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Improvement of the generalized quantal Bohr–Sommerfeld quantization condition

Published online by Cambridge University Press:  24 October 2008

Bengt Lundborg  and
Per Olof Fröman
Bengt Lundborg
Affiliation:
Institute of Theoretical Physics, University of Uppsala, Thunbergsvägen 3, S-752 38 Uppsala, Sweden
Per Olof Fröman
Affiliation:
Institute of Theoretical Physics, University of Uppsala, Thunbergsvägen 3, S-752 38 Uppsala, Sweden
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Abstract

The quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.

A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.

At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 3 , November 1988 , pp. 581 - 601
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Balian, R., Parisi, G. and Voros, A.. Discrepancies from asymptotic series and their relation to complex classical trajectories. Phys. Rev. Lett. 41 (1978), 1141–1144.CrossRefGoogle Scholar
[2]Fröman, N.. The energy levels of double-well potentials. Ark. Fys. 32 (1966), 79–97.Google Scholar
[3]Fröman, N.. Outline of a general theory for higher order approximations of the JWKB type. Ark. Fys. 32 (1966), 541–548.Google Scholar
[4]Fröman, N.. Connection formulas for certain higher order phase-integral approximations. Ann. Phys. (N.Y.) 61(1970), 451–464.CrossRefGoogle Scholar
[5]Fröman, N. and Fröman, P. O.. JWKB Approximation, Contributions to the Theory (North-Holland, 1965).Google Scholar
[6]Fröman, N. and Fröman, P. O.. Transmission through a real potential barrier treated by means of certain phase-integral approximations. Nuclear Phys. A 147 (1970), 606–626.CrossRefGoogle Scholar
[7]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.CrossRefGoogle Scholar
[8]Fröman, N. and Fröman, P. O.. On modifications of phase integral approximations of arbitrary order. Nuovo Cimento B 20 (1974), 121–132.CrossRefGoogle Scholar
[9]Fröman, N. and Fröman, P. O.. Arbitrary-order phase-integral approximation generated from an unspecified base function. In Méthodes Semi-Classiques en Mécanique Quantique (Publications de l'université de Nantes, Institut de Mathmatiques et d'informatiques, 1984), pp. 47–53.Google Scholar
[10]Fröman, N., Fröman, P. O. and Lundborg, B.. The Stokes constants for a cluster of transition points. Math. Proc. Cambridge Philos. Soc. 104 (1988), 153–179.CrossRefGoogle Scholar
[11]Fröman, N., Fröman, P. O. and Lundborg, B.. Symmetry relations for connection matrices in the phase-integral method. Math. Proc. Cambridge Philos. Soc. 104 (1988), 181–192.CrossRefGoogle Scholar
[12]Fröman, N., Froman, 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.CrossRefGoogle Scholar
[13]Fröman, N. and Myhrman, U.. The energy levels of double-well potentials. II. Ark. Fys. 40 (1970), 497–508.Google Scholar
[14]Lakshmanan, M., Karlsson, F. and Fröman, P. O.. Phase-integral calculation of the energy levels of a quantal anharmonic oscillator. Phys. Rev. D 24 (1981), 2586–2598.CrossRefGoogle Scholar
[15]Lundborg, B.. Phase-integral treatment of wave reflexion by real potentials. Math. Proc. Cambridge Philos. Soc. 85 (1979), 493–522.CrossRefGoogle Scholar
[16]Voros, A.. The return of the quartic oscillator. The complex WKB method. Ann. Inst. H. Poincaré Phys. Theor. 39(1983), 211–338.Google Scholar