Diffusive Stability of Turing Patterns via Normal Forms

Abstract

We investigate dynamics near Turing patterns in reaction–diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a “normal form” coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction–diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in $L^1\cap L^\infty$, with sharp rates, recovering and slightly improving on results in Johnson and Zumbrun (Ann Inst H Poincaré Anal Non Linéaire 28:471–483, 2011) and Schneider (Commun Math Phys 178:679–702, 1996).

Appendix

1.1 Estimates on Nonlinear Terms

In this section, we derive the estimates on the nonlinear terms $\mathbf{N}^\theta$ and $\mathbf{N}^\mathbf{w}$ in our normal form (2.21).

Lemma 6.1

For $\Vert \underline{\mathbf{W}}\Vert _{X_{\mathrm{ch}}}, \Vert \underline{\theta }\Vert _{\ell ^1}<\varepsilon$, where $\varepsilon$ is sufficiently small($0<\varepsilon \le \varepsilon _0$), there exists a nondecreasing function $C(\varepsilon )>0$ such that, for all $1\le p\le \infty$, the nonlinear terms in system (2.21) have the following estimates.

$$\begin{aligned} |\mathbf{N}^\theta _j|&\le C(\varepsilon )\bigg [\sum _{k=-1}^{0}|(\delta _+\underline{\theta })_{j+k}|^2+\left( \sum _{k=-1}^{1}|\theta _{j+k}|^3\right) \left( \sum _{k=-1}^{0}|(\delta _+\underline{\theta })_{j+k}|\right) \nonumber \\&+\left( \sum _{k=-1}^{1}|\theta _{j+k}|\right) \bigg (|(\delta _+\underline{\mathbf{W}})_j(-\pi )|+|(\delta _+\partial _x\underline{\mathbf{W}})_j(-\pi )| \bigg )\nonumber \\&+\Vert \mathbf{W}_j\Vert _{L^p}|(\delta _+\underline{\mathbf{W}})_j(-\pi )|+\Vert \mathbf{W}_j^2\Vert _{L^p}\bigg ],\nonumber \\ \Vert \mathbf{N}^\mathbf{w}_j\Vert _{L^p}&\le C(\varepsilon )\bigg [\left( \sum _{k=-1}^{0}|(\delta _+\underline{\theta })_{j+k}|\right) \left( \sum _{k=-1}^1|\theta _{j+k}|\right) + |\theta _j|\Vert \mathbf{W}_j\Vert _{L^p}\nonumber \\&+\left( \sum _{k=-1}^1|\theta _{j+k}|\right) \left( \sum _{k=-1}^1|(\delta _+\underline{\mathbf{W}})_{j+k}(-\pi )|+|(\delta _+\partial _x\underline{\mathbf{W}})_{j+k}(-\pi )|\right) \nonumber \\&+\Vert \mathbf{W}_j\Vert _{L^p}|(\delta _+\underline{\mathbf{W}})_{j}(-\pi )| \!+\!\Vert \mathbf{W}^2_j\Vert _{L^p}+ |\mathbf{N}^\theta _j|\!+\!|\mathbf{N}^\theta _{j+1}|+|\mathbf{N}^\theta _{j-1}|\bigg ].\quad \end{aligned}$$

(6.1)

Proof

We point out that throughout the proof, we repeatedly exploit the fact that the $L^2$ scalar product of an even function and an odd function are zero. We also recall that $\mathbf{u}_\star$ is even and $\mathbf{u}_{\mathrm{ad}}$ is odd. By equations (2.9),(2.10) and (2.21), we obtain

$$\begin{aligned} \mathbf{N}^\theta _j&= I_j+\left( II_j+III_j+IV_j+V_j\right) \fancyscript{S}_j \text { and }\\ \mathbf{N}^\mathbf{w}_j&= \left( \mathrm{\,id}\,-\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}\right) ^{-1}\left( VI_j+VII_j +VIII_j+IX_j+\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}X_j\right) , \text { where}\\ \fancyscript{S}_j&= \left( -1+\langle \mathbf{W}_j(x)+\mathbf{H}_j(x),\mathbf{u}_{\mathrm{ad}}^\prime (x-\theta _j)\rangle \right) ^{-1};\\ \mathbf{G}_j&= \mathbf{G}(\theta _j, \mathbf{W}_j)=\langle \mathbf{W}_j(x),\mathbf{u}_{\mathrm{ad}}(x-\theta _j)-\mathbf{u}_{\mathrm{ad}}(x)\rangle \mathbf{\psi }(x-\theta _j);\\ I_j&= (-\fancyscript{S}_j-1)(\delta _+\Gamma W)_j;\\ II_j&= -\left( \mathbf{W}_{j+1}(-\pi )-\mathbf{W}_j(-\pi ),D\left( \mathbf{u}_{\mathrm{ad}}^\prime (\pi -\theta _j)-\mathbf{u}_{\mathrm{ad}}^\prime (\pi )\right) \right) ;\\ III_j&= \left( \partial _x\mathbf{W}_{j+1}(-\pi )-\partial _x\mathbf{W}_{j}(-\pi ),D\mathbf{u}_{\mathrm{ad}}(\pi -\theta _j)\right) ;\\ IV_j&= (\partial _x\mathbf{H}_{j}(\pi )-\partial _x\mathbf{H}_{j}(-\pi ),D\mathbf{u}_{\mathrm{ad}}(\pi -\theta _j))- (\mathbf{H}_{j}(\pi )-\mathbf{H}_{j}(-\pi ),D\mathbf{u}_{\mathrm{ad}}^\prime (\pi -\theta _j));\\ V_j&= \langle \tilde{g}(\theta _j,\mathbf{W}_j+\mathbf{H}_j),\mathbf{u}_{\mathrm{ad}}(x-\theta _j)\rangle ;\\ VI_j&= A(\mathbf{H}_j-(\underline{\mathbf{E}}*\underline{\theta })_j);\\ VII_j&= \!-\!\left( \left( \dot{\mathbf{H}}_j-\mathbf{u}^\prime _\star (x-\theta _j)\dot{\theta }_j\right) \!-\!(\underline{\mathbf{E}}*\underline{\dot{\theta }})_j+\langle \dot{\mathbf{W}}_j(x),\mathbf{u}_{\mathrm{ad}}(x\!-\!\theta _j)\!-\! \mathbf{u}_{\mathrm{ad}}(x)\rangle \mathbf{\psi }(x-\theta _j)\right) ;\\ VIII_j&= (\underline{\mathbf{E}}*(\delta _+\Gamma \underline{\mathbf{W}}-\underline{\dot{\theta }}))_j;\\ IX_j&= \tilde{g}(\theta _j,\mathbf{W}_j+\mathbf{H}_j)+\left[ \mathbf{f}^\prime (\mathbf{u}_\star (x-\theta _j))-\mathbf{f}^\prime (\mathbf{u}_\star (x))\right] (\mathbf{W}_j+\mathbf{H}_j);\\ X_j&= A(\underline{\mathbf{E}}*\underline{\theta })_j+A\mathbf{W}_j-(\underline{\mathbf{E}}*\delta _+\Gamma \underline{\mathbf{W}})_j.\\ \end{aligned}$$

We recall here that $\underline{\mathbf{E}}$ is defined in (2.18) and point out that the term in $VII_j$ involving $\dot{\mathbf{W}}_j$ in fact cancels with a contribution from $\dot{\mathbf{H}}_j$. We now prove the estimate of $\mathbf{N}^\theta _j$.

