## Dept. of Mathematics and Computer Science (Berlin)

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**reviewer:** Miloslav Znojil
**reviewernum:** 9689

**revieweremail:** znojil@ujf.cas.cz

**zblno:** DE014343024

**author:** L. Skala, J. Cizek and J. Zamastil

**shorttitle:** Strong coupling perturbation expansions for anharmonic oscillators

**source:** J. Phys. A: Math. Gen. 32, No 30, 5715 - 5734 (1999)

**rpclass:** 81Q15

**rsclass:** 47A55; 47N50; 65B10; 34L15; 34L40; 81T15

**keywords:** Rayleigh Schroedinger perturbation series, anharmonic oscillators

**revtext:** A typical user of quantum mechanics is given a phenomenological
Hamiltonian (say, quartic, sextic, octic or decadic anharmonic
oscillator) and constructs a low lying part of its spectrum via
the usual (and comparatively easily generated, so called
Rayleigh-Schroedinger) perturbation series. Such a user is given a
significant help by the present paper.
Firstly, he/she is advised to get rid of the divergence by moving
to the strong-coupling (SC) and renormalized strong-coupling (RSC)
regime via a suitable re-definition of the coupling strength. This
advice is accompanied by a number of formulae, Tables and Figures
which demonstrate the survival of an easy feasibility of the
recipe and illustrate the numerous specific merits of the RSC
approach.
One is particularly impressed by the observation that the RSC
perturbation series behaves as convergent for all the real values
of the renormalized coupling (i.e., in particular, far beyond the
single well regime: the next paper [20] by the same authors will
say more) and that it is able to provide the upper and upper
energy bounds in a way simplifying the current SC predictions
significantly.

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