A geometric approximation of δ-interactions by Neumann Laplacians

Throughout the paper we denote points in ${\mathbb{R}}^{2}$ by x = (x₁, x₂). Let ${{\Omega}}_{0}=({\ell }_{-},{\ell }_{+})\subset \mathbb{R}$ be an interval satisfying

$$\begin{equation}-\infty \leqslant {\ell }_{-}< 0< {\ell }_{+}\leqslant \infty .\end{equation} \tag{ 2.1 }$$

Let ɛ ∈ (0, ɛ₀] be a small parameter. We set

$$\begin{equation}{d}_{\varepsilon }=\gamma {\varepsilon }^{\alpha +1},\qquad {h}_{\varepsilon }={\varepsilon }^{\alpha },\qquad {b}_{\varepsilon }={\varepsilon }^{\beta }\quad \text{with}\enspace \alpha > 0,\enspace 0< \beta < \frac{1}{2},\enspace \gamma > 0.\end{equation} \tag{ 2.2 }$$

Moreover, we claim ɛ₀ to be sufficiently small, namely

$$\begin{equation}{\varepsilon }_{0}< \mathrm{min}\left\{{(2\gamma )}^{-1/\alpha },\vert {\ell }_{-}\vert ,{\ell }_{+},{\gamma }^{-1/(\alpha +1-\beta )},\gamma ,{\gamma }^{-1}\right\}.\end{equation} \tag{ 2.3 }$$

It follows from ${\varepsilon }_{0}< \mathrm{min}\left\{{(2\gamma )}^{-1/\alpha },\vert {\ell }_{-}\vert ,{\ell }_{+}\right\}$ that

$$\begin{equation}{d}_{\varepsilon }\leqslant \frac{\varepsilon }{2}\quad \text{and}\quad [-\varepsilon ,\varepsilon ]\subset {{\Omega}}_{0},\end{equation} \tag{ 2.4 }$$

from ɛ₀ < γ^{−1/(α+1−β)} we get

$$\begin{equation}{d}_{\varepsilon }\leqslant {b}_{\varepsilon },\end{equation} \tag{ 2.5 }$$

and ${\varepsilon }_{0}< \mathrm{min}\left\{\gamma ,{\gamma }^{-1}\right\}$ yields |ln γ| ⩽ |ln ɛ| (it will be used in the proof of lemma 4.3). Note that, since either γ ⩽ 1 or γ⁻¹ ⩽ 1, one has ɛ₀ < 1. Finally, we introduce the domains

$$\begin{equation*}\begin{aligned}{S}_{\varepsilon }& =\left\{\mathbf{x}\in {\mathbb{R}}^{2}:\enspace {x}_{1}\in {{\Omega}}_{0},\enspace {x}_{2}\in (-\varepsilon ,0)\right\}& (\text{straight strip}),\\ {P}_{\varepsilon }& =\left\{\mathbf{x}\in {\mathbb{R}}^{2}:\enspace \vert {x}_{1}\vert < \frac{{d}_{\varepsilon }}{2},\enspace {x}_{2}\in (0,{h}_{\varepsilon })\right\}& (\text{passage}),\\ {R}_{\varepsilon }& =\left\{\mathbf{x}\in {\mathbb{R}}^{2}:\enspace \vert {x}_{1}\vert < \frac{{b}_{\varepsilon }}{2},\enspace {x}_{2}\in ({h}_{\varepsilon },{h}_{\varepsilon }+{b}_{\varepsilon })\right\}& (\text{room}),\end{aligned}\end{equation*}$$

and the resulting domain Ω_ɛ given by

$$\begin{equation*}{{\Omega}}_{\varepsilon }=\mathrm{int}(\bar{{S}_{\varepsilon }\cup {P}_{\varepsilon }\cup {R}_{\varepsilon }})\end{equation*}$$

(here int(⋅) stands for the interior of a subset of ${\mathbb{R}}^{2}$). Due to (2.4) and (2.5) the geometry of Ω_ɛ is exactly as shown in figure 1, i.e. the bottom part (respectively, the top part) of ∂P_ɛ is contained in the top part of ∂S_ɛ (respectively, the bottom part of ∂R_ɛ ).

In the Hilbert space ${\mathcal{H}}_{\varepsilon }{:=}{\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$ we introduce the sesquilinear form

$$\begin{equation}{\mathfrak{a}}_{\varepsilon }[u,v]={\int }_{{{\Omega}}_{\varepsilon }}\left(\nabla u(\mathbf{x})\cdot \bar{\nabla v(\mathbf{x})}+{V}_{\varepsilon }(\mathbf{x})u(\mathbf{x})\bar{v(\mathbf{x})}\right)\enspace \mathrm{d}\mathbf{x},\quad \mathrm{dom}({\mathfrak{a}}_{\varepsilon })={\mathsf{H}}^{1}({{\Omega}}_{\varepsilon })\end{equation} \tag{ 2.6 }$$

with a real-valued potential ${V}_{\varepsilon }\in {\mathsf{L}}^{\infty }({{\Omega}}_{\varepsilon })$; later on we give more assumptions on V_ɛ , see (2.12). This form is densely defined in ${\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$, non-negative and closed, consequently [20, theorem VI.2.1] there is a unique non-negative self-adjoint operator ${\mathcal{A}}_{\varepsilon }$ acting in ${\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$ such that the domain inclusion $\mathrm{dom}({\mathcal{A}}_{\varepsilon })\subset \mathrm{dom}({\mathfrak{a}}_{\varepsilon })$ and the equality

$$\begin{equation*}{({\mathcal{A}}_{\varepsilon }u,v)}_{{\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })}={\mathfrak{a}}_{\varepsilon }[u,v],\quad \forall \enspace u\in \mathrm{dom}({\mathcal{A}}_{\varepsilon }),\enspace v\in \mathrm{dom}({\mathfrak{a}}_{\varepsilon })\end{equation*}$$

hold. Obviously, ${\mathcal{A}}_{\varepsilon }=-{{\Delta}}_{{{\Omega}}_{\varepsilon }}+{V}_{\varepsilon }$, where ${{\Delta}}_{{{\Omega}}_{\varepsilon }}$ is the Neumann Laplacian on Ω_ɛ .

The main goal of this work is to describe the behaviour of the resolvent and the spectrum of ${\mathcal{A}}_{\varepsilon }$ as ɛ → 0. In the next subsection we introduce the anticipated limiting operator.

Recall that ${{\Omega}}_{0}\subset \mathbb{R}$ is an open interval containing 0, see (2.1). We denote

$$\begin{equation*}{\mathcal{H}}_{0}{:=}{\mathsf{L}}^{2}({{\Omega}}_{0})\oplus \mathbb{C},\end{equation*}$$

i.e. ${\mathcal{H}}_{0}$ is a Hilbert space consisting of $f=({f}_{1},{f}_{2})\in {\mathsf{L}}^{2}({{\Omega}}_{0})\times \mathbb{C}$ equipped with the scalar product

$$\begin{equation*}{(f,g)}_{{\mathcal{H}}_{0}}={\int }_{{{\Omega}}_{0}}{f}_{1}\bar{{g}_{1}}\enspace \mathrm{d}x+{f}_{2}\bar{{g}_{2}},\quad f=({f}_{1},{f}_{2}),\enspace g=({g}_{1},{g}_{2}).\end{equation*}$$

In the space ${\mathcal{H}}_{0}$ we introduce the sesquilinear form ${\mathfrak{a}}_{0}$ defined by

$$\begin{equation}\begin{aligned}& {\mathfrak{a}}_{0}[f,g]={\int }_{{{\Omega}}_{0}}\left({f}_{1}^{\prime }(x)\bar{{g}_{1}^{\prime }(x)}+{V}_{0}(x){f}_{1}(x)\bar{{g}_{1}(x)}\right)\enspace \mathrm{d}x+\gamma \enspace {f}_{1}(0)\bar{{g}_{1}(0)},\\ & \mathrm{dom}({\mathfrak{a}}_{0})={\mathsf{H}}^{1}({{\Omega}}_{0})\times \mathbb{C},\end{aligned}\end{equation} \tag{ 2.7 }$$

where ${V}_{0}\in {\mathsf{L}}^{\infty }({{\Omega}}_{0})$. It is easy to see that the above form is densely defined in ${\mathcal{H}}_{0}$, non-negative and closed. We denote by ${\mathcal{A}}_{0}$ the self-adjoint operator associated with ${\mathfrak{a}}_{0}$. It is easy to show that its domain is given by

$$\begin{align*}& \mathrm{dom}({\mathcal{A}}_{0})=\left\{f=({f}_{1},{f}_{2})\in {\mathsf{H}}^{2}({{\Omega}}_{0}{\backslash}\left\{0\right\})\times \mathbb{C}:\enspace \begin{matrix}\hfill {f}_{1}(-0)={f}_{1}(+0),\hfill \\ \hfill {f}_{1}^{\prime }(+0)-{f}_{1}^{\prime }(-0)=\gamma {f}_{1}(\pm 0),\hfill \\ \hfill {f}_{1}^{\prime }({\ell }_{-})=0\;\text{provided}\;{\ell }_{-} > -\infty ,\hfill \\ \hfill {f}_{1}^{\prime }({\ell }_{+})=0\;\text{provided}\;{\ell }_{+}< \infty \hfill \end{matrix}\right\},\\ & {({\mathcal{A}}_{0}f)}_{1}(x)=-{f}_{1}^{{\prime\prime}}(x)+{V}_{0}(x){f}_{1}(x),\quad (x\ne 0),\qquad \qquad {({\mathcal{A}}_{0}f)}_{2}=0.\end{align*}$$