Estimate on $I_j$: $|I_j|\le C(\varepsilon )\left( |(\delta _+\underline{\theta })_j|+ |(\delta _-\underline{\theta })_j|+\Vert \mathbf{W}_j\Vert _{L^p}\right) |(\delta _+\underline{\mathbf{W}})_j(-\pi )|$.

We first recall that $\mathbf{H}_j$ is defined in (2.10) and (2.13). We claim that the number $c_j$, appearing in the definition of $\mathbf{H}_j^2$ as in (2.15) and (2.16), can be estimated as

$$\begin{aligned} |c_j|\le C(\varepsilon )\left[ |(\delta ^2\underline{\theta })_j|+\bigg (|(\delta _+\underline{\theta })_{j}|+ |(\delta _-\underline{\theta })_{j}|\bigg )\sum _{k=-1}^1\theta _{j+k}+|\theta _j|\Vert \mathbf{W}_j\Vert _{L^p}\right] , \end{aligned}$$

where we use notation $\delta ^2=\delta _+\delta _-$. In fact, we have

$$\begin{aligned}&|\langle \phi (x)(\mathbf{u}_\star (x+\theta _j-\theta _{j+1})-\mathbf{u}_\star (x+\theta _j-\theta _{j-1})),\mathbf{u}_{\mathrm{ad}}(x)\rangle | \le C(|(\delta _+\underline{\theta })_{j}|^2+|(\delta _-\underline{\theta })_{j}|^2);\\&|\langle (\phi (x+\theta _j)-\phi (x))(\mathbf{u}_\star (x+\theta _j-\theta _{j+1})-\mathbf{u}_\star (x+\theta _j-\theta _{j-1})),\mathbf{u}_{\mathrm{ad}}(x)\rangle |\\&\quad \le C|\theta _j|(|(\delta _+\underline{\theta })_{j}|+|(\delta _-\underline{\theta })_{j}|);\\&|\langle (\mathbf{u}_\star (x+\theta _j-\theta _{j+1})+\mathbf{u}_\star (x+\theta _j-\theta _{j-1})),\mathbf{u}_{\mathrm{ad}}(x)\rangle | \le C|(\delta ^2\underline{\theta })_j|. \end{aligned}$$

We also have $|\mathbf{H}_j(x)|\le C(\varepsilon )(|(\delta _+\underline{\theta })_j| +|(\delta _-\underline{\theta })_j|+|\theta _j|\Vert \mathbf{W}_j\Vert _{L^p})$, from which we obtain the estimate.

Estimate on $II_j$: $|II_j|\le C|\theta _j|^2|(\delta _+\underline{\mathbf{W}})_j(-\pi )|$.

This is straightforward.

Estimate on $III_j$: $|III_j|\le C|\theta _j||(\delta _+\partial _x\underline{\mathbf{W}})_j(-\pi )|$.

This is straightforward.

Estimate on $IV_j$: $|IV_j|\le C\left[ |(\delta _+\underline{\theta })_j|^2+ |(\delta _-\underline{\theta })_{j}|^2+|(\delta _+\underline{\theta })_{j}+(\delta _-\underline{\theta })_{j}| (|\theta _{j+1}|^3\right. \left. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +|\theta _{j}|^3+|\theta _{j-1}|^3)\right]$.

We first simplify $IV_j$ and obtain

$$\begin{aligned} IV_j&= \frac{1}{2}(\mathbf{u}_{\star }^\prime (\pi -\theta _{j+1})-\mathbf{u}_{\star }^\prime (\pi -\theta _{j-1}),D\mathbf{u}_{\mathrm{ad}}(\pi -\theta _j))\\&\quad - \frac{1}{2}(\mathbf{u}_\star (\pi -\theta _{j+1})-\mathbf{u}_\star (\pi -\theta _{j-1}),D\mathbf{u}_{\mathrm{ad}}^\prime (\pi -\theta _j)). \end{aligned}$$

Then, it is not hard to see that

$$\begin{aligned}&\left| \frac{1}{2}\bigg (\mathbf{u}_{\star }^\prime (\pi -\theta _{j+1})-\mathbf{u}_{\star }^\prime (\pi -\theta _{j-1}),D\mathbf{u}_{\mathrm{ad}}(\pi -\theta _j)\bigg )\right. \\&\qquad \left. - \frac{1}{2}\bigg (\mathbf{u}_{\star ,\theta \theta }(\pi )(\theta _{j-1}-\theta _{j+1}),-D\mathbf{u}_{\mathrm{ad}}^\prime (\pi )\theta _j\bigg )\right| \\&\quad \le C\bigg (|\theta _j||\theta _{j+1}^3-\theta _{j-1}^3|+|\theta _j|^3|\theta _{j+1}-\theta _{j-1}|\bigg ),\\&\left| \frac{1}{2}\bigg (\mathbf{u}_\star (\pi -\theta _{j+1})-\mathbf{u}_\star (\pi -\theta _{j-1}),D\mathbf{u}_{\mathrm{ad}}^\prime (\pi -\theta _j)\bigg )\right. \\&\qquad \left. -\frac{1}{2}\bigg (\frac{1}{2}\mathbf{u}_{\star ,\theta \theta }(\pi )(\theta _{j+1}^2-\theta _{j-1}^2),D\mathbf{u}_{\mathrm{ad}}^\prime (\pi )\bigg )\right| \\&\quad \le C\bigg (|\theta _{j+1}^4-\theta _{j-1}^4|+|\theta _j|^2|\theta _{j+1}^2-\theta _{j-1}^2|\bigg ),\\&\qquad \times \frac{1}{2}\bigg (\mathbf{u}_{\star ,\theta \theta }(\pi )(\theta _{j-1}\!-\!\theta _{j+1}),\!-\!D\mathbf{u}_{\mathrm{ad}}^\prime (\pi )\theta _j\bigg ) \!-\!\frac{1}{2}\bigg (\frac{1}{2}\mathbf{u}_{\star ,\theta \theta }(\pi )(\theta _{j+1}^2-\theta _{j-1}^2),D\mathbf{u}_{\mathrm{ad}}^\prime (\pi )\bigg )\\&\quad =\frac{1}{4}\bigg (\mathbf{u}_{\star ,\theta \theta }(\pi ),D\mathbf{u}_{\mathrm{ad}}^\prime (\pi )\bigg )\bigg [(\delta _-\underline{\theta })^2_j- (\delta _+\underline{\theta })^2_j\bigg ], \end{aligned}$$

which establishes the estimate on $IV_j$ as claimed.

Estimate on $V_j$: $|V_j|\le C(\varepsilon )\bigg ( |(\delta _+\underline{\theta })_j|^2+|(\delta _-\underline{\theta })_j|^2+\Vert \mathbf{W}_j^2\Vert _{L^p}\bigg )$.

Noting that $|V_j|\le C(\varepsilon )\Vert (\mathbf{W}_j+\mathbf{H}_j)^2\Vert _{L^p}$ and applying the estimate of $\mathbf{H}_j$ to the inequality lead to the above estimate.

