Anharmonic oscillator with polynomial self-interaction

Thomas Banks, Carl M. Bender

Research output: Contribution to journalArticle

43 Scopus citations


A quantum anharmonic oscillator with a polynomial self-interaction is defined in coordinate space by a Hamiltonian of the form H = -d 2/dx2 + 1/4x2 + g[(1/2x2) N + a(1/2x2)N-1 + b(1/2x2) N-2.ast; + ⋯]. Using WKB techniques we derive a secular equation which determines the eigenvalues of H for small |g|. We find that the qualitative analytic structure of these eigenvalues as functions of complex g remains unchanged for all fixed values of a, b, . . . , including a = b = ⋯ = 0. The secular equation also implies an elegant theorem which predicts how the a, b, ⋯ terms in H affect the large-order growth of perturbation theory. We use this theorem to compare the perturbative behavior of non-Wick-ordered and Wick-ordered field theories in one-dimensional space-time. In particular, we show that the perturbation series Σ Ang n; and Σ Bngn for the energy levels of the (gψ2n)1 and (:gψ2N:), field theories differ in large order by an over-all multiplicative constant lim n→∞ An/Bn = exp[N(2N - 1)/(2N - 2)].

Original languageEnglish (US)
Pages (from-to)1320-1324
Number of pages5
JournalJournal of Mathematical Physics
Issue number9
StatePublished - Jan 1 1972
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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