Evidently,

$$\begin{equation*}{\mathcal{A}}_{0}={\hat{\mathcal{A}}}_{0}\oplus {0}_{\mathbb{C}}\quad \text{in}\enspace {\mathsf{L}}^{2}({{\Omega}}_{0})\oplus \mathbb{C},\end{equation*}$$

where ${0}_{\mathbb{C}}$ is the null-operator in $\mathbb{C}$, and ${\hat{\mathcal{A}}}_{0}$ is defined by the operation

$$\begin{equation*}-\frac{\enspace {\mathrm{d}}^{2}}{\enspace \mathrm{d}{x}^{2}}+{V}_{0}\quad \text{on}\enspace ({\ell }_{-},0)\cup (0,{\ell }_{+})\end{equation*}$$

with Neumann conditions at ℓ₋ (provided ℓ₋ > −∞) and ℓ₊ (provided ℓ₊ < ∞) and δ-coupling with strength γ at x = 0. Consequently,

$$\begin{equation*}\sigma ({\mathcal{A}}_{0})=\sigma ({\hat{\mathcal{A}}}_{0})\cup \left\{0\right\}.\end{equation*}$$

Our first goal is to prove generalised norm resolvent convergence of the operator ${\mathcal{A}}_{\varepsilon }$ to the operator ${\mathcal{A}}_{0}$. Since these operators act in different Hilbert spaces ${\mathcal{H}}_{\varepsilon }={\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$ and ${\mathcal{H}}_{0}={\mathsf{L}}^{2}({{\Omega}}_{0})\oplus \mathbb{C}$, respectively, the standard definition of norm resolvent convergence cannot be applied here and should be modified in an appropriate way. The modified definition should be adjusted in such a way that it still implies the convergence of spectra as it takes place in the classical situation. The standard approach (see, e.g. the abstract scheme in [19] and its applications to homogenisation in perforated spaces [24, chapter III], [30, chapter XI]) is to treat the operator ${\mathcal{L}}_{\varepsilon }:{\mathcal{H}}_{0}\to {\mathcal{H}}_{\varepsilon }$,

$$\begin{equation*}{\mathcal{L}}_{\varepsilon }{:=}{\mathcal{R}}_{\varepsilon }{\mathcal{J}}_{\varepsilon }-{\mathcal{J}}_{\varepsilon }{\mathcal{R}}_{0},\end{equation*}$$

where ${\mathcal{R}}_{\varepsilon }$ and ${\mathcal{R}}_{0}$ are the resolvents of ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$, respectively, and ${\mathcal{J}}_{\varepsilon }:{\mathcal{H}}_{0}\to {\mathcal{H}}_{\varepsilon }$ is a suitable bounded linear operator being 'almost isometric' in a sense that

$$\begin{equation*}\forall \enspace f\in {\mathcal{H}}_{0}:\enspace \underset{\varepsilon \to 0}{\mathrm{lim}}{\Vert}{\mathcal{J}}_{\varepsilon }f{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}={\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}}.\end{equation*}$$

For the problem we deal in this paper we define the operator ${\mathcal{J}}_{\varepsilon }$ as follows:

$$\begin{equation}({\mathcal{J}}_{\varepsilon }f)(\mathbf{x})=\begin{cases}{\varepsilon }^{-1/2}{f}_{1}({x}_{1}),\quad & \mathbf{x}=({x}_{1},{x}_{2})\in {S}_{\varepsilon },\\ 0,\quad & \mathbf{x}\in {P}_{\varepsilon },\\ {({b}_{\varepsilon })}^{-1}{f}_{2},\quad & \mathbf{x}\in {R}_{\varepsilon },\end{cases}\quad f=({f}_{1},{f}_{2})\in {\mathcal{H}}_{0}={\mathsf{L}}^{2}({{\Omega}}_{0})\oplus \mathbb{C}.\end{equation} \tag{ 2.8 }$$

The operator ${\mathcal{J}}_{\varepsilon }$ is an isometry, namely one has

$$\begin{equation}\forall \enspace f\in {\mathcal{H}}_{0}:\quad {\Vert}{\mathcal{J}}_{\varepsilon }f{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}={\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}}.\end{equation} \tag{ 2.9 }$$

Along with ${\mathcal{J}}_{\varepsilon }$ we also introduce the operator ${\tilde{\mathcal{J}}}_{\varepsilon }:{\mathcal{H}}_{\varepsilon }\to {\mathcal{H}}_{0}$ by

$$\begin{align}& {\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })\ni u\enspace {\longmapsto}{\tilde{\mathcal{J}}}_{\varepsilon }u=({({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{1},{({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{2})\in {\mathsf{L}}^{2}({{\Omega}}_{0})\times \mathbb{C},\;\text{where}\;\\ & {({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{1}({x}_{1})={\varepsilon }^{-1/2}{\int }_{-\varepsilon }^{0}u({x}_{1},{x}_{2})\enspace \mathrm{d}{x}_{2}\quad ({x}_{1}\in {{\Omega}}_{0}),\qquad {({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{2}={({b}_{\varepsilon })}^{-1}{\int }_{{R}_{\varepsilon }}u(\mathbf{x})\enspace \mathrm{d}\mathbf{x}.\end{align} \tag{ 2.10 }$$

It is easy to see that ${\Vert}\tilde{\mathcal{J}}u{{\Vert}}_{{\mathcal{H}}_{0}}\leqslant {\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}$. Moreover, ${\tilde{\mathcal{J}}}_{\varepsilon }$ is the adjoint of ${\mathcal{J}}_{\varepsilon }$, as

$$\begin{equation}\forall \enspace f\in {\mathcal{H}}_{0},\enspace \forall \enspace u\in {\mathcal{H}}_{\varepsilon }:\quad {({\mathcal{J}}_{\varepsilon }f,u)}_{{\mathcal{H}}_{\varepsilon }}={(f,{\tilde{\mathcal{J}}}_{\varepsilon }u)}_{{\mathcal{H}}_{0}}.\end{equation} \tag{ 2.11 }$$

Remark 2.1. The above choice of the operators ${\tilde{\mathcal{J}}}_{\varepsilon }$ and ${\mathcal{J}}_{\varepsilon }$ is rather standard when studying convergence of differential operators on thin waveguide-like domains, cf [13, 25]. The first component of ${\tilde{\mathcal{J}}}_{\varepsilon }u$ is the transversal average of u on S_ɛ being scaled in such a way that the ${\mathsf{L}}^{2}$-norm is (asymptotically) preserved: if ${({u}_{\varepsilon }\in {\mathsf{H}}^{1}({S}_{\varepsilon }))}_{\varepsilon }$ with ${\Vert}{u}_{\varepsilon }{{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })}^{2}\leqslant C$, then

$$\begin{equation*}{\Vert}{({\tilde{\mathcal{J}}}_{\varepsilon }{u}_{\varepsilon })}_{1}{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}\leqslant {\Vert}{u}_{\varepsilon }{{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}\leqslant {\Vert}{({\tilde{\mathcal{J}}}_{\varepsilon }{u}_{\varepsilon })}_{1}{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}+o(\varepsilon )\quad \text{as}\enspace \varepsilon \to 0\end{equation*}$$

(this asymptotic behaviour follows easily from the Poincaré inequality on the cross-section of S_ɛ , see lemma 4.8 and its proof). Similarly, the second component of ${\tilde{\mathcal{J}}}_{\varepsilon }u$ is an appropriately scaled average of u on R_ɛ . The part of u on the passage P_ɛ has negligible impact on ${\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}$: if ${({u}_{\varepsilon }\in {\mathsf{H}}^{1}({S}_{\varepsilon }))}_{\varepsilon }$ with ${\Vert}{u}_{\varepsilon }{{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })}\leqslant C$, then ${\Vert}{u}_{\varepsilon }{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}\to 0$ as ɛ → 0 (see (4.6)).

The operator ${\mathcal{J}}_{\varepsilon }$ is constructed to be adjoint to ${\tilde{\mathcal{J}}}_{\varepsilon }$. With such a choice we automatically fulfil the condition (3.7) of the abstract theorem 3.1 playing a crucial role in the proof of the main results.