Estimate on $\fancyscript{S}_j$: $|\fancyscript{S}_j|\le C(\varepsilon )$.

This is straightforward.

Combining our estimates of $II_j-V_j$ and $\fancyscript{S}_j$, we obtain the first inequality in (6.1).

Now, we have to show that the estimate of $\mathbf{N}^\mathbf{w}_j$ in (6.1) is true.

Estimate on $VI_j$:

$$\begin{aligned} |VI_j|\le C(\varepsilon )\left[ \bigg (|(\delta _+\underline{\theta })_j|+ |(\delta _-\underline{\theta })_j|\bigg )\sum _{k=-1}^1|\theta _{j+k}|+|\theta _j|\Vert \mathbf{W}_j\Vert _{L^p}\right] . \end{aligned}$$

First, for $f$ $2\pi$-periodic and smooth, we have

$$\begin{aligned} |f(x-\theta _1)-f(x-\theta _2)-f^\prime (x)(\theta _2-\theta _1)|\le C\left( |\theta _2-\theta _1|^2+|\theta _2||\theta _2-\theta _1|\right) . \end{aligned}$$

If in addition, $f$ is odd, we have

$$\begin{aligned} |f(\theta _1)-f(\theta _2)-f^\prime (0)(\theta _1-\theta _2)|\le C|\theta _2^3-\theta _1^3|. \end{aligned}$$

The latter implies that

$$\begin{aligned} |c_j-\frac{1}{4}(\delta ^2\underline{\theta })_j|\le C(\varepsilon )\bigg (|(\delta _+\underline{\theta })_{j}|^2+ |(\delta _-\underline{\theta })_{j}|^2+|\theta _j|\Vert \mathbf{W}_j\Vert _{L^p}\bigg ). \end{aligned}$$

Moreover, by the former inequality, we have

$$\begin{aligned} |VI_j|&\le C\left( |(\delta _+\underline{\theta })_j|^2+|(\delta _-\underline{\theta })_j|^2+ |\theta _j||(\delta _+\underline{\theta })_j|+|\theta _j||(\delta _-\underline{\theta })_j|\right) \\&+\left| c_jA\psi (x-\theta _j)-\frac{1}{4}(\delta ^2\underline{\theta })_jA\psi (x)\right| \\&\le C(\varepsilon )\left[ \bigg (|(\delta _+\underline{\theta })_j|+ |(\delta _-\underline{\theta })_j|\bigg )\sum _{k=-1}^1|\theta _{j+k}|+|\theta _j|\Vert \mathbf{W}_j\Vert _{L^p}\right] . \end{aligned}$$

Estimate on $VII_j$:

$$\begin{aligned} |VII_j|\le C\left[ \left( \sum _{k=-1}^1|\theta _{j+k}|\right) \left( \sum _{k=-1}^1|\dot{\theta }_{j+k}|\right) +|\dot{\theta }_j|\Vert \mathbf{W}_j\Vert _{L^p}\right] . \end{aligned}$$

Noting that $(\underline{\mathbf{E}}*\underline{\theta })_j$ is the linear part of $\mathbf{H}_j+\mathbf{u}_\star (x-\theta _j)-\mathbf{u}_\star (x)$ and there is no term involving $\dot{\mathbf{W}}_j$ in $VII_j$, we have

$$\begin{aligned} |VII_j|&\le C\bigg (|\theta _{j+1}||\dot{\theta }_{j+1}|+|\theta _{j-1}||\dot{\theta }_{j-1}| +|\theta _{j}||\dot{\theta }_{j}|\bigg )\\&+\left| c_j\psi ^\prime (x-\theta _j)\dot{\theta }_j\right| +\left| \frac{1}{4}(\delta ^2\dot{\underline{\theta }})_j\psi (x)-\tilde{\dot{c}}_j\psi (x-\theta _j)\right| , \end{aligned}$$

where $\tilde{\dot{c}}_j=\dot{c}_j+\langle \dot{\mathbf{W}}_j(x),\mathbf{u}_{\mathrm{ad}}(x-\theta _j)-\mathbf{u}_{\mathrm{ad}}(x)\rangle$.

First, we note that

$$\begin{aligned} \left| c_j\psi ^\prime (x-\theta _j)\dot{\theta }_j\right| \le C|\dot{\theta }_j|\left[ |(\delta ^2\underline{\theta })_j| +\bigg (|(\delta _+\underline{\theta })_{j}|+ |(\delta _-\underline{\theta })_{j}|\bigg )\sum _{k=-1}^1\theta _{j+k}+|\theta _j|\Vert \mathbf{W}_j\Vert _{L^p}\right] . \end{aligned}$$

Moreover, we claim that

$$\begin{aligned}&|\tilde{\dot{c}}_j| \le C\left[ |(\delta _+\dot{\underline{\theta }})_j|+|(\delta _-\dot{\underline{\theta }})_j|+ \left( \sum _{k=-1}^1|\theta _{j+k}|\right) \left( \sum _{k=-1}^1|\dot{\theta }_{j+k}|\right) +|\dot{\theta }_j|\Vert \mathbf{W}_j\Vert _{L^p}\right] ,\\&\quad |\tilde{\dot{c}}_j-\frac{1}{4}(\delta ^2\dot{\underline{\theta }})_j| \le C\left[ \left( \sum _{k=-1}^1|\theta _{j+k}|\right) \left( \sum _{k=-1}^1|\dot{\theta }_{j+k}|\right) +|\dot{\theta }_j|\Vert \mathbf{W}_j\Vert _{L^p}\right] . \end{aligned}$$

In fact, we have

$$\begin{aligned}&\left| \left\langle \phi (x)(\dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j+1})- \dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j-1})),\mathbf{u}_{\mathrm{ad}}(x)\right\rangle \right| \\&\quad \le C\bigg (|(\delta _+\underline{\theta })_{j}||(\delta _+\dot{\underline{\theta }})_{j}|+ |(\delta _-\underline{\theta })_{j}||(\delta _-\dot{\underline{\theta }})_{j}|\bigg ),\\&|\langle \phi ^\prime (x+\theta _j)\dot{\theta }_j(\mathbf{u}_\star (x+ \theta _j-\theta _{j+1})-\mathbf{u}_\star (x+\theta _j-\theta _{j-1})), \mathbf{u}_{\mathrm{ad}}(x)\rangle |\\&\quad \le C|\dot{\theta }_j|\bigg (|(\delta _+\underline{\theta })_{j}|+|(\delta _-\underline{\theta })_{j}|\bigg ),\\&\left| \left\langle (\phi (x+\theta _j)-\phi (x))(\dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j+1})-\dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j-1})), \mathbf{u}_{\mathrm{ad}}(x)\right\rangle \right| \\&\quad \le C|\theta _j|\bigg (|(\delta _+\dot{\underline{\theta }})_{j}|+|(\delta _-\dot{\underline{\theta }})_{j}|\bigg ),\\&\left| \left\langle (\dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j+1})+ \dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j-1})),\mathbf{u}_{\mathrm{ad}}(x)\right\rangle \right| \\&\quad \le C\bigg (|(\delta _+\dot{\underline{\theta }})_j|+|(\delta _-\dot{\underline{\theta }})_j|\bigg ),\\&|\left\langle (\dot{\mathbf{u}}_\star (x+\theta _j-\theta _{j+1})+\dot{\mathbf{u}}_\star (x+ \theta _j-\theta _{j-1})),\mathbf{u}_{\mathrm{ad}}(x)\right\rangle +\delta ^2\underline{\dot{\theta }}_j|\\&\quad \le C\bigg (|(\delta _+\underline{\theta })_j||(\delta _+\dot{\underline{\theta }})_j|+ |(\delta _-\underline{\theta })_j||(\delta _-\dot{\underline{\theta }})_j|\bigg ), \end{aligned}$$

which establishes the claim and thus the estimate on $VII_j$.

