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Direct demonstration of the completeness of the eigenstates of the Schrodinger equation with local and non-local potentials bearing a Coulomb tail

Research paper by Nicolas Michel

Indexed on: 12 Feb '08Published on: 12 Feb '08Published in: Mathematical Physics



Abstract

Demonstrating the completeness of wave functions solutions of the radial Schrodinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schrodinger potentials. One can mention in particular unbounded potentials yielding a purely discrete spectrum and short-range potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their long-range character. The method proposed in this paper allows to treat them correctly, provided the non-Coulomb part of potentials vanishes after a finite radius. Non-locality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at k = 0 of long-range potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.