= Volterra analyticity sectors for reverse-dispersion mKdV
{c}
{title2=$D_j:\ (\operatorname{Im}k,\operatorname{Im}k^3)$}
For the <modified Korteweg-De Vries equation> with linear part $q_t-q_{xxx}$, the <Volterra integral equation> kernel is $e^{i[k(x-x')-4k^3(t-t')]\widehat\sigma_3}$. In column one its off-diagonal modulus is $e^{2\operatorname{Im}k(x-x')-8\operatorname{Im}k^3(t-t')}$; 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.
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