Bender-Wu Formulas and Generalized Nonanalytic Expansions for Odd Anharmonic Oscillators Ulrich D. Jentschura Missouri University of Science and Technology Rolla, Missouri, USA Approximation and extrapolation of convergent and divergent sequences and series CIRM, Luminy, Marseille 29-SEP-2009
Another Example Where We Would Expect Convergence The energy of the ground state of the harmonic oscillator is E = � . If we add a quartic perturbation, then we obtain the following perturbation series for the ground-state energy, For small coupling g , the subsequent terms in the series actually first decrease in magnitude up to order n where n=1/g , approximately, but then they grow bigger.
Quantifying the (Formal, Unphysical) Divergence Quartic Hamiltonian: Generalization of this formula for an even oscillator of arbitrary even degree and for an arbitrarily excited level: Bender-Wu formulas
Accurately Quantifying the Asymptotic Behaviour Quartic Hamiltonian:
Whether we like it or not… …the perturbation series for the anharmonic oscillator energy eigenvalue diverges eventually, no matter how small the coupling is! but not
The Borel Way Borel summation is a gem in the hand of physicists for summing divergent series. The similarity of the name “Borel” to “Beryl” may not be completely accidental. You can sum the divergent asymptotic perturbation series for the ground-state energy of the quartic anharmonic oscillator when you combine the Borel method with Padé approximants (in t ).
Padé Approximants in the “Borel Plane” The basic idea of Padé approximants: Borel-Padé resummation has long been some kind of “standard” for the summation of divergent perturbation series in physics. Variants and improvements of this method: Caliceti, Grecchi, Maioli, Brezinski, Fischer, Seznec and Zinn-Justin…
Mention Recursive Algorithms Well-known epsilon algorithm calculates Padé approximants Goal: Start with k=0 , then decrease M , increase k (Lozenge ) . For M=0 , we obtain so-called diagonal Padé approximants. The existence of the recursive scheme actually is quite surprising. Two reviews with an emphasis on this point: E. J. Weniger, Comput. Phys. Rep. 10 (1989) 189-371 E. Caliceti et al., Phys. Rep. 446 (2007) 1-96
Energy Levels of Anharmonic Oscillators of Even Degree… …for positive coupling are characterized by divergent, alternating, perturbative expansions that can be resummed by the Borel-Padé method and by other methods, e.g., summation methods based on factorial series [Weniger transformations, and/or Levin-Weniger transforms, see E. J. Weniger’s and A. Sidi’s talks] The perturbation series for the sextic and octic oscillators diverge even more wildly than for the quartic oscillator. Nonlinear sequence transformations based on factorial series are remarkably powerful in resumming these alternating series. [work of Weniger, � i � ek and Vinette]
Other methods… Sophisticated methods for Borel-summable series relevant to physics: Renormalized strong-coupling expansions [Weniger] Multi-stage transformations [Borel+Conformal+Padé] [Fischer, U.D.J.] Exponentially convergent strong-coupling expansions [Kleinert] Order-dependent mappings [Seznec and Zinn-Justin] But: e.g., perturbation series for the g x^3 oscillator at REAL COUPLING g is not Borel summable.
But [in French: bøt…] Not all perturbation series occuring in physics are Borel-summable, and they may not even be pure power series. WHY?
Model Example for Lamb-Shift Theory (Slide 1) “Innocent” Model Example: Expansion of the integrand leads to problems: [see also “asymptotic matching, Chapter 7 in Bender-Orszag]]
Model Example for Lamb-Shift Theory (Slide 2) “Innocent” Model Example: “High-Energy” Part: “Low-Energy” Part:
Model Example for Lamb-Shift Theory (Slide 3) “High-Energy” Part: “Low-Energy” Part:
Model Example for Lamb-Shift Theory (Slide 4) “Innocent” Model Example: Result:
But [in French: bøt…] There are physicists (including myself) who really spend a great deal of their time calculating actual higher-order terms of this sort (much more difficult than this example): Theory of the hydrogen spectrum has surpassed the anomalous magnetic moment of the electron As the most precise prediction of QED [Phys. Rev. Lett. 95 (2005) 163003] . Conclusion for us: We need LOGARITHMS in addition to POWERS in order to describe nature.
