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**reviewernum:** 9689

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

**zblno:** DE015065997

**author:** Ighezou, F.-Z.; Lombard, R. J.:

**shorttitle:** A new approximation method

**source:** Ann. Phys. 278, No. 2, 265 - 279 (1999).

**class:** 81V35; 81V45; 33F99; 34L16

**keywords:** Schroedinger equation, potentials of Hulthen, Poeschl-Teller and square well, moments, kinetic energy, monopole transition,critical couplings,inequalities

**revtext:**
A slightly annoying feature of many applications of quantum
mechanics (recall, for example, the hypernuclear or heavy-meson
models requiring a study of spectra in central potentials) lies in
the traumatizing dichotomy between the extreme user-friendliness
of a few available exactly solvable (ES -- say, Hulthen,
Poeschl-Teller and square well) Hamiltonians and a painful
computational clumsiness of the work with all their (usually, not
too essential but, still, phenomenologically well motivated)
unsolvable modifications (UM). The paper addresses this situation
and offers (or rather further develops) a compromizing approach
working (often, successfully) with the inequalities and estimates
of the Bertlmann and Martin type. Typically, they inter-relate the
moments and energies at different angular momenta so that one can
extend quite easily the available information by deducing, say,
the former quantities from the latter values as taken immediately
from an experiment (this was performed in detail in the preceding
work [6] and [7] by Lombard and Mare\v{s}) or vice versa (the
present, numerically less friendly and, hence, formally more
challenging case).
The paper's method is mainly heuristic. In the first step, the
approximative capacity of a few selected inequalities and
estimates is enhanced by an addition of a suitable correction
term. Next, the appropriate form and shape of the correction is
inferred semi-empirically from a few most common ES models.
Finally, the proposed range of practical applicability to the
``typical" phenomenological UM models is intuitively quantified by
its ``brute-force" determination using just another set of the ES
interactions as a test. In this way a reasonable success is shown
to be achieved, e.g., when estimating the so called yrast (i.e.,
quasi-ground) energy levels (possessing nodeless wave functions
and popular in the atomic, molecular as well as nuclear physics)
from the (more common) knowledge of the ground-state ``input"
information.

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