Geodesic
Solution of the stochastic Liouville equation for a particle in the field of variable scatterers
Teoretičeskaâ i matematičeskaâ fizika, Tome 29 (1976) no. 2, pp. 244-254 
Yu. N. Barabanenkov. Solution of the stochastic Liouville equation for a particle in the field of variable scatterers. Teoretičeskaâ i matematičeskaâ fizika, Tome 29 (1976) no. 2, pp. 244-254. http://geodesic.mathdoc.fr/item/TMF_1976_29_2_a10/
@article{TMF_1976_29_2_a10,
     author = {Yu. N. Barabanenkov},
     title = {Solution of the stochastic {Liouville} equation for a~particle in the field of variable scatterers},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {244--254},
     year = {1976},
     volume = {29},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1976_29_2_a10/}
}
TY  - JOUR
AU  - Yu. N. Barabanenkov
TI  - Solution of the stochastic Liouville equation for a particle in the field of variable scatterers
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1976
SP  - 244
EP  - 254
VL  - 29
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1976_29_2_a10/
LA  - ru
ID  - TMF_1976_29_2_a10
ER  - 
%0 Journal Article
%A Yu. N. Barabanenkov
%T Solution of the stochastic Liouville equation for a particle in the field of variable scatterers
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1976
%P 244-254
%V 29
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1976_29_2_a10/
%G ru
%F TMF_1976_29_2_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

The quantum-mechanical scattering of a particle on a number of scatterers is considered. It is assumed that the positions of the scatterers in space and times they are switched on are distributed randomly. The solution of the Liouville equation for the density matrix of a particle in the class of nuclear operators is represented in the form of an absolutely convergent series that admits termwise averaging with respect to the positions and switching on times of the scatterers. As a result of partial summation of this averaged series, an analog is obtained of the Dyson equation with “mass operator” in the first order in the density of scatterers. A rigorous estimate is obtained in the form of an inequality for the difference between the exact value of the averaged density matrix of the particle and the solution of the Dyson equation that is obtained. At the same time, the number of scatterers enters the right-hand side of this inequality only through their density.

[1] N. N. Bogolyubov, N. M. Krylov, Zapiski kafedry matematichnoi fiziki Institutu budivelnoi mekhaniki AN URSR, 4, 1939, 5

[2] N. N. Bogolyubov, Izbrannye trudy, t. 2, izd. «Naukova dumka», Kiev, 1970 | MR

[3] S. Bokhner, Lektsii ob integralakh Fure, Fizmatgiz, 1962

[4] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, 1962 | MR

[5] G. E. Shilov, Matematicheskii analiz, ch. 1 i 2, «Nauka», 1972 | MR | Zbl

[6] Yu. L. Daletskii, M. G. Krein, Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, «Nauka», 1970 | MR

[7] D. Ya. Petrina, TMF, 13 (1972), 391 | MR

[8] Yu. V. Prokhorov, Yu. A. Rozanov, Teoriya veroyatnostei, «Nauka», 1973 | MR

[9] L. Shvarts, Analiz, t. 1, «Mir», 1972

[10] F. Dyson, Phys. Rev., 75 (1949), 1736 ; Noveishee razvitie kvantovoi elektrodinamiki, Sb., IL, 1954 | DOI | MR | Zbl

[11] N. N. Bogolyubov, Problemy dinamicheskoi teorii v statisticheskoi fizike, Gostekhizdat, 1946 | MR