Volterra analyticity sectors for a half-line NLS Lax pair

For the normalization points $(0,T)$, $(0,0)$ and spatial infinity, the Volterra integral equation kernel is $e^{-i[k(x-x^{\prime })+2k^2(t-t^{\prime })]{\hat{\sigma }}_3}$. Its lower entry in column one has factor $e^{2i[k(x-x^{\prime })+2k^2(t-t^{\prime })]}$; column two has the opposite factor. With quadrants numbered counterclockwise, the column domains are $(D_2,D_3)$, $(D_1,D_4)$ and $({\mathbb{C}}_{-},{\mathbb{C}}_{+})$, respectively.