Eigenvalue bounds for polynomial central potentials in d dimensions

If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given approximately by the semi-classical expression E = \min_{r > 0}[\frac{1}{r^2} + \sum_{i = 1}^{k}a_i(P_ir)^{q_i}]. It is proved that this formula yields a lower bound if P_i = P_{n\ell}^{(d)}(q_1), an upper bound if$P_i = P_{n\ell}^{(d)}(q_k) and a general approximation formula if P_i = P_{n\ell}^{(d)}(q_i). For the quantum anharmonic oscillator f(r)=r^2+\lambda r^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(\lambda) is determined by the algebraic expression \lambda={1\over \beta}({2\alpha(m-1)\over mE-\delta})^m({4\alpha \over (mE-\delta)}-{E\over (m-1)}) where \delta={\sqrt{E^2m^2-4\alpha(m^2-1)}} and \alpha, \beta are constants. An improved lower bound to the lowest eigenvalue in each angular-momentum subspace is also provided. A comparison with the recent results of Bhattacharya et al (Phys. Lett. A, 244 (1998) 9) and Dasgupta et al (J. Phys. A: Math. Theor., 40 (2007) 773) is discussed.