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Bound states in the Kratzer plus polynomial potentials and the new form of perturbation theory

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Abstract

The Schrödinger equation with potentials of the Kratzer plus polynomial type (say, quartic V(r) = Ar 4 + Br 3 + Cr 2 + Dr + F/r + G/r 2 etc.) is considered and a new method of exact construction of some of its bound states is presented. Our approach is made feasible via a combination of the traditional use of the infinite series ψ(r)(terminated rigorously after N + 1 terms at certain specific couplings and energies) with several new ideas. We proceed in two steps. Firstly, in the strong coupling regime with G → ∞, we find the exact, complete and compact unperturbed solution of our N + 1 coupled and nonlinear algebraic conditions of the termination. Secondly, we adapt the current Rayleigh–Schrödinger perturbation theory to our nonlinear equations and define the general G < ∞ bound states via an innovated, triple perturbation series. In its tests we show how all the corrections appear in integer arithmetics and remain, therefore, exact.

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  1. OTF, Ústav jaderné fyziky AV ČR, 250 68 Řež, Czech Republic

    Miloslav Znojil

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  1. Miloslav Znojil
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Znojil, M. Bound states in the Kratzer plus polynomial potentials and the new form of perturbation theory. Journal of Mathematical Chemistry 26, 157–172 (1999). https://doi.org/10.1023/A:1019185911999

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