A new approximation method for the Schrödinger equation (Q1580516)

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 by Lombard and Mareš) 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.