Estimate on $VIII_j$:

$$\begin{aligned} |VIII_j|\le C\bigg (|\mathbf{N}^\theta _j|+|\mathbf{N}^\theta _{j+1}|+|\mathbf{N}^\theta _{j-1}|\bigg ). \end{aligned}$$

The calculation is straightforward using the expressions for $\mathbf{K}_j$ and $\dot{\theta }_j$.

Estimate on $IX_j$:

$$\begin{aligned} |IX_j|\le C(\varepsilon )\left[ \left( \sum _{k=0}^1|(\delta _-\underline{\theta })_{j+k}|\right) \left( \sum _{k=-1}^1|\theta _{j+k}\right) +|\theta _j||\mathbf{W}_j|+|\theta _j|^2\Vert \mathbf{W}_j\Vert _{L^p}+|\mathbf{W}_j|^2 \right] . \end{aligned}$$

The calculation is straightforward using the estimate on $\mathbf{H}_j$.

Estimate on $\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}X_j$:

$$\begin{aligned} |\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}X_j|&\le C(\varepsilon )\left( |\theta _j|\sum _{k=-1}^1\bigg (|(\delta _+\underline{\theta })_{j+k}|+|(\delta _+\underline{\mathbf{W}})_{j+k}(-\pi )|\bigg )\right. \\&\left. + |\langle A\mathbf{W}_j(x),\mathbf{u}_{\mathrm{ad}}(x-\theta _j)-\mathbf{u}_{\mathrm{ad}}(x)\rangle |\right) . \end{aligned}$$

Integrating by parts, we have

$$\begin{aligned}&\langle A\mathbf{W}_j(x),\mathbf{u}_{\mathrm{ad}}(x-\theta _j)-\mathbf{u}_{\mathrm{ad}}(x)\rangle = \langle \mathbf{W}_j(x),A^*\left( \mathbf{u}_{\mathrm{ad}}(x-\theta _j)-\mathbf{u}_{\mathrm{ad}}(x)\right) \rangle \\&\quad +(\partial _x\mathbf{W}_{j+1}(-\pi )-\partial _x\mathbf{W}_{j}(-\pi ), D(\mathbf{u}_{\mathrm{ad}}(\pi -\theta _j)-\mathbf{u}_{\mathrm{ad}}(\pi )))\\&\quad -(\mathbf{W}_{j+1}(-\pi )-\mathbf{W}_{j}(-\pi ), D(\mathbf{u}^\prime _{\mathrm{ad}}(\pi -\theta _j)-\mathbf{u}^\prime _{\mathrm{ad}}(\pi ))). \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\Big |\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}X_j\Big | \le C(\varepsilon )|\theta _j|\left( \sum _{k=-1}^1\bigg (|(\delta _+\underline{\theta })_{j+k}|+|(\delta _+\underline{\mathbf{W}})_{j+k}(-\pi )|\bigg )\right. \\&\quad \left. + \Vert \mathbf{W}_j\Vert _{L^p}+|(\delta _+\partial _x\underline{\mathbf{W}})_{j}(-\pi )|\right) . \end{aligned}$$

Estimate on $(\mathrm{\,id}\,-\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j})^{-1}$: For any $\underline{\theta }\in \ell ^\infty$ and $p\in [1, \infty ]$, there exists a constant $C>0$ such that

$$\begin{aligned} |||{\left( \mathrm{\,id}\,-\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}\right) ^{-1}}|||_{L^p}\le C. \end{aligned}$$

Combining estimates on $VI_j$ to $IX_j$, $\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j}$ and $(\mathrm{\,id}\,-\frac{\partial \mathbf{G}_j}{\partial \mathbf{W}_j})^{-1}$, we obtain the second inequality in (6.1). $\square$

Moreover, we have the following lemma.

Lemma 6.2

There exist $C>0$ and $\eta >0$ such that, for all $(\underline{\theta },\underline{\mathbf{W}})\in Y$ with its $Y$-norm smaller than $\eta$, we have

$$\begin{aligned} \Vert \mathbf{N}^\theta (s)\Vert _{\ell ^1}&\le \frac{C}{(1+s)^{\frac{3}{2}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2\\&+ \frac{C}{(1+s)^{\frac{5}{4}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y(1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2},\\ \Vert \mathbf{N}^\theta (s)\Vert _{\ell ^2}&\le \frac{C}{(1+s)^{\frac{3}{2}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2\\&+ \frac{C}{(1+s)^{\frac{3}{2}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y(1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2},\\ \Vert \mathbf{N}^{\mathbf{w}}(s)\Vert _{X_1}&\le \frac{C}{1+s}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2\\&+ \frac{C}{(1+s)^{\frac{5}{4}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y(1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2},\\ \Vert \mathbf{N}^{\mathbf{w}}(s)\Vert _{X_2}&\le \frac{C}{(1+s)^{\frac{5}{4}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2\\&+\frac{C}{(1+s)^{\frac{3}{2}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y(1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2}. \end{aligned}$$

Proof

The estimates are obtained through a direct calculation from the estimates in Lemma 6.1. We sketch the computation for $\Vert \mathbf{N}^\theta (s)\Vert _{\ell ^1}$, and the others follow similarly.

First, for terms only involving $\underline{\theta }$, we notice that

$$\begin{aligned}&\sum _{j\in \mathbb {Z}}|(\delta _+\underline{\theta })_j|^2=-\sum _{j\in \mathbb {Z}}\theta _j(\delta ^2\underline{\theta })_j\le \Vert \underline{\theta }\Vert _{\ell ^2}\Vert \delta ^2\underline{\theta }\Vert _{\ell ^2}\le \frac{1}{(1+s)^{\frac{3}{2}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2,\\&\sum _{j\in \mathbb {Z}}|\theta _j|^3|(\delta _+\underline{\theta })_j|\le \Vert \underline{\theta }\Vert _{\ell ^\infty }^2 \Vert \underline{\theta }\Vert _{\ell ^2}\Vert \delta _+\underline{\theta }\Vert _{\ell ^2}\le \Vert \underline{\theta }\Vert _{\ell ^\infty }^2 \Vert \underline{\theta }\Vert _{\ell ^2}^{\frac{3}{2}}\Vert \delta ^2\underline{\theta }\Vert _{\ell ^2}^{\frac{1}{2}}\\&\qquad \qquad \quad \qquad \qquad \le \frac{1}{(1+s)^2} \Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^4. \end{aligned}$$

