Wallis Products from the Four-Dimensional Singular Harmonic Oscillator

Abstract

We present a variational derivation of the Wallis product and its reciprocal from the four-dimensional singular harmonic oscillator. The inverse-square interaction is absorbed into an effective angular parameter $\nu$, so that the lowest exact energy in a fixed sector is $E_{4d,\mathrm{exact}}=\hbar\omega(\nu+2)$. Motivated by the radial Kustaanheimo--Stiefel relation $r=\rho^2$ between the four-dimensional oscillator and the three-dimensional Coulomb problem, we use the quartic trial family $R_a(\rho)=N\rho^\nu e^{-a\rho^4}$. The minimized variational energy yields an accuracy ratio governed by adjacent Gamma functions. In the large-$\nu$ semiclassical limit, this ratio approaches unity. Restricting $\nu$ to the odd sequence $\nu=2n-1$ gives the standard Wallis product, whereas the even sequence $\nu=2n$ gives its reciprocal form. The Coulomb-dual interpretation further relates the two branches to integer and half-integer effective angular sectors in the dual Coulomb/MICZ description. The result shows that Wallis-type infinite products persist under an inverse-square deformation of the oscillator and arise from a common Gamma-function structure in radial variational dynamics.