Proof of basic inequalities in the hyperspherical formalism for the<i>N</i>-body problem

Abstract

The nonrelativistic N-body problem can be solved in the hyperspherical formalism by the hyperspherical adiabatic approach (HAA). The ground-state energy \ensuremath{\varepsilon} is compared with three other eigenenergies obtained from the HAA: the coupled and uncoupled adiabatic approximations (CAA and UAA), and the extreme adiabatic approximation (EAA). The first two provide an upper bound and the last provides a lower bound to the ground-state energy, or ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{EAA}}$\ensuremath{\le}\ensuremath{\varepsilon}\ensuremath{\le}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{CAA}}$\ensuremath{\le}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{UAA}}$. A main point of the article lies in extending to hyperspherical formalism an inequality familiar for the Born-Oppenheimer approximation (BOA), namely, \ensuremath{\varepsilon}\ensuremath{\ge}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{BOA}}$.