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

This article is cited in 10 scientific papers (total in 10 papers)

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

Bibliographic databases:

Received: 15.12.2003
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
\by V.~V.~Kozlov, D.~V.~Treschev
\paper Polynomial Conservation Laws in Quantum Systems
\jour TMF
\yr 2004
\vol 140
\issue 3
\pages 460--479
\jour Theoret. and Math. Phys.
\yr 2004
\vol 140
\issue 3
\pages 1283--1298

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  mathscinet  zmath
    2. Kozlov VV, “Topological obstructions to the existence of quantum conservation laws”, Doklady Mathematics, 71:2 (2005), 300–302  mathnet  mathnet  mathscinet  zmath  isi  elib
    3. Rylov, AI, “Infinite set of polynomial conservation laws in gas dynamics”, Doklady Mathematics, 76:3 (2007), 962–964  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. Kozlov, VV, “Several problems on dynamical systems and mechanics”, Nonlinearity, 21:9 (2008), T149  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Kozlov V.V., “Conservation Laws of Generalized Billiards That Are Polynomial in Momenta”, Russ. J. Math. Phys., 21:2 (2014), 226–241  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46  mathnet  crossref  mathscinet
    7. V. V. Kozlov, “Linear systems with quadratic integral and complete integrability of the Schrödinger equation”, Russian Math. Surveys, 74:5 (2019), 959–961  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. Volovich I.V., “Complete Integrability of Quantum and Classical Dynamical Systems”, P-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334  crossref  mathscinet  isi
    9. V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics”, Russian Math. Surveys, 75:3 (2020), 445–494  mathnet  crossref  crossref  mathscinet  isi  elib
    10. I. V. Volovich, “On Integrability of Dynamical Systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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