0000000000362544
AUTHOR
Hagen Kleinert
Resummation of anisotropic quartic oscillator. Crossover from anisotropic to isotropic large-order behavior
We present an approximative calculation of the ground-state energy for the anisotropic anharmonic oscillator Using an instanton solution of the isotropic action $\delta = 0$, we obtain the imaginary part of the ground-state energy for small negative $g$ as a series expansion in the anisotropy parameter $\delta$. From this, the large-order behavior of the $g$-expansions accompanying each power of $\delta$ are obtained by means of a dispersion relation in $g$. These $g$-expansions are summed by a Borel transformation, yielding an approximation to the ground-state energy for the region near the isotropic limit. This approximation is found to be excellent in a rather wide region of $\delta$ aro…
Convergent Strong-Coupling Expansions from Divergent Weak-Coupling Perturbation Theory
Divergent weak-coupling perturbation expansions for physical quantities can be converted into sequences of uniformly and exponentially fast converging approximations. This is possible with the help of an additional variational parameter to be optimized order by order. The uniformity of the convergence for any coupling strength allows us to take all expressions directly to the strong-coupling limit, yielding a simple calculation scheme for the coefficients of convergent strong-coupling expansions. As an example, we determine these coefficients for the ground state energy of the anharmonic oscillator up to 22nd order with a precision of about 20 digits.
Scaling property of variational perturbation expansion for a general anharmonic oscillator with xp-potential
We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary $x^p$-potential. The scaling property greatly increases the accuracy of the results.