Planar scattering dbar equation and uniform parameter bounds

Put $e_k(z)=e^{i(kz+\bar{k}\bar{z})}$, $A_k={\mathrm{\partial }}_z+ik$, and define the complex-antilinear operator $L_{k}g=\mathcal{C}[\mu e_{-k}\overline{A_{k}g}]$ on the real Banach space $Y=W^{1,r}(\mathbb{C})\oplus \mathbb{C}$. It is bounded into $W^{1,r}$ and smooth in the two real coordinates of $k$, because its coefficient derivatives are bounded multiplications on the fixed disk. Both $I-L_k$ and $I+L_k$ are invertible on $Y$. To verify surjectivity rather than assume it, prescribe a right side $h+c\in Y$, put the desired unknown equal to $h+c+j$, and differentiate the equation for $j$. Its derivative equation is $(I-\mu e_{-k}\bar{\mathcal{B}})\bar{\mathrm{\partial }}j=\alpha \bar{j}+E$, with $\alpha =-i\bar{k}\mu e_{-k}$ and compact forcing $E=\mu e_{-k}\overline{\mathrm{\partial }h}+\alpha (\bar{h}+\bar{c})$. It reduces to the same compact Fredholm operator as [F1], with arbitrary compact $L^r$ forcing, so gives a unique $j\in W^{1,r}$. Replacing $\mu$ by $-\mu$ proves the other sign. The bounded inverse follows from this Fredholm solution formula. Resolvent identities therefore imply that $M_{\pm }=(I\mp L_k{)}^{-1}1$ depend smoothly on $k$ in $Y$, with constant component always $1$.

Differentiating the multiplier equation with respect to either real coordinate of $k$ gives an equation for $W={\mathrm{\partial }}_{k_j}{M}_{\pm }$ with principal coefficient of modulus at most $\kappa$, lower-order coefficient bounded by $C\mathrm{\mid }k\mathrm{\mid }$, and compact forcing bounded in $L^r$ by $C(1+\mathrm{\mid }k\mathrm{\mid })(1+\mathrm{\parallel }{M}_{\pm }-1{\mathrm{\parallel }}_{W^{1,r}})$. Coefficient derivatives contribute only bounded factors $z,\bar{z}$ on $D_1$. By [F1] this forcing is at most $C{e}^{C(1+\mathrm{\mid }k\mathrm{\mid })}$. The derivatives $W$ are continuous, in $W^{1,r}$, and tend to zero at infinity by the compact Cauchy representation and its differentiated data. Apply [F2] to get the claimed derivative bounds. The integral identity $\overline{b_{\pm }}={\pi }^{-1}\int (\pm \mu ){\mathrm{\partial }}_z(e_k{M}_{\pm })dA$ follows by integrating the compact Cauchy data for $M_{\pm }-1$ and conjugating, using that $\mu$ is real. Differentiation under this compact integral, justified by the just-proved $W^{1,r}$ bounds, gives the stated derivative bound on $\tau$.

Define $F_{+}=(M_{+}+M_{-})\mathrm{/}2$, $J=(M_{+}-M_{-})\mathrm{/}2$, and $F_{-}=i{e}_{-k}\bar{J}$. Adding and subtracting the multiplier equations gives $F_{+}=1+L_{k}J$ and $J=L_k{F}_{+}$, hence $(I-L_k^2)F_{+}=1$. The square is complex-linear and $I-L_k^2=(I-L_k)(I+L_k)$ is invertible. Put $Q_{k}h=e_{-k}\bar{\mathcal{C}}(e_{k}h)$, where $\bar{\mathcal{C}}$ has kernel $1\mathrm{/}(\pi (\bar{z}-\bar{w}))$. Direct kernel differentiation gives, for fixed compact data $h$, ${\bar{\mathrm{\partial }}}_k{Q}_{k}h=-i{e}_{-k}{\pi }^{-1}\int e_{k}hdA$: the derivative of the exponential cancels the kernel denominator. Composing the two antilinear factors of $L_k$ gives $L_k^{2}g=\mathcal{C}\mu {\bar{\mathrm{\partial }}}_z{Q}_k\mu A_{k}g$. Since ${\bar{\mathrm{\partial }}}_k{A}_k=0$, it follows for fixed $g\in Y$ that ${\bar{\mathrm{\partial }}}_k(L_k^2)g=-it(g;k)\mathcal{C}(\mu {\bar{\mathrm{\partial }}}_z{e}_{-k})$, where $t(g;k)={\pi }^{-1}\int \mu {\mathrm{\partial }}_z(e_{k}g)dA$. All these identities hold first for smooth compact data by the displayed integrals and then by $L^r$ approximation and boundedness of the composed operators.

