The anharmonic oscillator

Abstract
Accurate eigenvalues and eigenfunctions of the anharm onic oscillator ( H = p 2 + x 2 + λx 4 , λ > 0) and the quartic oscillator ( H = p 2 + x 4 ) are obtained in all regimes of the quantum num ber n and the anharm onicity constant λ. Transition moments of comparable accuracy are obtained for the quartic oscillator. The method, applicable quite generally for eigenvalue problems, is non-perturbative and involves the use of an appropriately scaled basis for the determ ination of each eigenvalue. The appropriate scaling formula for a given regime of ( n , λ) is constructed from the oscillation properties of the eigenfunctions. More general anharm onic oscillators are also discussed.

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