On the semiclassical expansion for 1-dim $x^N$ potentials
  In the present paper we study the structure of the WKB series for the polynomial potential $V(x)=x^N$ ($N$ even). In particular, we obtain relatively simple recurrence formula of the coefficients $\s'_k$ of the semiclassical approximation and of the WKB terms for the energy eigenvalues. 
Marko Robnik and Valery Romanovski