Eigenvalue bounds for polynomial central potentials in d dimensions

Abstract

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$,thentheeigenvaluesE=E_{n\ell }^{(d)}(\lambda )aregivenapproximatelybythesemi-classicalexpressionE={\min }_{r>0}[\frac{1}{r^2}+{\sum }_{i=1}^k{a}_i(P_{i}r{)}^{q_i}].Itisprovedthatthisformulayieldsalowerboundif{P}_i=P_{n\ell }^{(d)}(q_1),anupperboundif$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+λr^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(λ) is determined by the algebraic expression λ={1\over β}({2α(m-1)\over mE-δ})^m({4α\over (mE-δ)}-{E\over (m-1)}) where δ={\sqrt{E^2m^2-4α(m^2-1)}} and α, βare 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.

13 pages, no figures

References in corpus (1)

• Quantum anharmonic oscillators: a new approach