The spectral class of the quantum-mechanical harmonic oscillator

H. P. McKean, E. Trubowitz

Research output: Contribution to journalArticlepeer-review


The purpose of this paper is to study the so-called spectral classQ of anharmonic oscillators Q=-D2+q having the same spectrum λn=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}en0 being the nth eigenfunction Q0. The spectrum and norming constants are canonically conjugate relative to the bracket [F, G]=∫ΔFDΔGdx, to wit: [λi, λj=0, [ti, 2λj]=1 or 0 according to whether i=j or not, and [ti, tj]=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 languageEnglish (US)
Pages (from-to)471-495
Number of pages25
JournalCommunications In Mathematical Physics
Issue number4
StatePublished - Jan 1982

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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