The coefficient identity in step 2.1 gives $t(F_{+};k)=\frac{1}{2}({\bar{b}}_{+}-{\bar{b}}_{-})=\tau$. Also $J=L_{k}1+L_k^{2}J$ and $J=i{e}_{-k}{\bar{F}}_{-}$ imply $(I-L_k^2)(e_{-k}{\bar{F}}_{-})=-i\mathcal{C}(\mu {\bar{\mathrm{\partial }}}_z{e}_{-k})$. Differentiate $(I-L_k^2)F_{+}=1$ with respect to $\bar{k}$ and apply step 3.1 and complex linearity of the inverse. Uniqueness yields ${\bar{\mathrm{\partial }}}_k{F}_{+}=\tau e_{-k}{\bar{F}}_{-}$. The definitions further give $F_{+}=1-i\mathcal{C}(\mu {\bar{\mathrm{\partial }}}_z{F}_{-})$ and $F_{-}=i{Q}_k\mu A_k{F}_{+}$. Differentiating the latter identity, using the rank-one formula of step 3.1 and the first derivative identity just proved, gives ${\bar{\mathrm{\partial }}}_k{F}_{-}=\tau e_{-k}[1+i\bar{\mathcal{C}}(\mu {\mathrm{\partial }}_z{\bar{F}}_{-})]=\tau e_{-k}{\bar{F}}_{+}$. Since $u_{\gamma }=e^{ikz}(F_{+}-i{F}_{-})$, these two identities give ${\bar{\mathrm{\partial }}}_k{u}_{\gamma }=-i\tau \overline{u_{\gamma }}$. Division by the nonzero solution in [F3] gives the displayed phase equation.

By [F3], $\mathrm{\mid }J\mathrm{/}{F}_{+}\mathrm{\mid }<1$. Outside ${\bar{D}}_1$ this quotient is holomorphic, and it tends to zero at infinity. Thus $w\mapsto J(1\mathrm{/}w)\mathrm{/}{F}_{+}(1\mathrm{/}w)$ extends to a holomorphic disk self-map vanishing at zero. Its derivative there is $(b_{+}-b_{-})\mathrm{/}2$, so Schwarz's lemma in [F3] gives $\mathrm{\mid }\tau \mathrm{\mid }\le 1$. At $k=0$, $M_{+}=M_{-}=1$, giving $\tau (0)=0$. For $z\in D_S$, the pointwise rotated-coefficient representation in [F3] and the uniform principal deformation bound in [F1] give $\mathrm{\mid }{u}_{\gamma }(z,k)\mathrm{\mid }\ge e^{-C_S\mathrm{\mid }k\mathrm{\mid }}$. Step 2.1 bounds $\mathrm{\mid }{\mathrm{\nabla }}_k{u}_{\gamma }\mathrm{\mid }$ by $C{e}^{C_S(1+\mathrm{\mid }k\mathrm{\mid })}$. Dividing gives the asserted derivative bound for ${\delta }_{\gamma }$. The normalized multipliers tend to $1$ as $k\to 0$ by [F1], so the phase continues smoothly there with value zero. This completes all claims.