Analytic Continuation of Higher Angular Momentum States to D Dimensions and Interdimensional Degeneracies

Martin Dunn and Deborah K. Watson Department of Physics and Astronomy University of Oklahoma Norman OK 73019-0225, U.S.A. 

The difficulty in extending the techniques of dimensional scaling to states other than S-wave, centers on finding a way of factoring out the internal degrees of freedom to produce a tractable set of differential equations in the internal coordinates which admits analytic continuation in D. A solution to this problem for two electrons is discussed. This approach enables one to uncover all of the exact interdimensional degeneracies of the two electron system and is generalizable to more complicated problems. 

In this article we will take the opportunity to derive the expansion for the wavefunction in a heuristic, and we hope accessible, fashion. We refer the reader to references [8] and [9] for a more thorough treatment of the expansion of the wavefunction and further details about the interdimensional degeneracies. 

The current work is based on a generalization of Schwartz s [10] expansion for a two electron system to D dimensions and N electrons (although we will not pursue the latter here). Schwartz found that he could expand the wavefunction in the following fashion. 

The generalization of Schwartz s result to D dimensions requires the use of angular momentum algebra in D dimensions which can be developed using the method of irreducible Cartesian tensors. 

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Analytic Continuation of Higher Angular Momentum States to D Dimensions and Interdimensional Degeneracies... 

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