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MP: Theoretische und Mathematische Grundlagen der Physik

MP 3: Quantenmech., Schroed.+Dirac-Op, Vielteilchensysteme und QFT

MP 3.1: Talk

Wednesday, March 29, 2006, 14:00–14:30, P1-02-111

Test of resurgent expansion for the case of strong coupling — •Andrey Surzhykov and Ulrich D. Jentschura — Max–Planck–Institut für Kernphysik, Heidelberg, Germany

In quantum mechanics, one may consider a special class of (one–dimensional) potentials for which the standard Rayleigh–Schrödinger perturbation theory fails to explain the physical properties of the system even qualitatively. Particular interest has been devoted to the Fokker–Planck potential [1] whose eigenvalues cannot be described by perturbation theory series even if the perturbative series, which vanishes identically to all orders, is augmented by resummation prescriptions. Generalized perturbative expansions (resurgent expansion) have been proposed in order to evaluate the energy spectrum by taking into account instanton configurations, characterized by nonanalytic factors exp(−a/g), where a is the instanton action and g is the coupling parameter. Until now, the resurgent expansion has been applied mainly for the small values of g, i.e. for weak coupling, while much less attention has been paid to the strong coupling regime. In this contribution, we investigate the behaviour of the resurgent expansion in the transition from weak to strong coupling [2], on the basis of numerical calculations and a direct resummation of the generalized Bohr–Sommerfeld quantization condition, which enhances the convergence of the weak-coupling resurgent expansion for stronger couplings.

[1] U. D. Jentschura and J. Zinn–Justin, Phys. Lett. B 596, 138 (2004).

[2] U. D. Jentschura and A. Surzhykov, Phys. Rev B, to be submitted.