Logarithmic zero separation for perturbed scattering phase equations

Set $\lambda =\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$ and choose $R=\max \{1,(8C\mathrm{/}\lambda {)}^{1\mathrm{/}b}\}$. Then for $\mathrm{\mid }k\mathrm{\mid }\ge R$, $\mathrm{\mid }g(k)-\mathrm{\ell }k\mathrm{\mid }\le (\lambda \mathrm{/}8)\mathrm{\mid }k\mathrm{\mid }$, so $g$ is nonzero there and its restriction to $\mathrm{\mid }k\mathrm{\mid }=R$ has winding one, by the straight homotopy to $\mathrm{\ell }k$. Choose a smooth real cutoff ${\chi }_R$ equal to $1$ on $D_R$, supported in $D_{2R}$, with derivative bounded by $C\mathrm{/}R$, and set $s=\mathcal{C}(\sigma {\chi }_R)$. Direct integration of its kernel gives $\mathrm{\parallel }s{\mathrm{\parallel }}_{\mathrm{\infty }}\le C_{B}R$. The $L^4$ Cauchy estimate in [F1] gives a one-half Hölder bound at most $C_B{R}^{1\mathrm{/}2}$. These estimates hold independently of the regularity of $\sigma$.

Put $H=e^{-s}E{\chi }_R$ and $S=\mathcal{C}H-\mathcal{C}H(0)$. On $D_{2R}$ the hypotheses give $\mathrm{\parallel }E{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\delta e^{2CR}$ and $\mathrm{Lip}(E)\le C{e}^{C(1+2R)}$. The elementary estimate $\mathrm{\mid }E(x)-E(y)\mathrm{\mid }\le \min \{2\mathrm{\parallel }E{\mathrm{\parallel }}_{\mathrm{\infty }},\mathrm{Lip}(E)\mathrm{\mid }x-y\mathrm{\mid }\}$ implies $[E{]}_{C^{0,1\mathrm{/}2}}\le C{\delta }^{1\mathrm{/}2}{e}^{C^{\mathrm{\prime }}R}$ on that disk. Combine this with step 1.1 and the cutoff product estimate to get $\mathrm{\parallel }H{\mathrm{\parallel }}_{C^{0,1\mathrm{/}2}(\mathbb{C})}\le C{\delta }^{1\mathrm{/}2}{e}^{C^{\mathrm{\prime }}R}$; the smooth cutoff vanishes before leaving $D_{2R}$, so extension by zero retains this bound after increasing the constant. By [F1] and dilation, $\mathrm{\parallel }\mathrm{\nabla }S{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{\delta }^{1\mathrm{/}2}{e}^{C_{1}R}$, absorbing polynomial powers of $R$ into the exponential. Also $S(0)=0$, so $\mathrm{\mid }S(k)\mathrm{\mid }\le C{\delta }^{1\mathrm{/}2}{e}^{C_{1}R}\mathrm{\mid }k\mathrm{\mid }$. If $E=0$, take $S=0$ and these bounds hold with zero right side.

Define $F=e^{-s}g-S$. The distributional product rule, justified by the locally Sobolev Cauchy gauge and smooth approximation, gives $\bar{\mathrm{\partial }}F=e^{-s}(1-{\chi }_R)(\sigma g+E)$, hence $F$ is holomorphic on $D_R$ by [F2], and $F(0)=0$. Suppose $C{\delta }^{1\mathrm{/}2}{e}^{(C_1+C_B)R}\le \lambda \mathrm{/}8$. On $\mathrm{\mid }k\mathrm{\mid }=R$, step 1.1 gives $\mathrm{\mid }{e}^{-s}g\mathrm{\mid }\ge (7\mathrm{/}8)\lambda R{e}^{-C_{B}R}$ and step 2.1 makes $\mathrm{\mid }S\mathrm{\mid }\le (1\mathrm{/}8)\lambda R{e}^{-C_{B}R}$. Thus $F$ is nonzero on that boundary, has $\mathrm{\mid }F\mathrm{\mid }\ge (3\mathrm{/}4)\lambda R{e}^{-C_{B}R}$ there, and is homotopic there to $e^{-s}g$, whose winding is one: $e^{-s}$ has a continuous logarithm on the disk, so contributes winding zero. Since $F$ is continuous on ${\bar{D}}_R$, this positive boundary lower bound and compactness of the circle give a zero-free inner collar: choose $\rho <R$ so $F$ is nonzero on $\{\rho \le \mathrm{\mid }k\mathrm{\mid }\le R\}$. The circles $\mathrm{\partial }{D}_{\rho }$ and $\mathrm{\partial }{D}_R$ are homotopic through this collar in $\mathbb{C}\setminus \{0\}$, so $F{\mathrm{\mid }}_{\mathrm{\partial }{D}_{\rho }}$ also has winding one. The function $F$ is holomorphic on a neighborhood of ${\bar{D}}_{\rho }$, and [F2] applied to this inner circle gives exactly one zero counted with multiplicity in $D_{\rho }$. The collar has no zeros, so this is the full zero count in $D_R$. Since $F(0)=0$, it is simple and there are no others. Consequently $H(k):=F(k)\mathrm{/}k$, with its holomorphic value at $0$, is nonvanishing on $D_R$ and continuous and nonvanishing on ${\bar{D}}_R$. Its modulus attains a minimum on the compact closed disk. If that minimum occurs in the interior, the minimum-modulus principle [F2] makes $H$ constant; otherwise it occurs on the boundary. In either case the boundary lower bound gives $\mathrm{\mid }H(k)\mathrm{\mid }\ge (3\mathrm{/}4)\lambda e^{-C_{B}R}$ throughout $D_R$. Step 2.1 now yields $\mathrm{\mid }F(k)+S(k)\mathrm{\mid }\ge (5\mathrm{/}8)\lambda e^{-C_{B}R}\mathrm{\mid }k\mathrm{\mid }$ for $0<\mathrm{\mid }k\mathrm{\mid }<R$. Hence $g=e^s(F+S)$ has no zero except zero inside this disk, and step 1.1 excludes all the exterior zeros. If $E=0$, the same proof works for every $\lambda >0$.

It remains to ensure the displayed smallness condition. Write $L=\log (1\mathrm{/}\delta )$. If $\lambda \ge D{L}^{-b}$, then $R\le 1+(8C\mathrm{/}D{)}^{1\mathrm{/}b}L$. Choose $D$ large enough that $(C_1+C_B)(8C\mathrm{/}D{)}^{1\mathrm{/}b}\le 1\mathrm{/}4$. For large $L$, the left side of the smallness condition is at most $C^{\mathrm{\prime }}{e}^{-L\mathrm{/}4}$, while its right side is at least $(D\mathrm{/}8)L^{-b}$. Since $e^{-L\mathrm{/}4}{L}^b\to 0$, there is a fixed $L_0$ for which the condition holds whenever $L\ge L_0$. Put ${\delta }_0=\min \{1\mathrm{/}4,e^{-L_0}\}$. Step 3.1 then proves the asserted implication and its zero-error version.