Asymptotic solution of the Schrödinger equation for three charged particles

The Schr\"odinger equation for three charged particles in the continuum is considered in the region ${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\rightarrow}\ensuremath{\infty}, ${\mathit{r}}_{\mathrm{\ensuremath{\alpha}}}$/${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\rightarrow}0, where ${\mathit{r}}_{\mathrm{\ensuremath{\alpha}}}$ is the distance between the particles \ensuremath{\beta} and \ensuremath{\gamma}, and ${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\alpha}}}$ denotes the distance between the center of mass of the pair (\ensuremath{\beta},\ensuremath{\gamma}) and particle \ensuremath{\alpha}, \ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\beta}\ensuremath{\ne}\ensuremath{\gamma}. The asymptotic Schr\"odinger equation valid in this domain is found to have at least two types of solutions. The first, exact one which is of the familar product form however, does not connect to the known asymptotic expression for the solution of the original Schr\"odinger equation in the region where all interparticle distances ${\mathit{r}}_{1}$, ${\mathit{r}}_{2}$, and ${\mathit{r}}_{3}$ go to infinity. Therefore a second type of asymptotic wave function is derived which satisfies the asymptotic Schr\"odinger equation to leading order. It has a surprisingly simple form, being the product of asymptotic Coulomb distortion factors for the relative motion of the particles within each of the pairs (\ensuremath{\alpha},\ensuremath{\beta}) and (\ensuremath{\alpha},\ensuremath{\gamma}), and an ordinary two-particle scattering state for the third pair (\ensuremath{\beta},\ensuremath{\gamma}) but belonging to an effective two-particle relative momentum which depends on the relative coordinate ${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\alpha}}}$. This constitutes a genuine three-body effect. Based on this result, we present an expression for the asymptotic wave function which is the asymptotic solution of the three-charged-particle Schr\"odinger equation in all asymptotic regions: where all three interparticle distances are large where it goes over into the standard asymptotic wave function, as well as if only any two interparticle distances, say ${\mathit{r}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{r}}_{\ensuremath{\gamma}}$, are large and the third one satisfies the condition ${\mathit{r}}_{\mathrm{\ensuremath{\alpha}}}$/${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\rightarrow}0.

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