Skip to main content
SpringerLink
Log in

On Calculating the Coefficients in the Quantum Averaging Procedure for the Hamiltonian of the Resonance Harmonic Oscillator Perturbed by a Differential Operator with Polynomial Coefficients

  • Research Articles
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

The quantum averaging method is applied to the Hamiltonian of the multifrequency resonance harmonic oscillator perturbed by a differential operator with polynomial coefficients. The twisted product is used to transfer the averaging procedure to the space of graded algebra of symbols. As a result, the averaged Hamiltonian is expressed in terms of generators of the quantum algebra of symmetries of the harmonic part of the Hamiltonian. The proposed approach to the operator averaging is used to solve the spectral problem for the Hamiltonian of the cylindrical Penning trap.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. V. Karasev and E. M. Novikova, “Algebra and Quantum Geometry of Multifrequency Resonance”, Izv. Ross. Akad. Nauk Ser. Mat., 74:6 (2010), 55–106.

    Article  MathSciNet  Google Scholar 

  2. M. V. Karasev and E. M. Novikova, “Algebra of Symmetries of Three-Frequency Resonance: Reduction of a Reducible Case to an Irreducible Case”, Math. Notes, 104:6 (2018), 45–59.

    MathSciNet  MATH  Google Scholar 

  3. E. M. Novikova, “Algebra of Symmetries of Three-Frequency Hyperbolic Resonance”, Math. Notes, 106:6 (2019), 940–956.

    Article  MathSciNet  Google Scholar 

  4. L. Charles, S. V. Ngoc, “Spectral Asymptotics via the Semiclassical Birkhoff Normal Form”, Duke Mathematical Journal, 143:3 (2008), 463–511.

    Article  MathSciNet  Google Scholar 

  5. A. Yu. Anikin, “Quantum Birkhoff Normal Forms”, Teoret. Mat. Fiz., 160:3 (2009), 487–506.

    Article  MathSciNet  Google Scholar 

  6. K. Blaum and F. Herfurth (eds.), Trapped Charged Particles and Fundamental Interactions, Springer-Verlag, 2008.

    Google Scholar 

Download references

Acknowledgments

I am grateful to M. V. Karasev – my teacher and coauthor of the previous works on resonance algebras. I want to thank V. G. Danilov and V. E. Nazaikinskii for discussions of the work, advice, valuable remarks, and support. I am much obliged to R. K. Gaidukov for the help in mastering the Wolfram Mathematica package of symbol computations.

Funding

The study was implemented in the framework of the Basic Research Program at National Research University Higher School of Economics in 2020.

Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    E. M. Novikova

Authors
  1. E. M. Novikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. M. Novikova.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Novikova, E.M. On Calculating the Coefficients in the Quantum Averaging Procedure for the Hamiltonian of the Resonance Harmonic Oscillator Perturbed by a Differential Operator with Polynomial Coefficients. Russ. J. Math. Phys. 28, 406–410 (2021). https://doi.org/10.1134/S1061920821030134

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920821030134

Search

Navigation