical solutions in the $N$ 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/

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