Second, for terms involving $\underline{\mathbf{W}}$, we observe that

$$\begin{aligned} \sum _{j\in \mathbb {Z}}|\theta _j||(\delta _+\partial _x\underline{\mathbf{W}})_j(-\pi )|&\le \Vert \underline{\theta }\Vert _{\ell ^2}\left( \sum _{j\in \mathbb {Z}}\left( \int \limits _{-\pi }^\pi \left( \partial _{xx}\mathbf{W}_{j+1}(x) -\partial _{xx}\mathbf{W}_j(x)\right) \mathrm{d}x\right) ^2\right) ^{\frac{1}{2}}\\&\le \sqrt{2\pi }\Vert \underline{\theta }\Vert _{\ell ^2}\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}\Vert _{X_2}\\&\le \frac{\sqrt{2\pi }}{(1+s)^{\frac{5}{4}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y (1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2}. \end{aligned}$$

Similarly, for $\sum _{j\in \mathbb {Z}}|\theta _j||(\delta _+\underline{\mathbf{W}})_j(-\pi )|$, we have

$$\begin{aligned} \sum _{j\in \mathbb {Z}}|\theta _j||(\delta _+\underline{\mathbf{W}})_j(-\pi )|\le \sqrt{2\pi }\Vert \underline{\theta }\Vert _{\ell ^2}\Vert \delta _+\partial _{x}\underline{\mathbf{W}}\Vert _{X_2}. \end{aligned}$$

Using the “homogeneous matching boundary conditions” (2.11), we have

$$\begin{aligned} \Vert \delta _+\partial _{x}\underline{\mathbf{W}}\Vert _{X_2}&= \left( -\sum _{j\in \mathbb {Z}}\int \limits _{-\pi }^\pi \left( \delta _+\underline{\mathbf{W}}\right) _j(x) \left( \delta _+\partial _{xx}\underline{\mathbf{W}}\right) _j(x)\mathrm{d}x\right) ^{\frac{1}{2}}\\&\le \Vert \delta _+\underline{\mathbf{W}}\Vert _{X_2}^{\frac{1}{2}}\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}\Vert _{X_2}^{\frac{1}{2}}\\&\le \Vert \delta _+\underline{\mathbf{W}}\Vert _{X_2}+\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}\Vert _{X_2}. \end{aligned}$$

We plug the latter estimate into the former one and obtain that

$$\begin{aligned} \sum _{j\in \mathbb {Z}}|\theta _j||(\delta _+\underline{\mathbf{W}})_j(-\pi )|&\le \sqrt{2\pi }\Vert \underline{\theta }\Vert _{\ell ^2}\bigg ( \Vert \delta _+\underline{\mathbf{W}}\Vert _{X_2}+\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}\Vert _{X_2}\bigg )\\&\le \frac{\sqrt{2\pi }}{(1+s)^{\frac{3}{2}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2\\&+ \frac{\sqrt{2\pi }}{(1+s)^{\frac{5}{4}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y (1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2}. \end{aligned}$$

For $\sum _{j\in \mathbb {Z}}\Vert \mathbf{W}_j\Vert _{L^p}|(\delta _+\underline{\mathbf{W}})_j(-\pi )|$, we take $p=2$ and follow steps as above, obtaining the following estimate.

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\Vert \mathbf{W}_j\Vert _{L^2}|(\delta _+\underline{\mathbf{W}})_j(-\pi )|&\le \frac{\sqrt{2\pi }}{(1+s)^{2}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2\\&+ \frac{\sqrt{2\pi }}{(1+s)^{\frac{7}{4}}}\Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y (1+s)\Vert \delta _+\partial _{xx}\underline{\mathbf{W}}(s)\Vert _{X_2}. \end{aligned}$$

For $\sum _{j\in \mathbb {Z}}\Vert \mathbf{W}_j^2\Vert _{L^p}|$, we take $p=1$ and obtain that

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\Vert \mathbf{W}_j^2\Vert _{L^1}\le \Vert \underline{\mathbf{W}}\Vert _{X_2}^2\le \frac{1}{(1+s)^{\frac{3}{2}}} \Vert (\underline{\theta }(t), \underline{\mathbf{W}}(t))\Vert _Y^2. \end{aligned}$$

Combining the above estimate, we establish the first inequality in the lemma. $\square$

1.2 Bloch Wave Decomposition

In this section, we present the Bloch wave decomposition of the linear operator $\widetilde{A}$. We first recall that $\widetilde{A}$, as in (3.1), is defined as

$$\begin{aligned} \begin{array}{llll} \widetilde{A}:&{} (H^2(\mathbb {R}))^n&{}\longrightarrow &{}(L^2(\mathbb {R}))^n\\ &{}\mathbf{v}&{}\longmapsto &{}D\partial _{xx}\mathbf{v}-\mathbf{f}^\prime (\mathbf{u}_\star )\mathbf{v}. \end{array} \end{aligned}$$

We introduce the direct integral [17, XIII.16]

$$\begin{aligned} \begin{array}{llll} \fancyscript{B}: &{}L^2(\mathbb {T}_1, (L^2(\mathbb {T}_{2\pi }))^n)&{}\longrightarrow &{}(L^2(\mathbb {R}))^n\\ &{}\mathbf{U}(\sigma ,x)&{}\longmapsto &{} \int \limits _{\sigma \in \mathbb {T}_1}\mathrm{e}^{\mathrm{i}\sigma \cdot x}\mathbf{U}(\sigma ,\cdot )\mathrm{d}\sigma \end{array}. \end{aligned}$$

(6.2)

The direct integral is an isometric isomorphism with inverse

$$\begin{aligned} \begin{array}{llll} \fancyscript{B}^{-1}: &{}(L^2(\mathbb {R}))^n &{}\longrightarrow &{}L^2(\mathbb {T}_1, (L^2(\mathbb {T}_{2\pi }))^n)\\ &{}\mathbf{u}(x)&{}\longmapsto &{}\frac{1}{2\pi }\sum _{\ell \in \mathbb {Z}^m}\mathrm{e}^{\mathrm{i}\ell \cdot x}\widehat{\mathbf{u}}(\sigma +\ell ). \end{array} \end{aligned}$$

The following result from [14, 20] characterizes the Bloch wave decomposition of $\widetilde{A}$.

