Spatial dimension as an expansion parameter in quantum mechanics

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Abstract
For quantum-mechanical problems defined at arbitrary values of the spatial dimension D, dramatic simplifications occur for special values of D. These provide the basis for new expansions which are quite accurate in cases where more traditional expansions fail. This high accuracy originates in the fact that no dynamical parameters are modified, making the essential features of the solutions to the approximate problem the same as in the problem of interest. For example, the wave functions in the special dimensions have the same asymptotic properties and satisfy the same cusp conditions as the true wave function in the dimension of interest. We show how to identify those dimensions which may be useful and illustrate the expansion using the Yukawa potential and some anharmonic oscillators as examples. In the Yukawa problem, the leading correction to the energy in this expansion yields accuracy equal to second order in the usual perturbation theory and to sixth order in the 1/D series. For the oscillator problems, the accuracy of the zero-order wave function and first-order energy are nearly independent of the strength of the anharmonicity. The expansion is only developed for ground states but we conclude with a discussion of the special dimensions associated with excited states.