BOUNDS ON SCHRODINGER EIGENVALUES FOR POLYNOMIAL POTENTIALS IN N DIMENSIONS

If a single particle obeys nonrelativistic QM in R-N and has the Hamil tonian H = - Delta + Sigma(q>0) a(q)r(q), a(q)greater than or equal to 0, then the lowest eigenvalue E is given approximately by the semicla ssical expression E = min(r>0){(1/r(2)) + Sigma(q>0) a(q)(P(q,N)r)(q)} . It is proved that this formula yields a lower bound when P(q,N) = (N e/2)(1/2)(N/qe)(1/q)[Gamma(1 + N/2)/Gamma(1 + N/q)](1/N) and an upper bound when P(q,N) = (N/2)(1/2)[Gamma((N + q)/2)/Gamma(N/2)](1/q). An e xtension is made to allow for a Coulomb term when N>1. The general for mula is applied to the examples V(r) = r + r(2) + r(3) and V(r) = r(2) + r(4) + r(6) in dimensions 1 to 10, and the results are compared to accurate eigenvalues obtained numerically. (C) 1997 American Institute of Physics.