The FD-method in spectral problems for the Schrödinger operator with polynomial potential on (-∞, ∞)

TitleThe FD-method in spectral problems for the Schrödinger operator with polynomial potential on (-∞, ∞)
Publication TypeJournal Article
Year of Publication2015
AuthorsMakarov, VL
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published11/2015

The boundary-value problem under study has two distinctive features: its integration interval is infinite, and the polynomial potential is unbounded. As a consequence, there is no justified numerical solution methodology available in the literature. This article offers one. We apply the Functionally-Discrete (FD) method to the mentioned problem and supply the justification of its convergence. The proposed method enables one to obtain the numerical solution to the problem with an arbitrarily prescribed precision. Among other areas, the results of this work can be applied to calculate the quantum anharmonic oscillator energy states (ground and excited), as well as the energy states of the oscillators with double-well potential.

Keywordseigenvalues, exponentially convergent method, Kummer’s functions, Schrödinger operator, spectral problems
  1. Kao Y.-M., Jiang T.-F. Phys. Rev. E, 2005, 71, No 3: 036702, 7 p.
  2. Roy A. K., Gupta N., Deb B. M. Phys. Rev. A, 2001, 65, No 1: 012109, 7 p.
  3. Makarov V. L. Dokl. AN SSSR, 1991, 320, No 1: 34–39 (in Russian).
  4. Adomian G. Solving frontier problems of physics: the Decomposition method, Dordrecht: Kluwer, 1994.
  5. Olver F. W. J., Lozier D. W., Boisvert R. F., Clarc C. W., editors. NIST Handbook of Mathematical Functions, New York: Cambridge University Press, 2010,
  6. Makarov V. L., Romanyuk N. M. Dopov. Nac. akad. nauk Ukr., 2014, No 2: 26–31 (in Ukrainian).
  7. Makarov V. L. J. Comp. and Appl. Math., 1997, No 82: 69–74 (in Ukrainian).