Comparison of two semiclassical expansions for a family of PT-symmetric oscillators

Original · EN

We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form V(x)=(ix)²ⁿ⁺¹+bix does not agree with an earlier WKB expression for V(x)=-(ix)²ⁿ⁺¹ the case b=0. The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.