Theorem 2

(Bloch wave decomposition) The linear operator $\widetilde{A}$ is diagonal in Bloch wave space. To be precise,

$$\begin{aligned} \fancyscript{B}^{-1}\widetilde{A}\fancyscript{B}=\widehat{A}=\int \limits _{-\frac{1}{2}}^{\frac{1}{2}}B(\sigma )\mathrm{d}\sigma , \end{aligned}$$

(6.3)

where by $\widehat{A}=\int \limits _{-\frac{1}{2}}^{\frac{1}{2}}B(\sigma )\mathrm{d}\sigma$, we mean that, given any $\mathbf{u}\in L^2(\mathbb {T}_1, (L^2(\mathbb {T}_{2\pi }))^n)$,

$$\begin{aligned} (\widehat{A}\mathbf{u})(\sigma )=B(\sigma )\mathbf{u}(\sigma ),\text { a.e. }\sigma \in \left[ -\frac{1}{2},\frac{1}{2}\right] . \end{aligned}$$

Moreover, we have the following spectral mapping property.

$$\begin{aligned} \mathop {\mathrm{spec}}(\widetilde{A})=\mathop {\mathrm{spec}}(\widehat{A})=\bigcup _{\sigma \in [-\frac{1}{2}, \frac{1}{2}]}\mathop {\mathrm{spec}}(B(\sigma )). \end{aligned}$$

(6.4)

1.3 Spectral Properties of $\{\widehat{A}_{\mathrm{ch}}(\sigma )\} _{\sigma \in [-\frac{1}{2},\frac{1}{2}]}$

We recall that $\widehat{A}_{\mathrm{ch}}(\sigma )$ is defined in (3.9) as $\widehat{A}_{\mathrm{ch}}(\sigma )=\fancyscript{F}_nB(\sigma )\fancyscript{F}_n^{-1}$ and $Y_q$ in (3.15) for $1\le q\le \infty$. We are concerned with their spectral properties as unbounded operators in $Y_q$, which is useful for the derivation of the estimates for $M(t,\sigma )$ as defined in (3.23).

We first show the well-definedness of $\widehat{A}_{\mathrm{ch}}(\sigma )$ in $Y_q$ in the following lemma.

Lemma 6.3

For any given $\sigma \in [-\frac{1}{2},\frac{1}{2}]$, $\widehat{A}_{\mathrm{ch}}(\sigma )$ is an unbounded closed operator in $Y_2$, that is,

$$\begin{aligned} \begin{array}{lrll} \widehat{A}_{\mathrm{ch}}(\sigma ):&{}\fancyscript{D}_2(\widehat{A}_{\mathrm{ch}}(\sigma ))\subset Y_2&{}\longrightarrow &{}Y_2\\ &{}\underline{\mathbf{w}}&{}\longmapsto &{}\left\{ -(\sigma +\ell )^2D\mathbf{w}_\ell + \sum _{k\in \mathbb {Z}}\mathbf{h}_{\ell -k}\mathbf{w}_k\right\} _{\ell \in \mathbb {Z}}, \end{array} \end{aligned}$$

(6.5)

where $\fancyscript{D}_2(\widehat{A}_{\mathrm{ch}}(\sigma ))=\{\mathbf{w}\in Y_2\mid \{(1+m^2)\mathbf{w}_m\}_{m\in \mathbb {Z}}\in Y_2\}$ and $\underline{\mathbf{h}}=\{\mathbf{h}_\ell \}_{\ell \in \mathbb {Z}}= \frac{1}{2\pi }\int \limits _{-\pi }^{\pi }\mathbf{f}^\prime (\mathbf{u}_\star (x))\mathrm{e}^{-\mathrm{i}kx} \mathrm{d}x$. Moreover, $\widehat{A}_{\mathrm{ch}}(\sigma )$ can naturally be considered as an unbounded closed operator in $Y_q$, with $\fancyscript{D}_q(\widehat{A}_{\mathrm{ch}}(\sigma ))=\{\mathbf{w}\in Y_q\mid \{(1+m^2)\mathbf{w}_m\}_{m\in \mathbb {Z}}\in Y_q\}$, for all $1\le q \le \infty$.

Proof

The expression for $\widehat{A}_{\mathrm{ch}}(\sigma )$ in $Y_2$ follows from a direct calculation. The extension to $Y_q$ follows from the fact that the set $\{\underline{\mathbf{w}}\in Y_\infty \mid \underline{\mathbf{w}} \text { has finitely many nonzero entries}\}$ is dense in $Y_q$ and $\fancyscript{D}_q(\widehat{A}_{\mathrm{ch}}(\sigma ))$, for all $q\in [1,\infty ]$. $\square$

We then have the following proposition.

Proposition 6.4

For any fixed $\sigma \in [-\frac{1}{2},\frac{1}{2}]$ and $p\in [1,\infty ]$, $\widehat{A}_{\mathrm{ch}}(\sigma )$ defined in $Y_q$ is sectorial and has compact resolvent. In fact, there exist $C>0$, $\omega \in (\pi /2,\pi )$ and $\lambda _0\in \mathbb {R}$, independent of $\sigma$ and $q$, such that the sector $S(\lambda _0,\omega )=\{\lambda \in \mathbb {C}\mid 0\le |\arg (\lambda -\lambda _0)| \le \omega , \lambda \ne \lambda _0\}\subseteq \rho (\widehat{A}_{\mathrm{ch}}(\sigma ))$ and

$$\begin{aligned} |||{(\widehat{A}_{\mathrm{ch}}(\sigma )\!-\!\lambda )^{-1}}|||_{Y_q}\le C|\lambda \!-\!\lambda _0|^{-1},\quad \text {for all } \lambda \in S(\lambda _0,\omega ),\sigma \in \left[ \!-\!\frac{1}{2},\frac{1}{2}\right] \text { and }q\in [1,\infty ].\nonumber \\ \end{aligned}$$

(6.6)

Moreover, for any fixed $\sigma \in [-\frac{1}{2},\frac{1}{2}]$, the spectrum of $\widehat{A}_{\mathrm{ch}}(\sigma )$ is independent of the choice of its underlying space $Y_q$ and thus denoted as $\mathop {\mathrm{spec}}(\widehat{A}_{\mathrm{ch}}(\sigma ))$, for any $q\in [1,\infty ]$, with $\mathop {\mathrm{spec}}(\widehat{A}_{\mathrm{ch}}(\sigma ))=\mathop {\mathrm{spec}}(B(\sigma ))$, consisting only of isolated eigenvalues with finite multiplicity.

Proof

We view $\widehat{A}_{\mathrm{ch}}(\sigma )$ as a perturbation of the Laplacian in the discrete Fourier space, that is,

$$\begin{aligned} \widehat{A}_{\mathrm{ch}}(\sigma )=L(\sigma )+H, \end{aligned}$$

where $L(\sigma )\underline{\mathbf{w}}=\{-(\sigma +\ell )^2D\mathbf{w}_\ell \}_{\ell \in \mathbb {Z}}$ and $H\underline{\mathbf{w}}=\{\sum _{k\in \mathbb {Z}}\mathbf{h}_{\ell -k}\mathbf{w}_k\}_{\ell \in \mathbb {Z}}$. It is straightforward to verify that the proposition holds for the Laplacian $L(\sigma )$. We only have to show that the perturbation $H$ is good enough to preserve these properties. Noting that $H\in \fancyscript{L}((\ell ^p)^n)$ for any $p\in [1,\infty ]$ with its norm uniformly bounded, we have, for $\lambda \in \rho (L(\sigma ))$, $|\lambda |$ sufficiently large,

$$\begin{aligned} (\widehat{A}_{\mathrm{ch}}(\sigma )-\lambda )^{-1}=(L(\sigma )+H-\lambda )^{-1}=(L(\sigma )-\lambda )^{-1}(\mathrm{\,id}\,+H(L(\sigma )-\lambda )^{-1})^{-1}.\qquad \end{aligned}$$

(6.7)

All assertions in the proposition easily follows from this expression (6.7), except for the fact that the spectrum of $\widehat{A}_{\mathrm{ch}}(\sigma )$ is independent of $q$.

