Advertisement

Kantorovich Method for Solving the Multi-dimensional Eigenvalue and Scattering Problems of Schrödinger Equation

  • Ochbadrakh Chuluunbaatar
  • Michael S. Kaschiev
  • Vera A. Kaschieva
  • Sergey I. Vinitsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)

Abstract

A Kantorovich method for solving the multi-dimensional eigenvalue and scattering problems of Schrödinger equation is developed in the framework of a conventional finite element representation of smooth solutions over a hyperspherical coordinate space. Convergence and efficiency of the proposed schemes are demonstrated on an exactly solvable model of three identical particles on a line with pair attractive zero-range potentials below three-body threshold. It is shown that the Galerkin method has a rather low rate of convergence to exact result of the eigenvalue problem under consideration.

Keywords

Eigenvalue Problem Galerkin Method Helium Atom Identical Particle Hyperspherical Harmonic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrashkevich, A.G., Abrashkevich, D.G., Kaschiev, M.S., Puzynin, I.V.: Finite-element solution of the coupled channel Schrödinger equation using high-order accuracy approximation. Comput. Phys. Commun. 85 (1995) 40–64.zbMATHCrossRefGoogle Scholar
  2. 2.
    Abrashkevich, A.G., Abrashkevich, D.G., Kaschiev, M.S., Puzynin, I.V.: FESSDE, a program for finite-element solution of the coupled channel Schrödinger equation using high-order accuracy approximation. Comput. Phys. Commun. 85 (1995) 65–81.zbMATHCrossRefGoogle Scholar
  3. 3.
    Abrashkevich, A.G., Kaschiev, M.S., Vinitsky, S.I.: A new method for solving an eigenvalue problem for a system of three Coolomb particles within the hyperspherical adiabatic representation. J. Comp. Phys. 163 (2000) 328–348.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Amaya-Tapia, A., Larsen, S.Y., Popiel, J.J.: Few-Body Systems, 23 (1997) 87.CrossRefGoogle Scholar
  5. 5.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, New York (1982)Google Scholar
  6. 6.
    Chuluunbaatar, O., Puzynin, I.V., Pavlov, D.V., Gusev, A.A., Larsen, S.Y., and Vinitsky, S.I. Preprint JINR, P11-2001-255, Dubna, 2001 (in Russian)Google Scholar
  7. 7.
    Derbov, V.L., Melnikov, L.A., Umansky, I.M., and Vinitsky, S.I.: Multipulse laser spectroscopy of pHe+: Measurement and control of the metastable state population. Phys. Rev. 55 (1997) 3394–3400.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Gibson, W., Larsen, S.Y., Popiel, J.J.: Hyperspherical harmonics in one dimension. I. adiabatic effective potentials for three particles with delta-function interactions. Phys. Rev. A, 35 (1987) 4919.CrossRefGoogle Scholar
  9. 9.
    Holzscheiter, M.H., Charlton, M.: Ultra-low energy antihydrogen. Rep. Prog. Phys. 62 (1999) 1–60.CrossRefGoogle Scholar
  10. 10.
    Kantorovich, L.V., Krylov, V.I.: The approximation methods of higher analysis. Nauka, Moscow (1952) (in Russian)Google Scholar
  11. 11.
    Rudnev, V., Yakovlev, S. Chem. Phys. Lett. 328 (2000) 97.CrossRefGoogle Scholar
  12. 12.
    Strang, G., Fix, G.: An Analisys of the Finite Element Method. Printice-Hall, Englewood Cliffs, N.J. (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ochbadrakh Chuluunbaatar
    • 1
  • Michael S. Kaschiev
    • 2
  • Vera A. Kaschieva
    • 3
  • Sergey I. Vinitsky
    • 1
  1. 1.Joint Institute for Nuclear ResearchMoscow regionRussia
  2. 2.South - West University Neofit RilskiBlagoevgrad, Bulgaria
  3. 3.Department of MathematicsTechnical University - SofiaBulgaria

Personalised recommendations