Technique of the Comparison Equation Adapted to the Phase-Integral Method

  • Nanny Fröman
  • Per Olof Fröman
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 40)


For a cluster consisting of an unspecified number of transition points we develop a convenient comparison equation technique for obtaining the wave function within the cluster and the Stokes constants that determine the phase-integral solution outside the cluster. The resulting formulas remain valid even if the transition points approach each other. We perform the rather lengthy calculations in a general way and obtain formulas which can easily be particularized to the specific situations encountered in applications. The final analytic formulas for the Stokes constants are expressed in terms of phase integrals, in which the contributions to the integrands in successive orders of approximation are to be integrated along contours enclosing pairs of transition points. With the aid of the comparison equation technique we thus obtain supplementary quantities which, when incorporated into the general formulas and the connection formulas of the phase-integral method, are of decisive importance when the transition points lie close to each other, but which become very small when those points recede sufficiently far away from each other.

It is shown explicitly that, after appropriate asymptotic expansion in terms of a “small” bookkeeping parameter, the comparison equation solution constructed here yields locally, when there are no transition points in the region under consideration (i.e., sufficiently far away from transition points) the arbitrary-order phase-integral approximation generated from an unspecified base function, which has been described in Chapter 1 of the present monograph.

The comparison equation technique used is based on general ideas that are already known; however, their adaptation to our purpose requires a number of new steps in the procedure, and the resulting phase-integral formulas are new.


Transition Point Comparison Equation Connection Formula Unspecified Number Supplementary Quantity 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Cherry, T.M., Uniform asymptotic formulae for functions with transition points, Trans. Am. Math. Soc. 68(1950), 224–257.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Erdélyi, A., Asymptotic solutions of differential equations with transition points, Proc. Int. Congr. Math. 1954 III(1956), 92–101.Google Scholar
  3. [3]
    Erdélyi, A., Asymptotic solutions of differential equations with transition points or singularities, J. Math. Phys. 1(1960), 16–26.ADSzbMATHCrossRefGoogle Scholar
  4. [4]
    Fröman, N., Detailed analysis of some properties of the JWKB-approximation, Ark. Fys. 31(1966), 381–408.MathSciNetGoogle Scholar
  5. [5]
    Fröman, N., Outline of a general theory for higher order approximations of the JWKB-type, Ark. Fys. 32(1966), 541–548.Google Scholar
  6. [6]
    Fröman, N., Connection formulas for certain higher order phase-integral approximations, Ann. Phys. (N. Y.) 61 (1970), 451–464.ADSCrossRefGoogle Scholar
  7. [7]
    Fröman, N., A simple formula for calculating quantal expectation values without the use of wave functions, Phys. Lett. 48A (1974), 137–139.ADSGoogle Scholar
  8. [8]
    Fröman, N., Phase-integral formulas for level densities, normalization factors, and quantal expectation values, not involving wave functions, Phys. Rev. A17 (1978), 493–504.ADSGoogle Scholar
  9. [9]
    Fröman, N. and Fröman, P.O., JWKB Approximation, Contributions to the Theory, North-Holland Publishing Company, Amsterdam, 1965. (Russian translation: MIR, Moscow, 1967.)zbMATHGoogle Scholar
  10. [10]
    Fröman, N. and Fröman, P.O., Transmission through a real potential barrier treated by means of certain phase-integral approximations, Nucl. Phys. A147 (1970), 606–626.ADSGoogle Scholar
  11. [11]
    Fröman, N. and Fröman, P.O., A direct method for modifying certain phase-integral approximations of arbitrary order, Ann. Phys. (N. Y.) 83 (1974), 103–107.ADSzbMATHCrossRefGoogle Scholar
  12. [12]
    Fröman, N. and Fröman, P.O., On modifications of phase integral approximations of arbitrary order, Nuovo Cimento 20B (1974), 121–132.ADSGoogle Scholar
  13. [13]
    Fröman, N. and Fröman, P.O., Phase-integral calculation of quantal matrix elements without the use of wave functions, J. Math. Phys. 18 (1977), 903–906.ADSCrossRefGoogle Scholar
  14. [14]
    Fröman, N. and Fröman, P.O., Exact formulas and phase-integral formulas, not involving wave functions, for expectation values pertaining to general potentials, Ann. Phys. (N. Y) 163 (1985), 215–226.ADSCrossRefGoogle Scholar
  15. [15]
    Fröman, N. and Fröman, P.O., Phase-integral approximation of arbitrary order generated from an unspecified base function. Review article in: Forty More Years of Ramifications: Spectral Asymptotics and Its Applications, edited by S.A. Fulling and F.J. Narcowich, Discourses in Mathematics and Its Applications, No. 1, Department of Mathematics, Texas A & M University, College Station, Texas 1991, pp. 121–159. After minor changes reprinted as Chapter 1 in the present monograph.Google Scholar
  16. [16]
    Fröman, N., Fröman, P.O., and Karlsson, F., Phase-integral calculation of quantal matrix elements between unbound states, without the use of wave functions, Molec. Phys. 38 (1979), 749–767.ADSCrossRefGoogle Scholar
  17. [17]
    Fröman, N., Fröman, P.O., and Linnaeus, S., Phase-integral formulas for the regular wave function when there are turning points close to a pole of the potential. This is Chapter 6 in this monograph.Google Scholar
  18. [18]
    Fröman, N., Fröman, P.O., Myhrman, U., and Paulsson, R., On the quantal treatment of the double-well potential problem by means of certain phase-integral approximations, Ann. Phys. (N. Y) 74 (1972), 314–323.ADSCrossRefGoogle Scholar
  19. [19]
    Miller, Jr., S.C. and Good, Jr., R.H., A WKB-type approximation to the Schrödinger equation, Phys. Rev. 91 (1953), 174–179.MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. [20]
    Zauderer, E., A uniform asymptotic turning point theory for second order linear ordinary differential equations, Proc. Am. Math. Soc. 31 (1972), 489–494.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Zauderer, E., Miller-Good method for approximating bound states, J. Chem. Phys. 56 (1972), 5198.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Nanny Fröman
    • 1
  • Per Olof Fröman
    • 1
  1. 1.Department of Theoretical PhysicsUniversity of UppsalaUppsalaSweden

Personalised recommendations