To guarantee the closeness of the resolvents of the operators ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$, the potentials V_ɛ and V₀ have to be close in a suitable sense. Namely, we choose the family ${\left\{{V}_{\varepsilon }\right\}}_{\varepsilon > 0}$ as follows

$$\begin{equation}{V}_{\varepsilon }(\mathbf{x})=\begin{cases}{V}_{0}({x}_{1}),\quad & \mathbf{x}=({x}_{1},{x}_{2})\in {S}_{\varepsilon },\\ 0,\quad & \mathbf{x}\in {P}_{\varepsilon }\cup {R}_{\varepsilon }.\end{cases}\end{equation} \tag{ 2.12 }$$

In order to simplify the presentation (cf remark 4.7) we assume further that

$$\begin{equation}{V}_{0}(x)\geqslant 0,\end{equation} \tag{ 2.13 }$$

and hence both ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$ are non-negative operators. We denote by ${\mathcal{R}}_{\varepsilon }$ and ${\mathcal{R}}_{0}$ the resolvents of ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$, respectively:

$$\begin{equation}{\mathcal{R}}_{\varepsilon }{:=}{({\mathcal{A}}_{\varepsilon }+\mathrm{I})}^{-1},\qquad {\mathcal{R}}_{0}{:=}{({\mathcal{A}}_{0}+\mathrm{I})}^{-1}.\end{equation} \tag{ 2.14 }$$

We are now in position to formulate the first result of this work. Below, ||⋅||_X→Y stands for the norm of an operator acting between normed spaces X and Y.

Theorem 2.2. One has

$$\begin{equation}{\Vert}{\mathcal{R}}_{\varepsilon }{\mathcal{J}}_{\varepsilon }-{\mathcal{J}}_{\varepsilon }{\mathcal{R}}_{0}{{\Vert}}_{{\mathcal{H}}_{0}\to {\mathcal{H}}_{\varepsilon }}={\Vert}{\tilde{\mathcal{J}}}_{\varepsilon }{\mathcal{R}}_{\varepsilon }-{\mathcal{R}}_{0}{\tilde{\mathcal{J}}}_{\varepsilon }{{\Vert}}_{{\mathcal{H}}_{\varepsilon }\to {\mathcal{H}}_{0}}\leqslant {C}_{1}{\varepsilon }^{\mathrm{min}\left\{\alpha ,1/2-\beta \right\}},\end{equation} \tag{ 2.15 }$$

where C₁ > 0 is a constant independent of ɛ (see remark 4.6 for more details).

Our second result concerns Hausdorff convergence of spectra. Recall (see, e.g. [29]), that for closed sets $X,Y\subset \mathbb{R}$ the Hausdorff distance between X and Y is given by

$$\begin{equation}{d}_{\mathrm{H}}(X,Y){:=}\mathrm{max}\left\{\underset{x\in X}{\mathrm{sup}\;}\underset{y\in Y}{\mathrm{inf}}\vert x-y\vert ;\enspace \underset{y\in Y}{\mathrm{sup}\;}\underset{x\in X}{\mathrm{inf}}\vert y-x\vert \right\}.\end{equation} \tag{ 2.16 }$$

The notion of convergence provided by this metric is too restrictive for our purposes. Indeed, the closeness of $\sigma ({\mathcal{A}}_{\varepsilon })$ and $\sigma ({\mathcal{A}}_{0})$ in the metric d_H(⋅, ⋅) would mean that these spectra look nearly the same uniformly on all parts of [0, ∞)—a situation, which is not guaranteed by norm resolvent convergence. To overcome this difficulty, we introduced the new metric ${\tilde{d}}_{\mathrm{H}}(\cdot ,\cdot )$, which is given by

$$\begin{equation*}{\tilde{d}}_{\mathrm{H}}(X,Y){:=}{d}_{\mathrm{H}}\left(\bar{{(1+X)}^{-1}},\bar{{(1+Y)}^{-1}}\right),\enspace X,Y\subset [0,\infty ),\end{equation*}$$

where (1 + X)⁻¹ = {(1 + x)⁻¹: x ∈ X} and (1 + Y)⁻¹ = {(1 + y)⁻¹: y ∈ Y}. With respect to this metric two spectra can be close even if they differ significantly at high energies. Note (see, e.g. [18, lemma A.2]) that ${\tilde{d}}_{\mathrm{H}}\left({X}_{\varepsilon },X\right)\to 0\enspace \text{as}\enspace \varepsilon \to 0$ iff

• For each $x\in \mathbb{R}{\backslash}X$ there exists d > 0 such that X_ɛ ∩ {y: |y − x| < d} = Ø eventually (as ɛ → 0), and
• For any x ∈ X there exists a family ${\left\{{x}_{\varepsilon }\right\}}_{\varepsilon }$ with x_ɛ ∈ X_ɛ such that lim_ɛ→0 x_ɛ = x.

Note that by the spectral mapping theorem

$$\begin{equation}{\tilde{d}}_{\mathrm{H}}\left(\sigma ({\mathcal{A}}_{\varepsilon }),\sigma ({\mathcal{A}}_{0})\right)={d}_{\mathrm{H}}\left(\sigma ({\mathcal{R}}_{\varepsilon }),\sigma ({\mathcal{R}}_{0})\right).\end{equation} \tag{ 2.17 }$$

Theorem 2.3. One has

$$\begin{equation}{\tilde{d}}_{\mathrm{H}}\left(\sigma ({\mathcal{A}}_{\varepsilon }),\sigma ({\mathcal{A}}_{0})\right)\leqslant {C}_{2}{\varepsilon }^{\mathrm{min}\left\{\alpha ,1/2-\beta ,2\beta \right\}},\end{equation} \tag{ 2.18 }$$

where C₂ > 0 is a constant independent of ɛ, see (4.44).

Note that the best convergence rate in (2.18) is provided when α = 1/3 and β = 1/6.

Remark 2.4. We denote

$$\begin{equation}{D}_{\varepsilon }^{+}{:=}\left\{\mathbf{x}=({x}_{1},{x}_{2})\in \partial {P}_{\varepsilon }:\enspace {x}_{2}={h}_{\varepsilon }\right\},\qquad {D}_{\varepsilon }^{-}{:=}\left\{\mathbf{x}=({x}_{1},{x}_{2})\in \partial {P}_{\varepsilon }:\enspace {x}_{2}=0\right\},\end{equation} \tag{ 2.19 }$$

and ${F}_{\varepsilon }{:=}\mathrm{int}(\bar{{S}_{\varepsilon }\cup {P}_{\varepsilon }})={S}_{\varepsilon }\cup {P}_{\varepsilon }\cup {D}_{\varepsilon }^{-}$. Let ${\mathcal{A}}_{{F}_{\varepsilon }}$ be the operator in ${\mathsf{L}}^{2}({F}_{\varepsilon })$ acting as −Δ + V_ɛ and with Neumann boundary conditions on $\partial {F}_{\varepsilon }{\backslash}{D}_{\varepsilon }^{+}$ and Dirichlet conditions on ${D}_{\varepsilon }^{+}$. Using similar methods as in the proof of theorem 2.3, one can show that

$$\begin{equation}{\tilde{d}}_{\mathrm{H}}(\sigma ({\mathcal{A}}_{{F}_{\varepsilon }}),\sigma ({\hat{\mathcal{A}}}_{0}))\to 0\quad \text{as}\enspace \varepsilon \to 0;\end{equation} \tag{ 2.20 }$$

the δ-coupling at 0 is caused by the Dirichlet conditions on ${D}_{\varepsilon }^{+}$ (these boundary conditions can be regarded as an infinite potential). Also, let ${\mathcal{A}}_{{R}_{\varepsilon }}$ be the Neumann Laplacian on R_ɛ . The first eigenvalue of ${\mathcal{A}}_{{R}_{\varepsilon }}$ is zero for each ɛ > 0, while the next eigenvalues escape to infinity as ɛ → 0. Hence, we get

$$\begin{equation}{\tilde{d}}_{\mathrm{H}}(\sigma ({\mathcal{A}}_{{R}_{\varepsilon }}),\left\{0\right\})\to 0\quad \text{as}\enspace \varepsilon \to 0.\end{equation} \tag{ 2.21 }$$

Finally, in ${\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })={\mathsf{L}}^{2}({F}_{\varepsilon })\oplus {\mathsf{L}}^{2}({R}_{\varepsilon })$ we consider the operator ${\mathcal{A}}_{\varepsilon }^{\prime }{:=}{\mathcal{A}}_{{F}_{\varepsilon }}\oplus {\mathcal{A}}_{{R}_{\varepsilon }}$. It follows from (2.18), (2.20) and (2.21) that the spectra of ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{\varepsilon }^{\prime }$ are close in the ${\tilde{d}}_{\mathrm{H}}$-metric as ɛ → 0. The fact that the asymptotic behaviour of the spectrum does not change, if one detaches the passage from the room and changes the boundary conditions on the contact part of the passage boundary, is not surprising—see, e.g. proposition 1.5 in [17] (the authors named it the 'Organ pipe lemma' as a reflection of the known fact that closed and open organ pipes have opposite boundary conditions).

The first theorem below yields the estimate for the difference of the resolvents $\mathcal{R}$ and $\tilde{\mathcal{R}}$ in the operator norm. Since $\mathcal{R}$ and $\tilde{\mathcal{R}}$ act in different Hilbert spaces, we need to intertwine them with suitable operators $\mathcal{J}:\mathcal{H}\to \tilde{\mathcal{H}}$ and $\tilde{\mathcal{J}}:\tilde{\mathcal{H}}\to \mathcal{H}$. Usually in applications these operators appear in a natural way (as, for example, $\mathcal{J}$ defined in (2.8) and $\tilde{\mathcal{J}}$ defined in (2.10) in our case). We also require the other two operators ${\mathcal{J}}^{1}:{\mathcal{H}}^{1}\to {\tilde{\mathcal{H}}}^{1}$ and ${\tilde{\mathcal{J}}}^{1}:{\tilde{\mathcal{H}}}^{1}\to {\mathcal{H}}^{1}$, which should be constructed as 'almost' restrictions of $\mathcal{J}$ and $\tilde{\mathcal{J}}$ to ${\mathcal{H}}^{1}$ and ${\tilde{\mathcal{H}}}^{1}$, respectively, modified in such a way that they respect the form domains (see conditions (3.5) and (3.6) below).

