With this operator at hand, we are able to define a scale of associated Hilbert spaces called the Sobolev chain associated with $\partial_{t,\rho}$, [@Picard_McGhee] (see also [@Nagel1997; @engel2000one], where these spaces are called Sobolev towers).

Let $\rho>0.$ For $k\in{\mathbb{N}}$ we define the spaces $$H_{\rho}^{k}({\mathbb{R}};H)\coloneqq{\operatorname{dom}}(\partial_{t,\rho}^{k})$$ equipped with the inner product $$\langle u,v\rangle_{\rho,k}\coloneqq\langle\partial_{t,\rho}^{k}u,\partial_{t,\rho}^{k}v\rangle_{\rho}\quad(u,v\in{\operatorname{dom}}(\partial_{t,\rho}^{k})).$$ Moreover, we set $H_{\rho}^{-k}({\mathbb{R}};H)$ as the completion of $L_{2,\rho}({\mathbb{R}};H)$ with respect to the norm induced by $$\langle u,v\rangle_{\rho,-k}\coloneqq\langle\partial_{t,\rho}^{-k}u,\partial_{t,\rho}^{-k}v\rangle_{\rho}\quad(u,v\in L_{2,\rho}({\mathbb{R}};H)).$$ For $k,j\in\mathbb{Z}$ with $k\geq j$ we have that $$H_{\rho}^{k}({\mathbb{R}};H)\hookrightarrow H_{\rho}^{j}({\mathbb{R}};H)$$ and the operator $$\partial_{t,\rho}^{k-j}:C_{c}^{\infty}({\mathbb{R}};H)\subseteq H_{\rho}^{k}({\mathbb{R}};H)\to H_{\rho}^{j}({\mathbb{R}};H)$$ extends to a unitary operator again denoted by $\partial_{t,\rho}^{k-j}.$

We state a simple observation, which will be used several times later on.

\[lem:better\_limit\]Let $\rho>0$ and $k,j\in\mathbb{Z}$ with $k\geq j.$ Let $(u_{n})_{n\in\mathbb{N}}$ be a sequence in $H_{\rho}^{k}({\mathbb{R}};H)$ such that $u_{n}\to u$ in $H_{\rho}^{j}({\mathbb{R}};H)$ for some $u\in H_{\rho}^{j}({\mathbb{R}};H)$ and assume that $(u_{n})_{n}$ is bounded in $H_{\rho}^{k}({\mathbb{R}};H).$ Then $u\in H_{\rho}^{k}({\mathbb{R}};H).$

As $(u_{n})_{n}$ is bounded in $H_{\rho}^{k}({\mathbb{R}};H),$ we can assume without loss of generality that $u_{n}\rightharpoonup v$ for some $v\in H_{\rho}^{k}({\mathbb{R}};H).$ As $H_{\rho}^{k}({\mathbb{R}};H)\hookrightarrow H_{\rho}^{j}({\mathbb{R}};H)$ we infer that $u_{n}\rightharpoonup v$ in $H_{\rho}^{j}({\mathbb{R}};H)$ and hence, $u=v\in H_{\rho}^{k}({\mathbb{R}};H).$

Finally, we provide a useful characterisation for elements lying in $H_{\rho}^{1}({\mathbb{R}};H).$

\[lem:difference\_quotient\] Let $\rho\geq0$, $u\in L_{2,\rho}({\mathbb{R}};H)$ and for $h>0$ we define $$D_{h}u\coloneqq\frac{1}{h}(\tau_{h}u-u)\in L_{2,\rho}({\mathbb{R}};H),$$ 