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 TMF, 2004, Volume 140, Number 3, Pages 460–479 (Mi tmf103)

Polynomial Conservation Laws in Quantum Systems

V. V. Kozlov, D. V. Treschev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e. the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.

Keywords: Hamiltonian operator, polynomial differential operator, system with exponential interaction, potential spectrum

DOI: https://doi.org/10.4213/tmf103

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English version:
Theoretical and Mathematical Physics, 2004, 140:3, 1283–1298

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Revised: 02.02.2004

Citation: V. V. Kozlov, D. V. Treschev, “Polynomial Conservation Laws in Quantum Systems”, TMF, 140:3 (2004), 460–479; Theoret. and Math. Phys., 140:3 (2004), 1283–1298

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf/v140/i3/p460

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. V. Treschev, “Quantum Observables: An Algebraic Aspect”, Proc. Steklov Inst. Math., 250 (2005), 211–244
2. Kozlov VV, “Topological obstructions to the existence of quantum conservation laws”, Doklady Mathematics, 71:2 (2005), 300–302
3. Rylov, AI, “Infinite set of polynomial conservation laws in gas dynamics”, Doklady Mathematics, 76:3 (2007), 962–964
4. Kozlov, VV, “Several problems on dynamical systems and mechanics”, Nonlinearity, 21:9 (2008), T149
5. Kozlov V.V., “Conservation Laws of Generalized Billiards That Are Polynomial in Momenta”, Russ. J. Math. Phys., 21:2 (2014), 226–241
6. Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46
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