Theorem 3.1 ([25, theorem A.5]). Let

$$\begin{equation*}\mathcal{J}:\mathcal{H}\to \tilde{\mathcal{H}},\qquad \tilde{\mathcal{J}}:\tilde{\mathcal{H}}\to \mathcal{H},\qquad {\mathcal{J}}^{1}:{\mathcal{H}}^{1}\to {\tilde{\mathcal{H}}}^{1},\qquad {\tilde{\mathcal{J}}}^{1}:{\tilde{\mathcal{H}}}^{1}\to {\mathcal{H}}^{1}\end{equation*}$$

be linear operators satisfying the conditions

$$\begin{equation}{\Vert}\mathcal{J}f-{\mathcal{J}}^{1}f{{\Vert}}_{\tilde{\mathcal{H}}}\leqslant \delta {\Vert}f{{\Vert}}_{{\mathcal{H}}^{1}}\qquad \forall \enspace f\in {\mathcal{H}}^{1},\end{equation} \tag{ 3.5 }$$

$$\begin{equation}{\Vert}\tilde{\mathcal{J}}u-{\tilde{\mathcal{J}}}^{1}u{{\Vert}}_{\mathcal{H}}\leqslant \delta {\Vert}u{{\Vert}}_{{\tilde{\mathcal{H}}}^{1}}\qquad \forall \enspace u\in {\tilde{\mathcal{H}}}^{1},\end{equation} \tag{ 3.6 }$$

$$\begin{equation}\left\vert {(\mathcal{J}f,u)}_{\tilde{\mathcal{H}}}-{(f,\tilde{\mathcal{J}}u)}_{\mathcal{H}}\right\vert \leqslant \delta {\Vert}f{{\Vert}}_{\mathcal{H}}{\Vert}u{{\Vert}}_{\tilde{\mathcal{H}}}\qquad \forall \enspace f\in \mathcal{H},u\in \tilde{\mathcal{H}},\end{equation} \tag{ 3.7 }$$

$$\begin{equation}\left\vert \tilde{\mathfrak{a}}[{\mathcal{J}}^{1}f,u]-\mathfrak{a}[f,{\tilde{\mathcal{J}}}^{1}u]\right\vert \leqslant \delta {\Vert}f{{\Vert}}_{{\mathcal{H}}^{2}}{\Vert}u{{\Vert}}_{{\tilde{\mathcal{H}}}^{1}}\qquad \forall \enspace f\in {\mathcal{H}}^{2},u\in {\tilde{\mathcal{H}}}^{1}\end{equation} \tag{ 3.8 }$$

for some δ ⩾ 0. Then

$$\begin{equation}{\Vert}\tilde{\mathcal{R}}\mathcal{J}-\mathcal{J}\mathcal{R}{{\Vert}}_{\mathcal{H}\to \tilde{\mathcal{H}}}\leqslant 4\delta .\end{equation} \tag{ 3.9 }$$

Remark 3.2. It is well-known [20, theorem VI.3.6], that convergence of sesquilinear forms with common domain implies norm resolvent convergence of the associated operators (see the recent paper [6, theorem 2] for a quantitative version of this result); in these theorems the convergence of the forms ${\mathfrak{a}}_{\varepsilon }$ to the form $\mathfrak{a}$ means that the inequality

$$\begin{equation}\left\vert {\mathfrak{a}}_{\varepsilon }[f,f]-\mathfrak{a}[f,f]\right\vert \leqslant {\delta }_{\varepsilon }\left(\mathfrak{a}[f,f]+{\Vert}f{{\Vert}}_{\mathcal{H}}^{2}\right),\quad {\delta }_{\varepsilon }\to 0,\end{equation} \tag{ 3.10 }$$

holds for each $f\in \mathrm{dom}({\mathfrak{a}}_{\varepsilon })=\mathrm{dom}(\mathfrak{a})$. In this sense, theorem 3.1 can be regarded as a generalisation of [20, theorem VI.3.6], [6, theorem 2] to the setting of varying spaces.

Remark 3.3. It is easy to see from the proof above, that some of the conditions (3.5)–(3.8) can be weakened. For example, theorem 3.1 remains valid if (3.8) is substituted by

$$\begin{equation}\left\vert \tilde{\mathfrak{a}}[{\mathcal{J}}^{1}f,u]-\mathfrak{a}[f,{\tilde{\mathcal{J}}}^{1}u]\right\vert \leqslant \delta {\Vert}f{{\Vert}}_{{\mathcal{H}}^{2}}{\Vert}u{{\Vert}}_{{\tilde{\mathcal{H}}}^{2}},\quad \forall \enspace f\in {\mathcal{H}}^{2},\enspace u\in {\tilde{\mathcal{H}}}^{2}.\end{equation} \tag{ 3.11 }$$

Nevertheless, in most of the applications one is able to establish stronger estimate (3.8) (cf lemma 4.5). Moreover, sometimes (for example, when studying convergence of graph-like manifolds [25, 26]), one even can prove the stronger inequality

$$\begin{equation*}\left\vert \tilde{\mathfrak{a}}[{\mathcal{J}}^{1}f,u]-\mathfrak{a}[f,{\tilde{\mathcal{J}}}^{1}u]\right\vert \leqslant \delta {\Vert}f{{\Vert}}_{{\mathcal{H}}^{1}}{\Vert}u{{\Vert}}_{{\tilde{\mathcal{H}}}^{1}},\quad \forall \enspace f\in {\mathcal{H}}^{1},\enspace u\in {\tilde{\mathcal{H}}}^{1},\end{equation*}$$

which can be regarded as a counterpart to (3.10).

Recall that the Hausdorff distance d_H(⋅, ⋅) is defined via (2.16). It is well-known, that norm resolvent convergence of self-adjoint operators in a fixed Hilbert space implies Hausdorff convergence of spectra of their resolvents. Namely, if the Hilbert spaces $\mathcal{H}$ and $\tilde{\mathcal{H}}$ coincide, then

$$\begin{equation}{d}_{\mathrm{H}}(\sigma (\mathcal{R}),\sigma (\tilde{\mathcal{R}}))\leqslant {\Vert}\mathcal{R}-\tilde{\mathcal{R}}{{\Vert}}_{\mathcal{H}\to \mathcal{H}}.\end{equation} \tag{ 3.12 }$$

This estimate follows immediately from the following result [18, lemma A.1]: let B₁ and B₂ be bounded normal operators in a Hilbert space $\mathcal{H}$ , then ${d}_{\mathrm{H}}(\sigma ({B}_{1}),\sigma ({B}_{2}))\leqslant {\Vert}{B}_{1}-{B}_{2}{{\Vert}}_{\mathcal{H}\to \mathcal{H}}$ .

The theorem below is an analogue of (3.12) for the case of operators acting in different Hilbert spaces. Its slightly weaker version was proven in [9, theorem 3.4]. In what follows, we assume that our operators $\tilde{\mathcal{A}}$ and $\mathcal{A}$ are unbounded, whence, $0\in \sigma (\tilde{\mathcal{R}})\cap \sigma (\mathcal{R})$.

Theorem 3.4. Let $\mathcal{J}:\mathcal{H}\to \tilde{\mathcal{H}}$, $\tilde{\mathcal{J}}:\tilde{\mathcal{H}}\to \mathcal{H}$ be linear bounded operators satisfying

$$\begin{equation}{\Vert}\tilde{\mathcal{R}}\mathcal{J}-\mathcal{J}\mathcal{R}{{\Vert}}_{\mathcal{H}\to \tilde{\mathcal{H}}}\leqslant \eta ,\end{equation} \tag{ 3.13 }$$

$$\begin{equation}{\Vert}\tilde{\mathcal{J}}\tilde{\mathcal{R}}-\mathcal{R}\tilde{\mathcal{J}}{{\Vert}}_{\tilde{\mathcal{H}}\to \mathcal{H}}\leqslant \tilde{\eta },\end{equation} \tag{ 3.14 }$$

and, moreover,

$$\begin{equation}{\Vert}f{{\Vert}}_{\mathcal{H}}^{2}\leqslant \mu {\Vert}\mathcal{J}f{{\Vert}}_{\tilde{\mathcal{H}}}^{2}+\nu \enspace \mathfrak{a}[f,f],\quad \forall \enspace f\in \mathrm{dom}(\mathfrak{a}),\end{equation} \tag{ 3.15 }$$

$$\begin{equation}{\Vert}u{{\Vert}}_{\tilde{\mathcal{H}}}^{2}\leqslant \tilde{\mu }{\Vert}\tilde{\mathcal{J}}u{{\Vert}}_{\mathcal{H}}^{2}+\tilde{\nu }\enspace \tilde{\mathfrak{a}}[u,u],\quad \forall \enspace u\in \mathrm{dom}(\tilde{\mathfrak{a}}),\end{equation} \tag{ 3.16 }$$

for some positive constants η, μ, ν, $\tilde{\eta }$, $\tilde{\mu }$ and $\tilde{\nu }$. Then for any κ, $\tilde{\kappa }\in (0,1)$ we have

$$\begin{equation*}{d}_{\mathrm{H}}\left(\sigma (\mathcal{R}),\enspace \sigma (\tilde{\mathcal{R}})\right)\leqslant \mathrm{max}\left\{\eta \sqrt{\frac{\mu }{\kappa }};\enspace \frac{\nu }{1-\kappa };\enspace \tilde{\eta }\sqrt{\frac{\tilde{\mu }}{\tilde{\kappa }}};\enspace \frac{\tilde{\nu }}{1-\tilde{\kappa }}\right\}.\end{equation*}$$

Remark 3.5. 

