## Abstract

The purpose of this paper is to study the so-called spectral classQ of anharmonic oscillators Q=-D^{2}+q having the same spectrum λ_{n}=2 n (n≧0) as the harmonic oscillator Q^{0}=-D^{2}+x^{2}-1. The norming constants {Mathematical expression} of the eigenfunctions of Q form a complete set of coordinates in Q in terms of which the potential may be expressed as q=x^{2}-1-2 D^{2} ℓgθ{symbol} with {Mathematical expression}e_{n}^{0} being the n^{th} eigenfunction Q^{0}. The spectrum and norming constants are canonically conjugate relative to the bracket [F, G]=∫ΔFDΔGdx, to wit: [λ_{i}, λj=0, [t_{i}, 2λ_{j}]=1 or 0 according to whether i=j or not, and [t_{i}, t_{j}]=0. This prompts an investigation of the symplectic geometry of Q. The function θ{symbol} is related to the theta function of a singular algebraic curve. Numerical results are also presented.

Original language | English (US) |
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Pages (from-to) | 471-495 |

Number of pages | 25 |

Journal | Communications In Mathematical Physics |

Volume | 82 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1982 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics