 Volterra analyticity sectors for reverse-dispersion mKdV

For the modified Korteweg-De Vries equation with linear part $q_t-q_{xxx}$, the Volterra integral equation kernel is $e^{i[k(x-x^{\prime })-4k^3(t-t^{\prime })]{\hat{\sigma }}_3}$. In column one its off-diagonal modulus is $e^{2\mathrm{Im}k(x-x^{\prime })-8\mathrm{Im}{k}^3(t-t^{\prime })}$; column two has the reciprocal modulus. Define $D_1$ by signs $(+,+)$, $D_2$ by $(+,-)$, $D_3$ by $(-,+)$ and $D_4$ by $(-,-)$. The normalization points $(0,T)$, $(0,0)$ and $(\infty ,t)$ give column domains $(D_4,D_1)$, $(D_3,D_2)$ and $({\mathbb{C}}_{+},{\mathbb{C}}_{-})$, respectively. Finite-path eigenfunctions can be entire at fixed coordinates while failing to be bounded outside these spectral sectors.