C_1 >0.$$

We further assume that $s - \frac12- \beta     < 0$. Then for ${N_{\text{HL}}}\gg 1$ sufficiently large and $\delta$ sufficiently small, the difference equation has a solution in $H^s \times H^{\beta - \frac12  -}$. Furthermore, if we write $$\begin{aligned}
m(x,t) &= {\operatorname{W}}(t) m_0(x) + {\widetilde}{m}(x,t),\end{aligned}$$ then we have for $t \in [0,\delta]$ $$\| v \|_{H_x^s} + \| {\widetilde}{m}\|_{ L^2_x} \lesssim {N_{\text{HL}}}^{s-\beta - \frac12   } \ll 1.$$

The proof of this amounts to proving a contraction for the difference equation $$\begin{cases}
i{v} + \Delta v = {\operatorname{Re}}(n + {\widetilde}{m} + {\operatorname{W}}(t)m_0)(u + v ) - {\operatorname{Re}}(n) u, \\
i{\widetilde}{m}_t - D{\widetilde}{m} = D[|u + v |^2 - |u|^2] 
\end{cases}$$ with zero initial data on the interval $[0,\delta]$ in a ball of radius $\approx {N_{\text{HL}}}^{s -\frac12 -\beta   }$ in the space $X^{s,\frac12, 1}_{\delta} \times Y^{0,\frac12, 1}_{\delta}$.

The only problematic term to estimate is $${\operatorname{Re}}({\operatorname{W}}(t) m_0)(u + v).$$ All others are covered by existing bilinear estimates due to Kishimoto [@Kish1] as long as we take $\delta$ sufficiently small. Furthermore, since $u$ typically has much larger norm that $v$, it suffices to consider $${\operatorname{Re}}({\operatorname{W}}(t) m_0)u.$$ To close the contraction, it suffices to obtain a bound of the form $$\left\|  \eta_\delta(t) \int_0^t e^{i(t-t')\Delta}\Bigl({\operatorname{Re}}({\operatorname{W}}(t) m_0)u 