Eigenvalue bounds for polynomial central potentials in d dimensions

If a single particle obeys non-relativistic QM in R and has the Hamiltonian H = −∆ + f(r), where f(r) = ∑k i=1 air qi , 2 ≤ qi < qi+1, ai ≥ 0, then the eigenvalues E = E (d) nl (λ) are given approximately by the semi-classical expression E = min r>0 { 1 r2 + ∑k i=1 ai(Pir) qi } . It is proved that this formula yields a lower bound if Pi = P (d) nl (q1), an upper bound if Pi = P (d) nl (qk) and a general approximation formula if Pi = P (d) nl (qi). For the quantum anharmonic oscillator f(r) = r2 + λr2m,m = 2, 3, . . . in d dimension, for example, E = E (d) nl (λ) is determined by the algebraic expression λ = 1 β ( 2α(m−1) mE−δ m ( 4α (mE−δ) − E (m−1) )