To prove this property, we denote the spectrum of $\widehat{A}_{\mathrm{ch}}(\sigma )$ defined on $Y_q$ as $\mathop {\mathrm{spec}}(\widehat{A}_{\mathrm{ch}}(\sigma ),q)$, which consists of eigenvalues with finite multiplicity, accumulating at infinity, only. Given any eigenfunction $\underline{\mathbf{v}}=\{\mathbf{v}_j\}_{j\in \mathbb {Z}}$, $\underline{\mathbf{v}}$ belongs to $\bigcap _{q\in [1,\infty ]}Y_q$ since $\underline{\mathbf{v}}$ are smooth, that is, $\mathbf{v}_j$ decays algebraically with any rate. This establishes $\mathop {\mathrm{spec}}(\widehat{A}(\sigma ),q)=\mathop {\mathrm{spec}}(\widehat{A}(\sigma ),p)$, for any $p, q\in [1,\infty ]$. $\square$

1.4 Perturbation Results

We apply perturbation theory to the Bloch wave operator $B(\sigma )$ for $\sigma$ near $0$ and obtain more detailed spectral information, including the Taylor expansion of $d$ in Hypotheses 1.2.

To this end, we define

$$\begin{aligned} \begin{array}{llll} F: &{}[-\frac{1}{2},\frac{1}{2}]\times \mathbb {C}\times H^2_{\perp }&{}\longrightarrow &{}L^2\\ &{}(\sigma , \lambda , \mathbf{w})&{}\longmapsto &{}(B(\sigma )-\lambda )(\mathbf{w}+\mathbf{u}_\star ^\prime ), \end{array} \end{aligned}$$

where $H^2_{\perp }=\{\mathbf{w}\in (H^2(\mathbb {T}_{2\pi }))^n\mid \langle \mathbf{w}, \mathbf{u}_\star ^\prime \rangle =0\}.$ A standard implicit-function-theorem argument shows that there are a small neighborhood of $\sigma$ at the origin and a smooth function $(\lambda (\sigma ), \mathbf{w}(\sigma ))$ with $(\lambda (\sigma ), \mathbf{w}(\sigma ))=0$ on this neighborhood such that $F(\sigma , \lambda (\sigma ),\mathbf{w}(\sigma ))=0$. We denote $\mathbf{e}(\sigma )=\mathbf{u}_\star ^\prime +\mathbf{w}(\sigma )$. Similarly, replacing $B(\sigma )$ with its adjoint $B^*(\sigma )$, we obtain a smooth continuation of $\mathbf{u}_{\mathrm{ad}}$, denoted as $\mathbf{e}^*(\sigma )$. Without loss of generality, we can assume that $\langle \mathbf{e}(\sigma ),\mathbf{e}^*(\sigma )\rangle =1$. Moreover, we have the following proposition.

Proposition 6.5

There exist positive numbers $\gamma _0$ and $\gamma _1$ such that for any $|\sigma |\le \gamma _0$ in $\mathbb {R}$, $B(\sigma )$ has only one simple eigenvalue within the strip $|\mathop {\mathrm{Re}}\lambda |\le \gamma _1$ in $\mathbb {C}$, which is exactly the continuation $\lambda (\sigma )$ of the eigenvalue $\lambda (0)=0$. Moreover, $\lambda (\sigma )$ has the Taylor expansion,

$$\begin{aligned} \lambda (\sigma )=-d\sigma ^2+\mathrm{O}(|\sigma |^3), \end{aligned}$$

where $-\gamma _1/4\le -2d\sigma ^2<Re\lambda (\sigma )<-\frac{d}{2}\sigma ^2$, for all $\sigma \in [-\gamma _0,\gamma _0]$ and

$$\begin{aligned} d=-\langle 2\mathrm{i}\frac{\partial ^2\mathbf{e}(0,x)}{\partial x\partial \sigma } -\mathbf{u}_\star ^\prime (x),D\mathbf{u}_{\mathrm{ad}}(x)\rangle . \end{aligned}$$

Proof

We first derive the explicit expression of $d$. To do that, taking first and second derivative with respect to $\sigma$ of $F(\sigma ,\lambda (\sigma ),\mathbf{w}(\sigma ))=0$, taking the inner product of the derivatives with $\mathbf{u}_{\mathrm{ad}}$ and letting $\sigma =0$, we have

$$\begin{aligned} \lambda ^\prime (0)&= \langle B(0)\partial _\sigma \mathbf{e}(0,x)+ 2\mathrm{i}D\mathbf{u}_\star ^{\prime \prime }(x),\mathbf{u}_{\mathrm{ad}}(x)\rangle ,\\ \lambda ^{\prime \prime }(0)&= \langle B(0)\partial _\sigma ^2\mathbf{e}(0,x)+(4\mathrm{i}D\partial _x-2\lambda ^\prime (0))\partial _\sigma \mathbf{e}(0,x)- 2D\mathbf{u}_\star ^\prime (x),\mathbf{u}_{\mathrm{ad}}(x)\rangle . \end{aligned}$$

Noting that $\mathop {\mathrm{span}}\{\mathbf{u}_{\mathrm{ad}}\}\perp Rg(B(0))$ and the inner product of an even function and an odd function is always 0, we have

$$\begin{aligned} \lambda ^\prime (0)=0, \quad \lambda ^{\prime \prime }(0) =2\langle 2\mathrm{i}\frac{\partial ^2\mathbf{e}(0,x)}{\partial x\partial \sigma } -\mathbf{u}_\star ^\prime (x),D\mathbf{u}_{\mathrm{ad}}(x)\rangle . \end{aligned}$$

It remains to prove the uniqueness of the eigenvalue of $B(\sigma )$ in a vertical strip centered at the origin for sufficiently small $\sigma$. First, there is no eigenvalue within the strip far away from the origin due to the fact that, by Proposition 6.4, $\mathop {\mathrm{spec}}(B(\sigma ))$ is in the same sector for every $\sigma \in [-\frac{1}{2}, \frac{1}{2}]$. Secondly, the uniqueness within a small neighborhood of the origin follows from the above perturbation results. For the region inbetween, compactness and the local robustness of resolvent guarantee the absence of eigenvalues within this area. $\square$

Remark 6.6

• (i) We stress that we may choose $\gamma _0$ as small as desired.

• (ii) The uniqueness implies that, for $|\sigma |$ sufficiently small, $\lambda (\sigma )$ is a real number since its complex conjugate is also an eigenvalue.

1.5 Properties of Analytic Semigroups $\{\mathrm{e}^{\widehat{A}_{\mathrm{ch}}(\sigma ) t}\} _{\sigma \in [-\frac{1}{2},\frac{1}{2}]}$

In this section, we will derive various estimates on $\mathrm{e}^{\widehat{A}_{\mathrm{ch}}(\sigma ) t}$. We first note that by [5, 1.4] the interpolation space $\fancyscript{D}_q(\widehat{A}_{\mathrm{ch}}(\sigma )^\alpha )$ is independent of $\sigma$,

$$\begin{aligned} \fancyscript{D}_q\left( \widehat{A}_{\mathrm{ch}}(\sigma )^\alpha \right)&= \left\{ \underline{\mathbf{w}}\in Y_q\mid \left\{ (1+m^2)^\alpha \mathbf{w}_m\right\} _{m\in \mathbb {Z}}\in Y_q\right\} =:Y^\alpha _q,\\ \Vert \underline{\mathbf{w}}\Vert _{Y^\alpha _q}&= \Vert \left\{ (1+m^2)^\alpha \mathbf{w}_m\right\} _{m\in \mathbb {Z}}\Vert _{Y_q}. \end{aligned}$$

We then recall the definitions of $Y_{q,\mathrm{c}}(\sigma )$, $Y_{q,\mathrm{s}}(\sigma )$, $\widehat{A}_{\mathrm{c}}(\sigma )$ and $\widehat{A}_{\mathrm{s}}(\sigma )$ from (3.18). We now have the following proposition.

