Complex-time approach for semi-classical quantum tunneling
  The complex-time method for quantum tunneling is studied. In one-dimensional quantum mechanics, we construct a reduction formula for a Green function in the number of turning points based on the WKB approximation. This formula yields a series, which can be interpreted as a sum over the complex-time paths. The weights of the paths are determined. 
Hideaki Aoyama and Toshiyuki Harano