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

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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 
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     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},
     publisher = {mathdoc},
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     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a5/}
}
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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/