JETP Letters

, 94:243 | Cite as

On a two-dimensional Schrödinger equation with a magnetic field with an additional quadratic integral of motion

  • V. G. MarikhinEmail author


The problem of commuting quadratic quantum operators with a magnetic field has been considered. It has been shown that any such pair can be reduced to the canonical form, which makes it possible to construct an almost complete classification of the solutions of equations that are necessary and sufficient for a pair of operators to commute with each other. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables; this change is similar to that in the theory of integrable tops. As an example, this procedure has been considered for the two-dimensional Schrödinger equation with the magnetic field; this equation has an additional quantum integral of motion.


Magnetic Field Canonical Form JETP Letter Classical Case Static Magnetic Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    E. V. Ferapontov and A. P. Veselov, J. Math. Phys. 42, 590 (2001).MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Dokl. Akad. Nauk SSSR 229, 15 (1976).MathSciNetGoogle Scholar
  3. 3.
    S. P. Novikov and A. P. Veselov, Am. Math. Soc. Transl. 179, 109 (1997).MathSciNetGoogle Scholar
  4. 4.
    J. Berube and P. Winternitz, J. Math. Phys. 45, 1959 (2004).MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    V. G. Marikhin and V. V. Sokolov, Theor. Math. Phys. 149, 1425 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    L. P. Eisenhart, Ann. Math. 35, 284 (1934).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. G. Marikhin and V. V. Sokolov, Reg. Chaot. Dynamics 10, 59 (2005).MathSciNetzbMATHCrossRefADSGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow regionRussia

Personalised recommendations