Potential envelopes and the large-N approximation.

If E is an eigenvalue of the quantum-mechanical Hamiltonian H= 1/2 Δ+V(r) in N spatial dimensions, then large-N theory, the potential-envelope method, and scale-optimized variational energies all lead to quasiclassical approximations having the same form given by E(α) = minr>0[ 1/2 αr-2+V(r)], where α depends on the quantum numbers and on N. Energy bounds provided by the envelope method allow us to prove that in many cases the large-N results are lower energy estimates. For pure power-law potentials all these energies approach the exact eigenvalue in either of the limits l → ∞ or N → ∞. [ABSTRACT FROM AUTHOR]