Analytical Evaluation of Energy Eigenvalues for a Class of Anharmonic Oscillators

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Abstract
For the family of potentials V(x) = Axm + Bx2m, B > 0; m = 2, 4, 6, ···, this paper derives analytical expressions for eigenvalues of the one-dimensional time-independent Schrödinger equation. The required eigenvalues are first defined implicitly through the authors' form of the asymptotic expansion of the quantum condition given originally by Dunham. For the stated family of potentials, the integrals which appear in this expansion are evaluated in terms of hypergeometric functions, and expanded in series. A technique is then developed for reversion of the resulting series, by which the eigenvalues are given directly as an expansion in powers of two well-defined variables. This technique is used to exhibit 13 terms of the resulting series and is shown to yield as many terms as may be desired. A second-order approximation obtained from this double series is evaluated for the special potentials V(x) = Ax2 + Bx4 and V(x) = Bx2m with m = 2, 4, 6. Our analytical results, known in principle to be increasingly accurate for higher eigenvalues, are then compared with the numerical or seminumerical computations of other authors and found in practice to yield excellent agreement as low as the third and fourth eigenvalues.

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