• (a)  
  In a typical application, μ ⩾ 1 is close to 1, and ν > 0 is small, as if $\mathcal{J}$is an isometry, then μ = 1 and ν = 0. A similar remark holds for $\tilde{\mu }$ and $\tilde{\nu }$. In our application later, we have μ = 1 and ν = 0 (hence, we are allowed to take κ = 1 by a limit argument), and $\tilde{\mu }={\mu }_{\varepsilon }=1+{C}_{6}{\varepsilon }^{2\alpha }$ and $\tilde{\nu }={\nu }_{\varepsilon }={C}_{17}{\varepsilon }^{2\mathrm{min}\left\{\alpha ,\beta \right\}}$ (see lemma 4.8).
• (b)  
  Note that the upper bound $\eta \sqrt{\mu /\kappa }$ arises from values of $\sigma (\mathcal{R})$ closely below 1, i.e. from small values of $\sigma (\mathcal{A})$ whereas the upper bound ν/(1 − κ) arises from values of $\sigma (\mathcal{R})$ near 0, i.e. from large values of $\sigma (\mathcal{A})$. A similar remark holds for $\tilde{\mathcal{R}}$ and $\tilde{\mathcal{A}}$.
• (c)  
  The role of κ (and $\tilde{\kappa }$) is as follows: one can, of course, fix $\kappa =\tilde{\kappa }=1/2$, then the error is the maximum of $\eta \sqrt{2\mu }$, 2ν, $\tilde{\eta }\sqrt{2\tilde{\mu }}$ and $2\tilde{\nu }$. Herbst and Nakamura proved in the classical case an estimate of the Hausdorff distance of the spectra in terms of ${{\Vert}\mathcal{R}-\tilde{\mathcal{R}}{\Vert}}_{\mathcal{H}\to \mathcal{H}}$, cf (3.12). If we strive to derive a similar result in the case $\mathcal{H}\ne \tilde{\mathcal{H}}$, we have to use the two norms in (3.13) and (3.14). The constants κ and $\tilde{\kappa }$ allow to estimate the Hausdorff distance of the spectra in terms of η and $\tilde{\eta }$ with a constant as close to 1 as wanted. The price of this factor to be close to 1 is then a worse estimate in the second term, namely ν/(1 − κ).

Proof of Theorem  3.4 .For each $z\in \mathbb{C}$ one has the estimate

$$\begin{equation}\forall \enspace \phi \in \tilde{\mathcal{H}}{\backslash}\left\{0\right\}:\quad \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\sigma (\tilde{\mathcal{R}}))\leqslant \frac{{\Vert}(\tilde{\mathcal{R}}-z\mathrm{I})\phi {{\Vert}}_{\tilde{\mathcal{H}}}}{{\Vert}\phi {{\Vert}}_{\tilde{\mathcal{H}}}}\end{equation} \tag{ 3.17 }$$

(hereinafter for $x\in \mathbb{R}$ and a compact set $Y\subset \mathbb{R}$ we denote dist(x, Y) := inf_y∈Y |x − y|). Indeed, for $z\in \sigma (\tilde{\mathcal{R}})$ estimate (3.17) is trivial, while for $z\in \mathbb{C}{\backslash}\sigma (\tilde{\mathcal{R}})$ it follows easily from

$$\begin{equation*}{\Vert}{(\tilde{\mathcal{R}}-z\mathrm{I})}^{-1}{{\Vert}}_{\tilde{\mathcal{H}}}=\frac{1}{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\enspace \sigma (\tilde{\mathcal{R}}))}.\end{equation*}$$

In what follows we assume that $z\in \sigma (\mathcal{R})\cap [{L}_{\nu },1]$, where

$$\begin{equation}{L}_{\nu }{:=}\frac{\nu }{\nu +(1-\kappa )}\in (0,1).\end{equation} \tag{ 3.18 }$$

We denote ${\lambda }_{z}{:=}\frac{1-z}{z}$. It is easy to see that the following identity holds:

$$\begin{equation}(\mathcal{R}-z\mathrm{I})\psi =-z\enspace \mathcal{R}(\mathcal{A}-{\lambda }_{z}\mathrm{I})\psi ,\quad \psi \in \mathrm{dom}(\mathcal{A}).\end{equation} \tag{ 3.19 }$$

Moreover, by the spectral mapping theorem ${\lambda }_{z}\in \sigma (\mathcal{A})$ and hence

$$\begin{equation}\forall \enspace \rho > 0\quad \exists {\psi }_{\rho }\in \mathrm{dom}(\mathcal{A}):\quad {\Vert}{\psi }_{\rho }{{\Vert}}_{\mathcal{H}}=1,\quad {\Vert}(\mathcal{A}-{\lambda }_{z}\mathrm{I}){\psi }_{\rho }{{\Vert}}_{\mathcal{H}}\leqslant \rho .\end{equation} \tag{ 3.20 }$$

Taking into account that z ∈ (0, 1] and ${\Vert}\mathcal{R}{{\Vert}}_{\mathcal{H}\to \mathcal{H}}\leqslant 1$, one gets from (3.19) and (3.20)

$$\begin{equation}{\Vert}(\mathcal{R}-z\mathrm{I}){\psi }_{\rho }{{\Vert}}_{\mathcal{H}}\leqslant \rho .\end{equation} \tag{ 3.21 }$$

Using (3.15) and (3.20) and taking into account that

$$\begin{equation}{\lambda }_{z}\leqslant \frac{1-{L}_{\nu }}{{L}_{\nu }},\end{equation} \tag{ 3.22 }$$

one can prove that $\mathcal{J}{\psi }_{\rho }\ne 0$ for small enough ρ. Indeed, we get

$$\begin{align}\mu {\Vert}\mathcal{J}{\psi }_{\rho }{{\Vert}}_{\tilde{\mathcal{H}}}^{2}& \stackrel{(3.15)}{\geqslant }{\Vert}{\psi }_{\rho }{{\Vert}}_{\mathcal{H}}^{2}-\nu \enspace \mathfrak{a}[{\psi }_{\rho },{\psi }_{\rho }]=(1 - {\lambda }_{z}\nu ){\Vert}{\psi }_{\rho }{{\Vert}}_{\mathcal{H}}^{2}-\nu {(\mathcal{A}{\psi }_{\rho }-{\lambda }_{z}{\psi }_{\rho },{\psi }_{\rho })}_{\mathcal{H}}\\ & \stackrel{(3.20),\enspace (3.22)}{\geqslant }1-\frac{1-{L}_{\nu }}{{L}_{\nu }}\nu -\rho \nu =\kappa -\rho \nu \end{align} \tag{ 3.23 }$$

(we use (3.18) for the last equality). Hence $\mathcal{J}{\psi }_{\rho }\ne 0$ as ρ < κ/ν.

For $z\in \sigma (\mathcal{R})\cap [{L}_{\nu },1]$, ρ ∈ (0, κ/ν) we obtain using (3.17), (3.21) and (3.23):

$$\begin{align*}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\enspace \sigma (\tilde{\mathcal{R}}))& \leqslant \frac{{\Vert}(\tilde{\mathcal{R}}-z\mathrm{I})\mathcal{J}{\psi }_{\rho }{{\Vert}}_{\tilde{\mathcal{H}}}}{{\Vert}\mathcal{J}{\psi }_{\rho }{{\Vert}}_{\tilde{\mathcal{H}}}}\\ & \leqslant \frac{{\Vert}(\tilde{\mathcal{R}}\mathcal{J}-\mathcal{J}\mathcal{R}){\psi }_{\rho }{{\Vert}}_{\tilde{\mathcal{H}}}+{\Vert}\mathcal{J}(\mathcal{R}-z\mathrm{I}){\psi }_{\rho }{{\Vert}}_{\tilde{\mathcal{H}}}}{{\Vert}\mathcal{J}{\psi }_{\rho }{{\Vert}}_{\tilde{\mathcal{H}}}}\leqslant \frac{\eta +{\Vert}\mathcal{J}{{\Vert}}_{\mathcal{H}\to \tilde{\mathcal{H}}}\cdot \rho }{\sqrt{{\mu }^{-1}\left(\kappa -\rho \nu \right)}}.\end{align*}$$

Passing to the limit ρ → 0 we arrive at the estimate

$$\begin{equation}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\sigma (\tilde{\mathcal{R}}))\leqslant \eta \sqrt{\frac{\mu }{\kappa }},\quad \forall \enspace z\in \sigma (\mathcal{R})\cap [{L}_{\nu },1].\end{equation} \tag{ 3.24 }$$

Finally, taking into account that $0\in \sigma (\tilde{\mathcal{R}})$ we also get

$$\begin{equation}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\sigma (\tilde{\mathcal{R}}))\leqslant \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,0)\leqslant {L}_{\nu },\quad \forall \enspace z\in \sigma (\mathcal{R})\cap [0,{L}_{\nu }].\end{equation} \tag{ 3.25 }$$

Combining (3.24) and (3.25) and L_ν ⩽ ν/(1 − κ) we obtain

$$\begin{equation}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\sigma (\tilde{\mathcal{R}}))\leqslant \mathrm{max}\left\{\eta \sqrt{\frac{\mu }{\kappa }};\enspace \frac{\nu }{1-\kappa }\right\},\quad \forall \enspace z\in \sigma (\mathcal{R}).\end{equation} \tag{ 3.26 }$$

Repeating verbatim the above arguments we also obtain the estimate

$$\begin{equation}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,\sigma (\mathcal{R}))\leqslant \mathrm{max}\left\{\tilde{\eta }\sqrt{\frac{\tilde{\mu }}{\tilde{\kappa }}};\enspace \frac{\tilde{\nu }}{1-\tilde{\kappa }}\right\},\quad \forall \enspace z\in \sigma (\tilde{\mathcal{R}}).\end{equation} \tag{ 3.27 }$$

The statement of the theorem follows immediately from (2.16), (3.26) and (3.27). □

Let us here finally comment on the concept originally introduced in [25, 26]:

Definition 3.6. We say that $\mathcal{J}$ and $\tilde{\mathcal{J}}$ are δ -quasi-unitary for some δ ⩾ 0 if

$$\begin{equation}{{\Vert}f-\tilde{\mathcal{J}}\mathcal{J}f{\Vert}}_{\mathcal{H}}\leqslant \delta {{\Vert}f{\Vert}}_{{\mathcal{H}}^{1}},\quad \forall \enspace f\in {\mathcal{H}}^{1},\end{equation} \tag{ 3.28 }$$

$$\begin{equation}{{\Vert}u-\mathcal{J}\tilde{\mathcal{J}}u{\Vert}}_{\tilde{\mathcal{H}}}\leqslant \delta {{\Vert}u{\Vert}}_{{\tilde{\mathcal{H}}}^{1}},\quad \forall \enspace u\in {\tilde{\mathcal{H}}}^{1},\end{equation} \tag{ 3.29 }$$

and also (3.7) holds. We say that $\mathcal{A}$ and $\tilde{\mathcal{A}}$ are δ -quasi-unitarily equivalent, if (additionally to (3.28) and (3.29)) $\mathcal{J}$ and $\tilde{\mathcal{J}}$ also fulfil

$$\begin{equation*}{\Vert}\tilde{\mathcal{R}}\mathcal{J}-\mathcal{J}\mathcal{R}{{\Vert}}_{\mathcal{H}\to \tilde{\mathcal{H}}}\leqslant \delta ,\qquad {\Vert}\tilde{\mathcal{J}}\tilde{\mathcal{R}}-\mathcal{R}\tilde{\mathcal{J}}{{\Vert}}_{\tilde{\mathcal{H}}\to \mathcal{H}}\leqslant \delta .\end{equation*}$$

The above concept allows to generalise norm resolvent convergence in the sense that ${\mathcal{A}}_{\varepsilon }$ converges to ${\mathcal{A}}_{0}$ in generalised norm resolvent sense (with convergence speed δ_ɛ ) if ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$ are δ_ɛ -quasi-unitarily equivalent with δ_ɛ → 0 as ɛ → 0 (cf (4.1)). This concept generalises the classical norm resolvent convergence in the sense that if ${\mathcal{H}}_{\varepsilon }={\mathcal{H}}_{0}=\mathcal{H}$ and choosing $\mathcal{J}=\tilde{\mathcal{J}}=\mathrm{I}$ the identity operator on $\mathcal{H}$, then the generalised norm resolvent convergence is just the classical norm resolvent convergence ${{\Vert}{\mathcal{R}}_{\varepsilon }-{\mathcal{R}}_{0}{\Vert}}_{\mathcal{H}\to \mathcal{H}}\leqslant {\delta }_{\varepsilon }\to 0$ as ɛ → 0.

As for the classical norm resolvent convergence we have [25, 26] (see also [28] for a brief up-to-date version and more details):

Proposition 3.7 ([28, section 1.3], [25, appendices A.4 and A.5]). Let Ψ be a measurable function such that f is continuous in a neighbourhood U of $\sigma ({\mathcal{A}}_{0})$ and such that Ψ(λ)(λ + 1)^1/2 → 0 as λ → ∞. If ${\mathcal{A}}_{\varepsilon }$ converges to ${\mathcal{A}}_{0}$ in generalised norm resolvent sense with convergence speed δ_ɛ , then we have

$$\begin{equation*}{{\Vert}{\Psi}({\mathcal{A}}_{\varepsilon })-{\mathcal{J}}_{\varepsilon }{\Psi}({\mathcal{A}}_{0}){\tilde{\mathcal{J}}}_{\varepsilon }{\Vert}}_{{\mathcal{H}}_{\varepsilon }\to {\mathcal{H}}_{\varepsilon }}\to 0\end{equation*}$$

as ɛ → 0. If Ψ is holomorphic on U, then the above norm of the resolvent difference is of order δ_ɛ .

The above proposition applies in particular to the heat operator with Ψ = Ψ_t and Ψ_t (λ) = e^−λt or spectral projections Ψ = 1_I and $\partial I\cap \sigma ({\mathcal{A}}_{0})=\varnothing$. We also showed spectral convergence, see [25, 26] for details. Nevertheless, the result in theorem 3.4 is more explicit as it imitates the proof of (3.12) of Herbst and Nakamura [18] and gives better error estimates.

The above concept of quasi-unitary operators implies the spectral convergence as in theorem 3.4:

Proposition 3.8. Assume that $\mathcal{J}$ and $\tilde{\mathcal{J}}$ are δ-quasi-unitary with δ < 2/3. Then the assumptions (3.15) and (3.16) in theorem 3.4 are fulfilled with

$$\begin{equation*}\mu =\tilde{\mu }=1+\frac{4\delta }{2-3\delta }\quad \text{and}\quad \nu =\tilde{\nu }=\frac{\delta }{2-3\delta }.\end{equation*}$$

Proof. We have

$$\begin{align*}{{\Vert}f{\Vert}}_{\mathcal{H}}^{2}-{{\Vert}\mathcal{J}f{\Vert}}_{\tilde{\mathcal{H}}}^{2}& ={(f-\tilde{\mathcal{J}}\mathcal{J}f,f)}_{\mathcal{H}}+{(\tilde{\mathcal{J}}\mathcal{J}f,f)}_{\mathcal{H}}-{(\mathcal{J}f,\mathcal{J}f)}_{\tilde{\mathcal{H}}}& (\text{using}\;(\text{3.28})\;\text{and}\;(\text{3.7}))\\ & \leqslant \delta {{\Vert}f{\Vert}}_{{\mathcal{H}}^{1}}{{\Vert}f{\Vert}}_{\mathcal{H}}+\delta {{\Vert}f{\Vert}}_{\mathcal{H}}{\Vert}\mathcal{J}f{{\Vert}}_{\tilde{\mathcal{H}}}& \\ & \leqslant \frac{\delta }{2}\left({{\Vert}f{\Vert}}_{{\mathcal{H}}^{1}}^{2}+{{\Vert}f{\Vert}}_{\mathcal{H}}^{2}\right)+\frac{\delta }{2}\left({{\Vert}f{\Vert}}_{\mathcal{H}}^{2}+{\Vert}\mathcal{J}f{{\Vert}}_{\tilde{\mathcal{H}}}^{2}\right)& \\ & =\frac{3\delta }{2}{{\Vert}f{\Vert}}_{\mathcal{H}}^{2}+\frac{\delta }{2}\mathfrak{a}[f,f]+\frac{\delta }{2}{\Vert}\mathcal{J}f{{\Vert}}_{\tilde{\mathcal{H}}}^{2},& \end{align*}$$

whence we get the desired estimate

$$\begin{equation*}{{\Vert}f{\Vert}}_{\mathcal{H}}^{2}\leqslant \left(1+\frac{4\delta }{2-3\delta }\right){{\Vert}\mathcal{J}f{\Vert}}_{\tilde{\mathcal{H}}}^{2}+\frac{\delta }{2-3\delta }\mathfrak{a}[f,f]\end{equation*}$$

provided δ < 2/3. The estimate (3.16) follows similarly. □

Remark 3.9. In our concrete example (cf (4.1)), $\mathcal{J}={\mathcal{J}}_{\varepsilon }$ is an isometry (cf (2.9)), hence (3.15) follows with μ = 1 and ν = 0.

Note that showing (3.16) directly (as in lemma 4.8), we obtain $\tilde{\mu }={\mu }_{\varepsilon }=1+{C}_{6}{\varepsilon }^{2\alpha }$ and $\tilde{\nu }={\nu }_{\varepsilon }={C}_{17}{\varepsilon }^{2\mathrm{min}\left\{\alpha ,\beta \right\}}$, whereas applying proposition 3.8 together with lemma 4.9, we only have $\tilde{\mu }={\mu }_{\varepsilon }=1+\mathcal{O}({\varepsilon }^{\mathrm{min}\left\{\alpha ,\beta \right\}})$ and $\tilde{\nu }={\nu }_{\varepsilon }=\mathcal{O}({\varepsilon }^{\mathrm{min}\left\{\alpha ,\beta \right\}})$, hence a worse estimate for the spectral convergence.

For the proof of theorems 2.2 and 2.3 we will use abstract results given in section 3 (theorems 3.1 and 3.4, respectively). Recall, that these abstract results serve to compare the resolvents and spectra of self-adjoint non-negative operators $\tilde{\mathcal{A}}$ and $\mathcal{A}$ acting in different Hilbert spaces $\tilde{\mathcal{H}}$ and $\mathcal{H}$, respectively. We will apply these abstract theorems for

$$\begin{equation}\tilde{\mathcal{H}}={\mathcal{H}}_{\varepsilon },\qquad \mathcal{H}={\mathcal{H}}_{0},\qquad \tilde{\mathcal{A}}={\mathcal{A}}_{\varepsilon },\quad \mathcal{A}={\mathcal{A}}_{0}\end{equation} \tag{ 4.1 }$$

(recall that ${\mathcal{H}}_{\varepsilon }={\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$ and ${\mathcal{H}}_{0}={\mathsf{L}}^{2}({{\Omega}}_{0})\oplus \mathbb{C}$, and ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$ are the self-adjoint operators acting in these spaces associated with the sesquilinear forms ${\mathfrak{a}}_{\varepsilon }$ (2.6) and ${\mathfrak{a}}_{0}$ (2.7)).

Similarly to (3.1) and (3.2) we introduce Hilbert spaces ${\mathcal{H}}_{\varepsilon }^{k}$ and ${\mathcal{H}}_{0}^{k}$, k = 1, 2, consisting of functions $u\in \mathrm{dom}({\mathcal{A}}_{\varepsilon }^{k/2})$ and $f\in \mathrm{dom}({\mathcal{A}}_{0}^{k/2})$, respectively, equipped with the norms

$$\begin{equation*}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{k}}={\Vert}{({\mathcal{A}}_{\varepsilon }+\mathrm{I})}^{k/2}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }},\qquad {\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{k}}={\Vert}{({\mathcal{A}}_{0}+\mathrm{I})}^{k/2}f{{\Vert}}_{{\mathcal{H}}_{0}}.\end{equation*}$$

Note that for Sobolev spaces we use a sans serif font, e.g. ${\mathsf{H}}^{1}({{\Omega}}_{\varepsilon })$, ${\mathsf{H}}^{2}({{\Omega}}_{0})$, etc.

Recall, that the sets ${D}_{\varepsilon }^{\pm }$ are given in (2.19). We introduce several other subsets of Ω_ɛ :

$$\begin{equation*}{D}_{\varepsilon }^{0}{:=}\left\{\mathbf{x}=({x}_{1},{x}_{2})\in {S}_{\varepsilon }:\enspace {x}_{1}=0\right\},\qquad {Y}_{\varepsilon }{:=}\left\{\mathbf{x}=({x}_{1},{x}_{2})\in {S}_{\varepsilon }:\enspace \vert {x}_{1}\vert < \frac{\varepsilon }{2}\right\}.\end{equation*}$$

Note that Y_ɛ ⊂ S_ɛ due to (2.3). We also denote

$$\begin{equation*}{{\Omega}}_{0}^{\pm }{:=}\left\{x\in {{\Omega}}_{0}:\enspace \pm x > 0\right\},\qquad {\tilde{{\Omega}}}_{0}{:=}{{\Omega}}_{0}\cap \left[-\frac{1}{2},\frac{1}{2}\right],\qquad {\tilde{{\Omega}}}_{0}^{\pm }{:=}{{\Omega}}_{0}^{\pm }\cap {\tilde{{\Omega}}}_{0}.\end{equation*}$$

By ⟨u⟩_D we denote the mean value of the function u(x) in the domain D, i.e.

$$\begin{equation*}{\langle u\rangle }_{D}=\vert D{\vert }^{-1}{\int }_{D}u(\mathbf{x})\enspace \mathrm{d}\mathbf{x},\end{equation*}$$

where $\left\vert D\right\vert$ denotes the area of D. Also we keep the same notation if D is a segment (for example, ${D}_{\varepsilon }^{0}$); in this case we integrate with respect to the natural coordinate on this segment, and |D| denotes its length.

In the following, we need the standard Sobolev inequality.

Lemma 4.1. Let $\mathcal{I}$ be a bounded interval. One has

$$\begin{equation}\forall \enspace f\in {\mathsf{H}}^{1}(\mathcal{I}):\quad {\Vert}f{{\Vert}}_{{\mathsf{L}}^{\infty }(\mathcal{I})}^{2}\leqslant {\ell }_{\mathcal{I}}{\Vert}f{{\Vert}}_{{\mathsf{H}}^{1}(\mathcal{I})}^{2},\end{equation} \tag{ 4.2 }$$

where the constant ${\ell }_{\mathcal{I}} > 1$ depends only on the length $\vert \mathcal{I}\vert$ of $\mathcal{I}$.

Remark 4.2. One can prove, using arguments as in [27, section 6.1], that (4.2) holds with ${\ell }_{\mathcal{I}}=\mathrm{coth}(\ell /2)$.

The following auxiliary estimates will be used further in the proof of theorems 2.2 and 2.3. Similar estimates can be found in [7, lemmatas 3.1 and 3.3] and [8, lemmatas 3.1 and 5.2].

Lemma 4.3. For any $u\in {\mathsf{H}}^{1}({{\Omega}}_{\varepsilon })$ one has

$$\begin{equation}\left\vert {\langle u\rangle }_{{D}_{\varepsilon }^{+}}-{\langle u\rangle }_{{R}_{\varepsilon }}\right\vert \leqslant {C}_{3}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })},\end{equation} \tag{ 4.3 }$$

$$\begin{equation}\vert {\langle u\rangle }_{{D}_{\varepsilon }^{-}}-{\langle u\rangle }_{{Y}_{\varepsilon }}\vert \leqslant {C}_{4}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })},\end{equation} \tag{ 4.4 }$$

$$\begin{equation}\left\vert {\langle u\rangle }_{{D}_{\varepsilon }^{-}}-{\langle u\rangle }_{{D}_{\varepsilon }^{0}}\right\vert \leqslant {C}_{5}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })},\end{equation} \tag{ 4.5 }$$

$$\begin{equation}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\leqslant {C}_{6}{\varepsilon }^{2\alpha }\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}+{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}+{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\right).\end{equation} \tag{ 4.6 }$$

Proof. The following estimates were established in [8, Ineqs. (3.13), (3.14) and (5.16)]:

$$\begin{equation}\vert {\langle u\rangle }_{{D}_{\varepsilon }^{+}}-{\langle u\rangle }_{{R}_{\varepsilon }}\vert \leqslant {C}_{7}{\left\vert \mathrm{ln}\enspace {d}_{\varepsilon }\right\vert }^{1/2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })},\end{equation} \tag{ 4.7 }$$

$$\begin{equation}\vert {\langle u\rangle }_{{D}_{\varepsilon }^{-}}-{\langle u\rangle }_{{Y}_{\varepsilon }}\vert \leqslant {C}_{8}{\left\vert \mathrm{ln}\enspace {d}_{\varepsilon }\right\vert }^{1/2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })},\end{equation} \tag{ 4.8 }$$

$$\begin{equation}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\leqslant {C}_{9}{h}_{\varepsilon }\left({d}_{\varepsilon }{\varepsilon }^{-2}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}^{2}+{d}_{\varepsilon }\left\vert \mathrm{ln}\enspace {d}_{\varepsilon }\right\vert {{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}^{2}\right)+2{({h}_{\varepsilon })}^{2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\end{equation} \tag{ 4.9 }$$

for any $u\in {\mathsf{H}}^{1}({{\Omega}}_{\varepsilon })$. Note that [8] deals with 'rooms' and 'passages' of a different size than in the current work, however, the estimates (4.7)–(4.9) were established for arbitrary ɛ, d_ɛ , h_ɛ , b_ɛ (such that d_ɛ < ɛ ⩽ ɛ₀ < 1, d_ɛ < b_ɛ ⩽ 1), and the constants C₇, C₈ and C₉ depend only on ɛ₀.

Due to ɛ ⩽ min{γ, γ⁻¹} (cf (2.3)), one has |ln γ| ⩽ |ln ɛ|. Using this and taking into account that d_ɛ = γɛ^α+1 with α > 0, we deduce from (4.7) the required estimate (4.3) with C₃ = (α + 2)^1/2 C₇. Similarly, (4.4) holds with C₄ = (α + 2)^1/2 C₈.

