Proof of basic inequalities in the hyperspherical formalism for theN-body problem

The nonrelativistic N-body problem can be solved in the hyperspherical formalism by the hyperspherical adiabatic approach (HAA). The ground-state energy {ε} 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 εEAA{≤}{ε}{≤}εCAA{≤}εUAA. A main point of the article lies in extending to hyperspherical formalism an inequality familiar for the Born-Oppenheimer approximation (BOA), namely, {ε}{≥}εBOA.