Anharmonic oscillator with polynomial self-interaction

A quantum anharmonic oscillator with a polynomial self-interaction is defined in coordinate space by a Hamiltonian of the form H = −d2/dx2 + ¼x2 + g[(½x2)N + a(½x2)N−1 + b(½x2)N−2 + ⋯]. 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 ∑Angn and ∑Bngn for the energy levels of the (gψ2N)1 and (:gψ2N:)1 field theories differ in large order by an over-all multiplicative constant limn→∞An/Bn = exp[N(2N − 1)/(2N − 2)].