The spectral class of the quantum-mechanical harmonic oscillator

The purpose of this paper is to study the so-called spectral classQ of anharmonic oscillators Q=-D²+q having the same spectrum λ_n=2 n (n≧0) as the harmonic oscillator Q⁰=-D²+x²-1. 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=x²-1-2 D² ℓgθ{symbol} with {Mathematical expression}e_n⁰ being the n^th eigenfunction Q⁰. 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.