Proposition 6.7

For every $q\in [1, +\infty ]$ and $\alpha >0$, there exist positive constants $\epsilon \in (0,1)$, $\gamma _2$, $C(q)$, $C(\alpha )$ and $C(\alpha , q)$ such that

$$\begin{aligned} |||{\mathrm{e}^{\widehat{A}_{\mathrm{c}}(\sigma )t}}|||_{Y_{q,\mathrm{c}}(\sigma )}&\le \mathrm{e}^{-\frac{d}{2}\sigma ^2 t}, \quad \text {for all }|\sigma |\le \gamma _0,t\ge 0,\\ |||{\mathrm{e}^{\widehat{A}_{\mathrm{c}}(\sigma )t}}|||_{Y_{q,\mathrm{c}(\sigma )}\rightarrow Y_q^\alpha }&\le C(\alpha )\mathrm{e}^{-\frac{d}{2}\sigma ^2 t}, \text { for all }|\sigma |\le \gamma _0,t\ge 0,\\ |||{\mathrm{e}^{\widehat{A}_{\mathrm{s}}(\sigma )t}}|||_{Y_{q,\mathrm{s}}(\sigma )}&\le C(q)e^{-\frac{\gamma _1}{2} t},\quad \text {for all }|\sigma |\le \gamma _0,t\ge 0,\\ |||{\mathrm{e}^{\widehat{A}_{\mathrm{s}}(\sigma )t}}|||_{Y_{q,\mathrm{s}}(\sigma )\rightarrow Y_q^\alpha }&\le C(\alpha ,q)t^{-\alpha }\mathrm{e}^{-\gamma _1 t/2},\quad \text {for all }|\sigma |\le \gamma _0,t>0,\\ |||{\mathrm{e}^{\widehat{A}_{\mathrm{ch}}(\sigma )t}}|||_{Y_q}&\le C(q)\mathrm{e}^{-\epsilon d\sigma ^2 t},\quad \text {for all }|\sigma |\le \gamma _0,t\ge 0,\\ |||{\mathrm{e}^{\widehat{A}_{\mathrm{ch}}(\sigma )t}}|||_{Y_q}&\le C(q)\mathrm{e}^{-\gamma _2 t},\quad \text {for all }\gamma _0\le |\sigma |\le \frac{1}{2}, t\ge 0,\\ |||{\mathrm{e}^{\widehat{A}_{\mathrm{ch}}(\sigma )t}}|||_{Y_q\rightarrow Y_q^\alpha }&\le C(\alpha ,p)t^{-\alpha }\mathrm{e}^{-\gamma _2 t}, \quad \text {for all }\gamma _0\le |\sigma |\le \frac{1}{2}, t> 0. \end{aligned}$$

Proof

We first derive estimates for the case $|\sigma |\le \gamma _0$. For $\widehat{A}_{\mathrm{c}}(\sigma )$, we have $\mathrm{e}^{\widehat{A}_{\mathrm{c}}(\sigma )t}=\mathrm{e}^{\lambda (\sigma )t}$. The first two inequalities follow directly from the fact that $\mathop {\mathrm{Re}}\lambda (\sigma )<-\frac{d}{2}\sigma ^2$ and $\mathbf{e}(\sigma )$ is smooth, by Proposition 6.5, for $|\sigma |\le \gamma _0$.

For $\widehat{A}_{\mathrm{s}}(\sigma )$, by Propositions 6.4 and 6.5, for any $\sigma \in (-\gamma _0,\gamma _0)$ and $q\in [1,\infty ]$,

$$\begin{aligned} \mathop {\mathrm{spec}}(\widehat{A}_{\mathrm{s}}(\sigma ),q)\subset \mathbb {C}\backslash S\left( -\frac{\gamma _1}{2},\tilde{\omega }\right) , \quad \text {where }\tilde{\omega }\in \left( \frac{\pi }{2},\pi \right) . \end{aligned}$$

Moreover, for every $q\in [1, +\infty ]$, there exists a positive constant $C(q)$ such that

$$\begin{aligned} |||{(\widehat{A}_{\mathrm{s}}(\sigma )-\lambda )^{-1}}|||_{Y_{q,\mathrm{s}}(\sigma )}\le C(q)|\lambda +\frac{\gamma _1}{2}|^{-1}, \quad \text {for all }|\sigma |\le \gamma _0\text { and }\lambda \in S\left( -\frac{\gamma _1}{2},\tilde{\omega }\right) . \end{aligned}$$

Thus, by [5, Thms. 1.3.4, 1.4.3], we immediately obtain the two inequalities for $\widehat{A}_{\mathrm{s}}(\sigma )$. The first inequality on $\widehat{A}_{\mathrm{ch}}(\sigma )$ follows directly by combining the first inequality for $\widehat{A}_{\mathrm{c}}(\sigma )$ and the first inequality for $\widehat{A}_{\mathrm{s}}(\sigma )$.

We now derive the estimates for the case $\gamma _0<|\sigma |\le \frac{1}{2}$. By a similar analysis as in Proposition 6.5, there exists a positive constant $\gamma _2$ such that

$$\begin{aligned} \mathop {\mathrm{Re}}\left( \mathop {\mathrm{spec}}\widehat{A}_{\mathrm{ch}}(\sigma )\right) <-2\gamma _2, \text { for all }\gamma _0<|\sigma |\le \frac{1}{2}. \end{aligned}$$

It is then not hard to conclude that

$$\begin{aligned} \mathop {\mathrm{spec}}\left( \widehat{A}_{\mathrm{ch}}(\sigma )\right) \subset \mathbb {C}\backslash S(-\gamma _2,\tilde{\omega }_1),\quad \text {where } \tilde{\omega }_1\in \left( \frac{\pi }{2},\pi \right) . \end{aligned}$$

Moreover, for every $q\in [1, +\infty ]$, there exists a positive constant $C(q)$ such that

$$\begin{aligned} |||{(\widehat{A}_{\mathrm{ch}}(\sigma )-\lambda )^{-1}}|||_{Y_q}\le C(q)|\lambda +\gamma _2|^{-1}, \quad \text {for all }\gamma _0<|\sigma |\le \frac{1}{2}\text { and }\lambda \in S(-\gamma _2,\tilde{\omega }_1). \end{aligned}$$

Therefore, again by [5, Thms. 1.3.4, 1.4.3], we immediately obtain the last two inequalities for $\widehat{A}_{\mathrm{ch}}(\sigma )$, which concludes the proof. $\square$