EXACT QUANTIZATION CONDITION FOR ANHARMONIC-OSCILLATORS (IN ONE-DIMENSION)

An exact quantization condition is given for the one-dimensional Schrodinger operator with a homogeneous anharmonic potential q2M. It has the form of an explicit mapping from level sequences to level sequences, involving a Bohr-Sommerfeld-like quantization step, and having the exact spectrum as fixed point. Numerical tests and an approximate linear theory both suggest, at least for the few lowest M, that the mapping has a contractive region: when an initial level sequence is only asymptotically correct to lowest order, its iterates are seen to converge term by term towards the exact eigenvalues. This type of approach ought to extend to general polynomial potentials.