Perturbative Scattering Phase Shifts in One-Dimension: Closed-form   Results
  A simple closed form expression is obtained for the scattering phase shift perturbatively to any given order in effective one-dimensional problems. The result is a hierarchical scheme, expressible in quadratures, requiring only knowledge of the zeroth order solution and the perturbation potential. 
C.K. Au, Chi-Keung Chow and Chong-Sun Chu