X.$$ We define for $w\in H$

$$(P_\lambda w)(t)=w(t)-(1-t)(I_{2n}-\hat{P}_\lambda)w(0)-t(I_{2n}-\tilde{P}_\lambda)w(1).$$ It is easily seen that $P^2_\lambda w=P_\lambda w$, as well as $P_\lambda w\in\mathcal{D}(\mathcal{A}_\lambda)$ for all $w\in H$ and $\lambda\in X$, which shows that

$$\begin{aligned}
\label{gapcontII}
\inf_{v\in\mathcal{D}(\mathcal{A}_{\lambda_0})}\|u-v\|_{H}\leq\|u-P_{\lambda_0}u\|_{H}.\end{aligned}$$

As

$$u(t)-(P_{\lambda_0}u)(t)=(1-t)(I_{2n}-\hat{P}_{\lambda_0})u(0)+t(I_{2n}-\tilde{P}_{\lambda_0})u(1),$$ it follows for $u\in\mathcal{D}(\mathcal{A}_\lambda)$ that

$$\begin{aligned}
\label{gapcontIII}
\begin{split}
\|u-P_{\lambda_0}u\|_{H}&\leq 2(\|(I_{2n}-\hat{P}_{\lambda_0})u(0)\|+\|(I_{2n}-\tilde{P}_{\lambda_0})u(1)\|)\\
&= 2(\|(I_{2n}-\hat{P}_{\lambda_0})\hat{P}_\lambda u(0)\|+\|(I_{2n}-\tilde{P}_{\lambda_0})\tilde{P}_\lambda u(1)\|)\\
&\leq 2(\|(I_{2n}-\hat{P}_{\lambda_0})\hat{P}_\lambda\|\|u(0)\|+\|(I_{2n}-\tilde{P}_{\lambda_0})\tilde{P}_\lambda\|\|u(1)\|),
\end{split}\end{aligned}$$

where we have used that $u\in\mathcal{D}(\mathcal{A}_\lambda)$ and so $\hat{P}_\lambda u(0)=u(0)$ and $\tilde{P}_\lambda u(1)=u(1)$. Let us note that the factor $2$ appears in the previous estimate as we are dealing with the norm on $H$ and so we also need to take into account the derivatives of $u-P_{\lambda_0}u$ with respect to $t$.\
Since the point evaluation is continuous in $H$, there is a constant $\alpha>0$ such that for $t=0$ and $t=1$

$$\begin{aligned}
\label{gapcontIV}
\|u(t)\|\leq\alpha\|u\|_{H}=\alpha\left(\|u\|^2_{E}+\|u'\|^2_{E}\right)^\frac{1}{2}=\alpha\left(\|u\|^2_{E}+\|Ju'\|^2_{E}\right)^\frac{1}{2},\end{aligned}$$

where we use that $J$ is an isometry on $\mathbb{R}^{2n}$. Hence, by –,

$$\begin{aligned}
d((u,\mathcal{A}_\lambda u),\operatorname{graph}(\mathcal{A}_{\lambda_0}))&=\inf_{v\in\mathcal{D}(\mathcal{A}_{\lambda_0})}\|(u,\mathcal{A}_\lambda u)-(v,\mathcal{A}_{\lambda_0}v)\|_{E\oplus E}\\
&\leq \inf_{v\in\mathcal{D}(\mathcal{A}_{\lambda_0})}\|u-v\|_{H}\leq\|u-P_{\lambda_0}u\|_{H}\\
&\leq 2(\|(I_{2n}-\hat{P}_{\lambda_0})\hat{P}_\lambda\|\|u(0)\|+\|(I_{2n}-\tilde{P}_{\lambda_0})\tilde{P}_\lambda\|\|u(1)\|)\\
&\leq 2\alpha(\|(I_{2n}-\hat{P}_{\lambda_0})\hat{P}_\lambda\|+\|(I_{2n}-\tilde{P}_{\lambda_0})\tilde{P}_\lambda\|)(\|u\|^2_{E}+\|Ju'\|^2_{E})^\frac{1}{2}. \end{aligned}$$

As the unit sphere in $\operatorname{graph}(\mathcal{A}_\lambda)$ is given by

$$\{(u,\mathcal{A}_\lambda u):\, u\in\mathcal{D}(\mathcal{A}_\lambda),\, \|u\|^2_{E}+\|Ju'\|^2_{E}=1\},$$ we finally get

$$\begin{aligned}
\label{gapfinalI}
\begin{split}
\delta(\operatorname{graph}(\mathcal{A}_\lambda),\operatorname{graph}(\mathcal{A}_{\lambda_0}))&=\sup\{d((u,\mathcal{A}_\lambda u),\operatorname{graph}(\mathcal{A}_{\lambda_0})):\,u\in\mathcal{D}(\mathcal{A}_\lambda),\,\|u\|^2+\|Ju'\|^2=1\}\\
&\leq 2\alpha(\|(I_{2n}-\hat{P}_{\lambda_0})\hat{P}_\lambda\|+\|(I_{2n}-\tilde{P}_{\lambda_0})\tilde{P}_\lambda\|).
\end{split}\end{aligned}$$

Note that if we swap $\lambda$ and $\lambda_0$ and repeat the above argument, we also have

$$\begin{aligned}
\label{gapfinalII}
\delta(\operatorname{graph}(\mathcal{A}_{\lambda_0}),\operatorname{graph}(\mathcal{A}_{\lambda}))\leq 2\alpha(\|(I_{2n}-\hat{P}_{\lambda})\hat{P}_{\lambda_0}\|+\|(I_{2n}-\tilde{P}_{\lambda})\tilde{P}_{\lambda_0}\|).\end{aligned}$$

To finish the proof, we 