Potentials with Zero Ground State Energy The ground-state energy is exactly zero.
Potentials with Almost Zero Ground State Energy Ira Herbst and Barry Simon [Phys. Lett. B 78 (1978) 304]: The energy of the ground-state of the anharmonic ocillator vanishes to all orders in perturbation theory. The eigenvector is not normalizable [exp(-x 3 )]. We have a nonvanishing ground-state energy: Yet we do not know what are A and B, and what are the higher-order corrections?
Exponentials, Powers and Logarithms are Busy Describing the Energy Eigenvalue Jean Zinn-Justin and U.D.J. [Phys. Lett. B 596 (2004) 138] derive a triple expansion: Dominant contribution: Put n=1 and sum L=1 to L=8 :
A Perhaps More Approachable, and Also True Statement How good can it be? Numerical Result ( g = 0.007, this number is shaken, not stirred): Just add up the analytic terms given above
We Need to Go Forward Concept of Resurgent Expansions or Generalized Nonanalytic Expansions Involving POWERS, LOGARITHMS and EXPONENTIALS
Now as promised: Bender-Wu Formulas For Odd Anharmonic Oscillators
Connection of Stable and Unstable Values of the Couplings for the Quartic [ C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973) ]
Connection of Stable and Unstable Values of the Couplings for the Quartic [ C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973) ]
Connection of Stable and Unstable Values of the Couplings for the Quartic [ C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973) ]
Connection of Unstable and PT-Symmetric Values of the Coupling for the Cubic Oscillator [ C. M. Bender and G. V. Dunne, J. Math. Phys. 40, 4616 (1999) ]
Connection of Unstable and PT-Symmetric Values of the Coupling for the Cubic Oscillator Complex phase and magnitude of x 2 + i x 3 [ C. M. Bender and G. V. Dunne, J. Math. Phys. 40, 4616 (1999) ]
Connection of Unstable and PT-Symmetric Values of the Coupling for the Cubic Oscillator [ C. M. Bender and G. V. Dunne, J. Math. Phys. 40, 4616 (1999) ]
Imaginary Part of Resonance Energies for the Unstable Domains The results of [ C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973) ] are generalized to the case of odd anharmonic oscillators . [U. D. Jentschura, A. Surzhykov and J. Zinn-Justin, Phys. Rev. Lett. 102 , 011601 (2009)]
Generalization of a Result of Carl Bender and Tai-Tsung Wu (1971) Phys. Rev. D {\bf 7}, 1620 (1973) The results of [ C. M. Bender and T. T. Wu, Phys. Rev. Lett. 27, 461 (1971) ] are generalized to the case of odd anharmonic oscillators . [U. D. Jentschura, A. Surzhykov and J. Zinn-Justin, Phys. Rev. Lett. 102 , 011601 (2009)]
Higher-Order Analysis of the Cubic Potential: Decay Rates Particle can escape to infinity: Decay rate is generated.
Generalized Nonanalytic Expansions for Decay Rates Importance of higher-order terms
Decay Rates for the Quintic Potential
Decay Rates for the Seventh-Degree Oscillator The Golden Ratio Enters the Result for the Seventh-Degree Anharmonic Oscillator
Conclusions Concept of Generalized NonanalytIc Expansions Unification of Even and Odd Anharmonic Oscillators Higher-Order Formulas Exponentials, Powers, and Logarithms are necessary in order to describe nature Raises fundamental mathematical questions about summability which have not yet been fully and satisfactorily addressed in the literature.
Thank You Very Much For Your Attention
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