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 ${\mathrm{\sum A}}_n{g}^n$ and ${\mathrm{\sum B}}_n{g}^n$ for the energy levels of the (gψ^2N)₁ and (:gψ^2N:)₁ field theories differ in large order by an over‐all multiplicative constant lim_n→∞A_n/B_n = exp[N(2N − 1)/(2N − 2)].