Adiabatic Approximation and Necessary Conditions for the Existence of Bound States

Abstract

Let $H(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ represent the ($M\ensuremath{-}1$)-body Hamiltonian that results from fixing the center of mass $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$ of an $M$-body system, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ is the relative coordinate of a particular pair of particles, and $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}$ represents the $M\ensuremath{-}2$ remaining internal coordinates. With ${E}_{\mathrm{thr}}$ the lowest continuum threshold associated with $H$, the number of bound states of the system is the number of negative eigenvalues of $H\ensuremath{-}{E}_{\mathrm{thr}}$. A simple lower bound on $H$ was derived by Hahn and Spruch through the use of an adiabatic-like approximation in which the ($M\ensuremath{-}1$)-body problem is attacked by considering first an ($M\ensuremath{-}2$)-body problem and then a one-body problem. With ${E}_{\mathrm{a}0}(\mathcal{r})$ the lowest energy of the system for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ fixed, one finds $H(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})\ensuremath{-}{E}_{\mathrm{thr}}>~\stackrel{^}{1}(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}){H}^{(1)}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}), \mathrm{where} {H}^{(1)}\ensuremath{\equiv}{T}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}+{E}_{\mathrm{a}0}(\mathcal{r})\ensuremath{-}{E}_{\mathrm{thr}}\ensuremath{\equiv}{T}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}+{V}^{(1)}(\mathcal{r}).$ ${H}^{(1)}$ is a one-body Hamiltonian, ${T}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}$ is the kinetic energy operator for the relative motion of the particular pair, and $\stackrel{^}{1}(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}})$ is the unit operator in the space of quadratically integrable functions of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}$. The adiabatic potential ${E}_{\mathrm{a}0}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathcal{r}})$ has been tabulated for a number of systems, primarily atomic and molecular. A necessary condition for the existence of a bound state of $H$ is that the lowest eigenvalue of ${H}^{(1)}$ be negative.The method is formulated fairly generally and discussed in some detail for the case of Coulomb interactions. It is shown that neither He+${e}^{+}$ nor $\ensuremath{\alpha}+p+{e}^{\ensuremath{-}}$ can form bound states. We also find lower bounds of -0.068 and -0.065 eV on the energy of the ground state and the first excited state, respectively, of the H+${e}^{+}$ system; presumably there is no bound state of this system, but we are unable to prove it. The system H+${e}^{\ensuremath{-}}$ is also considered.