Uniformly Valid Asymptotic Approximation for the Quantized Anharmonic Oscillator at High Energies

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Abstract
A uniformly valid asymptotic expansion is obtained for the solution of the one‐dimensional time‐independent Schrödinger equation which describes an anharmonic oscillator. As is well known this leads to a two‐turning‐point problem in asymptotic analysis. The expansion itself is determined for large values of an appropriately constructed parameter λ which can be specialized to 4J(E) / 2πℏ . It is shown to be valid for high energies which, in turn, implies that the turning points are well separated. The technique employed is an extension of that used previously by the authors for low energies, i.e., for nearby turning points. The energy levels are obtained implicitly to all orders in λ −1 by a relation which generalizes the traditional Bohr–Wilson–Sommerfeld quantum condition. For analytic potentials the leading terms in the asymptotic expansion of this relation are obtained in terms of contour integrals and are found to agree with the corresponding higher‐order JWKB results of Dunham. Finally, the normalization of the eigen‐solutions is discussed and a procedure for obtaining the asymptotic expansion of the normalization constant is indicated.