Higher-order JWKB approximations for radial problems. III. The r2moscillator

Abstract

For pt.II see ibid., vol.17, p.2493, 1984. Higher-order JWKB approximations are applied to the calculation of energy levels of an oscillator with the potential V(r)=r^2m (m integer). The JWKB quantisation condition for the energy W is shown to be expressible as (n+3/2) pi =AX+BX^-1+CX^-3+. . . where X=W^(m+1)2m/. The l-dependent coefficients A, B, C are determined exactly by taking into account contributions from all orders. On inversion the above series yields an explicit analytical formula for the energy levels. Extension of the result to d dimensions is immediate.