Geodesic
Integrable potentials by Darboux transformations in rings and quantum and classical problems
Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 108-123 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study a problem in associative rings of left and right factorization of a polynomial differential operator regarded as an evolution operator. In a direct sum of rings, the polynomial arising in the problem of dividing an operator by an operator for two commuting operators leads to a time-dependent left/right Darboux transformation based on an intertwining relation and either Miura maps or generalized Burgers equations. The intertwining relations lead to a differential equation including differentiations in a weak sense. In view of applications to operator problems in quantum and classical mechanics, we derive the direct quasideterminant or dressing chain formulas. We consider the transformation of creation and annihilation operators for specified matrix rings and study an example of the Dikke model.
Keywords: factoring a polynomial differential operator, Darboux–Matveev transformation, generalized Miura transformation, Burgers equation, chain in a ring, dressing the Dicke Hamiltonian.
Mots-clés : Darboux transformation 
@article{TMF_2018_197_1_a5,
     author = {S. B. Leble},
     title = {Integrable potentials by {Darboux} transformations in rings and quantum and classical problems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {108--123},
     year = {2018},
     volume = {197},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a5/}
}
TY  - JOUR
AU  - S. B. Leble
TI  - Integrable potentials by Darboux transformations in rings and quantum and classical problems
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 108
EP  - 123
VL  - 197
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a5/
LA  - ru
ID  - TMF_2018_197_1_a5
ER  - 
%0 Journal Article
%A S. B. Leble
%T Integrable potentials by Darboux transformations in rings and quantum and classical problems
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 108-123
%V 197
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a5/
%G ru
%F TMF_2018_197_1_a5
S. B. Leble. Integrable potentials by Darboux transformations in rings and quantum and classical problems. Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 108-123. http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a5/

[1] V. B. Matveev, “Darboux transformations in associative rings and functional-difference equations”, The Bispectral Problem (Montreal, PQ, March 1997), CRM Proceedings and Lecture Notes, 14, eds. J. Harnad, A. Kasman, AMS, Providence, RI, 1998, 211–226 | DOI | MR

[2] E. V. Doktorov, S. B. Leble, A Dressing Method in Mathematical Physics, Mathematical Physics Studies, 28, Springer, Dordrecht, 2007 | MR

[3] N. N. Bogolyubov, N. N. Bogolyubov (ml.), Vvedenie v kvantovuyu statisticheskuyu mekhaniku, Nauka, M., 1984 | MR

[4] S. B. Leble, “Otklik klassicheskoi mnogochastichnoi sistemy na dvukhimpulsnoe vozbuzhdenie”, Kogerentnoe vozbuzhdenie kondensirovannykh sred, ed. U. Kh. Kopvillem, Dalnevost. nauch. tsentr AN SSSR, Vladivostok, 1979, 51–74; С. Б. Лебле, А. И. Иванов, Метод вторичного квантования, Калинингр. гос. ун-т, Калининград, 1981

[5] S. B. Leble, “Metod odevaniya v kvantovykh modelyakh vzaimodeistviya izlucheniya s veschestvom”, TMF, 152:1 (2007), 118–132 | DOI | DOI | MR | Zbl

[6] L. I. Menshikov, “Sverkhizluchenie i nekotorye rodstvennye yavleniya”, UFN, 169:2 (1999), 113–154 | DOI | DOI

[7] N. M. Bogolyubov, P. P. Kulish, Zap. nauch. sem. POMI, 398 (2012), 26–54 | DOI | MR | Zbl

[8] V. B. Matveev, “Darboux transformation and explicit solutions of the Kadomtcev–Petviaschvily equation, depending on functional parameters”, Lett. Math. Phys., 3:3 (1979), 213–216 | DOI | MR

[9] S. Leble, A. Zaitsev, “Division of differential operators, intertwine relations and Darboux transformations”, Rep. Math. Phys., 46:1–2 (2000), 165–174 | DOI | MR

[10] P. Dubard, V. B. Matveev, “Multi-rogue waves solutions: from the NLS to the KP-I equation”, Nonlinearity, 26:12 (2013), R93–R125 | DOI | MR