The spectral class of the quantum-mechanical harmonic oscillator

Abstract

The purpose of this paper is to study the so-calledspectral class Q of anharmonic oscillatorsQ=−D ²+q having the same spectrum λ _n =2n (n≧0) as the harmonic oscillatorQ ⁰=−D ²+x ²−1. Thenorming constants $t_n = \mathop {\lim }\limits_{x \uparrow \infty } \ell g[( - 1)^n {{e_n (x)} \mathord{\left/ {\vphantom {{e_n (x)} {e_n }}} \right. \kern-\nulldelimiterspace} {e_n }}( - x)]$ of the eigenfunctions ofQ form a complete set of coordinates inQ in terms of which the potential may be expressed asq=x ²−1−2D ² ℓgϑ with

$$\theta = \det \left[ {\delta _{ij} + (e^{ti} - 1)\int\limits_x^\infty {e_i^0 e_j^0 :0 \leqq i,j,< \infty } } \right],$$

e ⁰ _n being then ^th eigenfunctionQ 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 whetheri=j or not, and [t _i,t _j]=0. This prompts an investigation of the symplectic geometry ofQ. The function ϑ is related to the theta function of a singular algebraic curve. Numerical results are also presented.