The continuum Schrödinger-Coulomb and Dirac-Coulomb Sturmian functions
Journal of Physics A: Mathematical and General
Spherical continuum Sturmian functions for the Schrödinger-Coulomb and Dirac-Coulomb problems are constructed by solving appropriate Sturm-Liouville systems. It is proved that in the non-relativistic case a spectrum of potential strengths is continuous and covers the whole real axis. In the relativistic case two Sturmian sets may be derived. For the relativistic Sturm-Liouville problems their eigenvalue spectra consist of the real axes with zero excluded plus circumferences in the complex plane
... n the complex plane centred at zero. It is shown that, as a consequence of a relationship existing between the two families of the continuum Dirac-Coulomb Sturmians, each family obeys two orthogonality and two closure relations. R Szmytkowski constructed according to the domain from which the energy parameter E is chosen. In the non-relativistic case, if E < 0, it appears that the Schrödinger-Coulomb Sturmians form a discrete set [2, 3] . Properties of the discrete Sturmians have been thoroughly investigated and the functions are widely used in atomic physics (for a comprehensive bibliography, see [2, 4] ). In contrast, studies concerning the Schrödinger-Coulomb Sturmians for which E > 0 are scarce. Searching through the literature we found that the problem of constructing non-relativistic positive-energy Coulomb Sturmian states was considered, very briefly, by Khristenko , Blinder [5, 6] and more recently, in a different way, by Ovchinnikov and Macek  . Khristenko  defined positive-energy Sturmian functions as those solutions of the Schrödinger-Coulomb equation with E > 0 which, after multiplication by the radius r, remained bounded for 0 r ∞. He found that the spectrum of potential strengths was continuous and covered the whole real axis. In two works concerning Sturmian propagators for the non-relativistic attractive Coulomb problem Blinder [5, 6] postulated the same form of the positive-energy Schrödinger-Coulomb Sturmians as had been found before by Khristenko but, in disagreement with Khristenko's results, claimed that the spectrum of potential strengths was limited to the positive real half-axis. A different approach to the problem was proposed by Ovchinnikov and Macek  . In a work concerning positive-energy Sturmians for two-Coulomb-centre problems these authors defined the non-relativistic one-centre positive-energy Coulomb Sturmian states, as those solutions of the Schrödinger-Coulomb wave equation with E > 0 which were regular at the origin and behaved as purely outgoing waves for r → ∞. Functions defined in that manner are analytic continuation of the negative-energy Sturmians to the positive-energy domain. The potential-strength spectrum for this problem was found to be discrete and purely imaginary. The Sturmians obtained in that way possess, however, a deficiency owing to a long-range nature of the Coulomb field: they become unbounded as r increases to infinity. For that reason, in spite of claims to the contrary , Ovchinnikov and Macek's Sturmians fail to obey a simple orthogonality relation and it is difficult to infer anything about the completeness of this set. We have found the situation to be unsatisfactory and decided to reinvestigate the subject. Because of the difficulty encountered by the method of Ovchinnikov and Macek, in this work we have adopted the approach of Khristenko. We present a method of construction of a set of the non-relativistic positive-energy (or continuum) Schrödinger-Coulomb Sturmians and discuss their basic properties. It is shown that the spectrum of potential-strength eigenvalues coincides with the whole real axis. This resolves the disagreement between Khristenko's and Blinder's results in Khristenko's favour. We also discuss important problems concerning orthogonality and normalization of the positive-energy Schrödinger-Coulomb Sturmians. In recent years one observes a rapid growth of interest in developing mathematical tools suitable for the use in relativistic theoretical atomic physics [8, 9] . Therefore, it is natural to ask whether it would be possible to find a relativistic analogue of the continuum Schrödinger-Coulomb Sturmians: the continuum Dirac-Coulomb Sturmian functions. Such a set (or sets) should be useful in the analysis of those atomic continuum processes where the relativity is expected to play an important role. We have studied the subject in a manner similar to that used in our study of discrete Dirac-Coulomb Sturmians  and found that the answer is positive: by solving suitable boundary-value problems with a fixed value of the energy parameter selected so that |E| > mc 2 and with cleverly chosen eigenvalues it is possible to construct two different sets of continuum Dirac-Coulomb Sturmian functions. A detailed analysis of properties of these functions, presented in sections 4 and 5, leads to the conclusion that the eigenvalue spectrum for each problem is continuous and consists of the real axis with zero excluded, plus a circumference in the complex plane centred at zero.