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Anharmonic Oscillator with Polynomial Self‐Interaction
1.The nth Rayleigh‐Schrödinger coefficient is the coefficient of in the perturbation series.
2.A discussion of the numerical methods used may be found in C. M. Bender and T. T. Wu, Phys. Rev. 184, 1231 (1969). Appendix E.
3.C. M. Bender, J. Math. Phys. 11, 796 (1970).
4.For a complete description of level crossing in the anharmonic oscillator see Ref. 2, Sec. VI. For a numerical study of level crossing in other models see Th. W. Ruijgrok, Ref. TH. 1393‐CERN (August, 1971).
5.C. M. Bender and T. T. Wu, Phys. Rev. Letters 27, 461 (1971).
5.These dispersion techniques were discovered independently by B. Simon, Ann. Phys. (N.Y.) 58, 79 (1970).
6.Although the term in Eq. (1) is a singular perturbation of the harmonic oscillator (the reason for the divergence of perturbation theory), the terms are regular perturbations of the term and, therefore, cause only small changes in large‐order growth of the Rayleigh‐Schrödinger coefficients.
7.For a schematic representation of this phenomenon, see Ref. 3.
8.The particular choice of is not significant. What is important is that must be very far from both and The geometric mean between and is a satisfactory choice for
9.C. M. Bender and T. T. Wu, “Anharmonic Oscillator. II” and “Generalized Anharmonic Oscillator. II” (submitted to Phys. Rev.).
10.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Nat. Bur. Stds., Washington, D.C., 1964), p. 801.
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