Exact Anharmonic Quantization Condition (In One Dimension)

  • André Voros
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 95)


An exact version of the Bohr-Sommerfeld quantization scheme is constructed for any one-dimensional homogeneous anharmonic oscillator (having the potential q 2M ). It identifies the exact spectrum as the fixed point of a nonlinear transformation acting upon level sequences within a suitable domain. This mapping itself is explicitly given as a combination of a standard Bohr-Sommerfeld quantization step with a feedback operation from the resulting spectrum. An approximate linear theory suggests, and numerical tests confirm, that our mapping is contractive up to very high (possibly all) degrees of anharmonicity. The exact spectrum is then constructively specified as the attractor of semiclassically correct level sequences. (This type of approach ought to extend to general polynomial potentials.)


Spectral Function Zeta Function Exact Quantization Flux Operator Exact Spectrum 
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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • André Voros
    • 1
  1. 1.CEA — Service de Physique ThéoriqueCE-SaclayGif-sur-Yvette CedexFrance

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