Generalized Anharmonic Oscillator

TLDR: In this paper, the generalized anharmonic oscillator is defined by the Hamiltonian HN, which in the coordinate space representation is given by HN = −d2/dx2 + ¼x2 + g(½x2)N.

Abstract: The generalized anharmonic oscillator is defined by the Hamiltonian HN, which in the coordinate space representation is given by HN = −d2/dx2 + ¼x2 + g(½x2)N. The analytic properties of the energy levels of HN as functions of complex coupling g are derived and described. Zeroth‐order WKB techniques are used in the mathematical analysis. For all N, the results are qualitatively similar to those for the ordinary anharmonic oscillator in which N = 2 and, thus, the results are model independent for this wide class of models. The limiting case N → ∞ is solved exactly without using WKB techniques. The exact solution agrees with the WKB solution to zeroth order. This agreement is most impressive and testifies to the accuracy and utility of WKB methods.