Before proving the estimate (4.5) we need an additional step. Let $u\in {C}^{\infty }(\bar{{Y}_{\varepsilon }})$. One has:

$$\begin{equation*}u({x}_{1},{x}_{2})=u(0,{x}_{2})+{\int }_{0}^{{x}_{1}}{\partial }_{1}u(\tau ,{x}_{2})\enspace \mathrm{d}\tau ,\end{equation*}$$

where ${x}_{1}\in \left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)$, x₂ ∈ (−ɛ, 0), and ∂₁ u denotes the partial derivative of u with respect to the first variable. Integrating the above equality with respect to x₁ over $(-\frac{\varepsilon }{2},\frac{\varepsilon }{2})$, with respect to x₂ over (−ɛ, 0), and then dividing by ɛ², we get

$$\begin{equation*}{\langle u\rangle }_{{Y}_{\varepsilon }}={\langle u\rangle }_{{D}_{\varepsilon }^{0}}+{\varepsilon }^{-2}{\int }_{-\varepsilon }^{0}{\int }_{-\varepsilon /2}^{\varepsilon /2}{\int }_{0}^{{x}_{1}}{\partial }_{1}u(\tau ,{x}_{2})\enspace \mathrm{d}\tau \enspace \enspace \mathrm{d}{x}_{1}\enspace \enspace \mathrm{d}{x}_{2},\end{equation*}$$

whence, using the Cauchy–Schwarz inequality, we deduce the estimate

$$\begin{equation}\vert {\langle u\rangle }_{{Y}_{\varepsilon }}-{\langle u\rangle }_{{D}_{\varepsilon }^{0}}\vert \leqslant \frac{1}{2}{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}.\end{equation} \tag{ 4.10 }$$

By standard density arguments (4.10) holds not only for smooth functions, but also for any $u\in {\mathsf{H}}^{1}({Y}_{\varepsilon })$. The estimate (4.5) follows from (4.4) and (4.10) with ${C}_{5}={C}_{4}+\frac{1}{2}{\left\vert \mathrm{ln}\enspace {\varepsilon }_{0}\right\vert }^{-1/2}$.

It remains to prove the estimates (4.6). Let $u\in {C}^{\infty }(\bar{{S}_{\varepsilon }})$. By lemma 4.1 one has

$$\begin{equation*}\vert u({x}_{1},{x}_{2}){\vert }^{2}\leqslant {\ell }_{{\tilde{{\Omega}}}_{0}}{\Vert}u(\cdot ,{x}_{2}){{\Vert}}_{{\mathsf{H}}^{1}({\tilde{{\Omega}}}_{0})}^{2},\end{equation*}$$

where ${x}_{1}\in {\tilde{{\Omega}}}_{0}$, x₂ ∈ (−ɛ, 0). Integrating the above inequality with respect to x₁ over $(-\frac{\varepsilon }{2},\frac{\varepsilon }{2})$, and with respect to x₂ over (−ɛ, 0), we obtain

$$\begin{equation}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}^{2}\leqslant {\ell }_{{\tilde{{\Omega}}}_{0}}\varepsilon {\Vert}u{{\Vert}}_{{\mathsf{H}}^{1}({\tilde{{\Omega}}}_{0}\times (-\varepsilon ,0))}^{2}\leqslant {\ell }_{{\tilde{{\Omega}}}_{0}}\varepsilon {\Vert}u{{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })}^{2};\end{equation} \tag{ 4.11 }$$

by density arguments the estimate (4.11) holds for any $u\in {\mathsf{H}}^{1}({S}_{\varepsilon })$. Combining (4.9), (4.11) and using (2.2), we arrive at the desired estimate (4.6) with

$$\begin{equation}{C}_{6}{:=}\mathrm{max}\left\{{C}_{9}\gamma ({\ell }_{{\tilde{{\Omega}}}_{0}}+\alpha +2);\enspace 2\right\}.\end{equation} \tag{ 4.12 }$$

□

In order to utilise theorem 3.1 we need to construct suitable operators

$$\begin{equation*}{\mathcal{J}}_{\varepsilon }^{1}:{\mathcal{H}}_{0}^{1}\to {\mathcal{H}}_{\varepsilon }^{1},\qquad {\tilde{\mathcal{J}}}_{\varepsilon }^{1}:{\mathcal{H}}_{\varepsilon }^{1}\to {\mathcal{H}}_{0}^{1},\end{equation*}$$

where ${\mathcal{H}}_{0}^{1}$ and ${\mathcal{H}}_{\varepsilon }^{1}$ are the energy spaces associated with the forms ${\mathfrak{a}}_{0}$ and ${\mathfrak{a}}_{\varepsilon }$, i.e.

$$\begin{equation*}\begin{matrix}\hfill {\mathcal{H}}_{0}^{1}=\mathrm{dom}({\mathfrak{a}}_{0})\hfill & \hfill \text{equipped with the norm}\enspace \hfill & \hfill {\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}={({\mathfrak{a}}_{0}[f,f]+{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}}^{2})}^{1/2},\hfill \\ \hfill {\mathcal{H}}_{\varepsilon }^{1}=\mathrm{dom}({\mathfrak{a}}_{\varepsilon })\hfill & \hfill \text{equipped with the norm}\enspace \hfill & \hfill {\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}={({\mathfrak{a}}_{\varepsilon }[u,u]+{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}^{2})}^{1/2}.\hfill \end{matrix}\end{equation*}$$

We define the operator ${\mathcal{J}}_{\varepsilon }^{1}$ as follows. Let $f=({f}_{1},{f}_{2})\in \mathrm{dom}({\mathfrak{a}}_{0})={\mathsf{H}}^{1}({{\Omega}}_{0})\times \mathbb{C}$, we set

$$\begin{equation}({\mathcal{J}}_{\varepsilon }^{1}f)(\mathbf{x})=\begin{cases}{\varepsilon }^{-1/2}{f}_{1}({{\Phi}}_{\varepsilon }({x}_{1})),\quad & \mathbf{x}=({x}_{1},{x}_{2})\in {S}_{\varepsilon },\\ {\varepsilon }^{-1/2}{f}_{1}(0)+\frac{1}{{h}_{\varepsilon }}\left({({b}_{\varepsilon })}^{-1}{f}_{2}-{\varepsilon }^{-1/2}{f}_{1}(0)\right){x}_{2},\quad & \mathbf{x}=({x}_{1},{x}_{2})\in {P}_{\varepsilon },\\ {({b}_{\varepsilon })}^{-1}{f}_{2},\quad & \mathbf{x}\in {R}_{\varepsilon },\end{cases}\end{equation} \tag{ 4.13 }$$

where ${{\Phi}}_{\varepsilon }:\mathbb{R}\to \mathbb{R}$ is a continuous and piecewise linear function given by

$$\begin{equation*}{{\Phi}}_{\varepsilon }(x)=\begin{cases}x,\quad & \vert x\vert \geqslant \frac{\varepsilon }{2},\\ \frac{2x+{d}_{\varepsilon }}{2(\varepsilon -{d}_{\varepsilon })}\varepsilon ,\quad & -\frac{\varepsilon }{2}< x< -\frac{{d}_{\varepsilon }}{2},\\ \frac{2x-{d}_{\varepsilon }}{2(\varepsilon -{d}_{\varepsilon })}\varepsilon ,\quad & \frac{{d}_{\varepsilon }}{2}< x< \frac{\varepsilon }{2},\\ 0,\quad & \vert x\vert \leqslant \frac{{d}_{\varepsilon }}{2}.\end{cases}\end{equation*}$$

The operator ${\tilde{\mathcal{J}}}_{\varepsilon }^{1}$ is introduced as a restriction of ${\tilde{\mathcal{J}}}_{\varepsilon }$ onto $\mathrm{dom}({\mathfrak{a}}_{\varepsilon })$:

$$\begin{equation}{\tilde{\mathcal{J}}}_{\varepsilon }^{1}={\tilde{\mathcal{J}}}_{\varepsilon }{\upharpoonright }_{\mathrm{dom}({\mathfrak{a}}_{\varepsilon })}.\end{equation} \tag{ 4.14 }$$

It is easy to see that (4.14) correctly defines a linear operator from ${\mathsf{H}}^{1}({{\Omega}}_{\varepsilon })$ to ${\mathsf{H}}^{1}({{\Omega}}_{0})\times \mathbb{C}$; here we use the fact that for $u\in {\mathsf{H}}^{1}({S}_{\varepsilon })$ the function $v({x}_{1}){:=}{\int }_{-\varepsilon }^{0}u({x}_{1},{x}_{2})\enspace \mathrm{d}{x}_{2}$ belongs to ${\mathsf{H}}^{1}({{\Omega}}_{0})$, namely

$$\begin{align}{v}^{\prime }({x}_{1})& ={\int }_{-\varepsilon }^{0}{\partial }_{1}u({x}_{1},{x}_{2})\enspace \mathrm{d}{x}_{2},\enspace {\Vert}v{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}\\ & \leqslant {\varepsilon }^{1/2}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })},\enspace {\Vert}{v}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}\leqslant {\varepsilon }^{1/2}{\Vert}{\partial }_{1}u{{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}.\end{align} \tag{ 4.15 }$$

Lemma 4.4. One has

$$\begin{equation}\forall \enspace f\in {\mathcal{H}}_{0}^{1}:\quad {\Vert}{\mathcal{J}}_{\varepsilon }f-{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}\leqslant {C}_{10}{\varepsilon }^{\mathrm{min}\left\{1,\alpha \right\}}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}.\end{equation} \tag{ 4.16 }$$

Proof. It is easy to see (cf (2.8) and (4.13)) that

$$\begin{equation}{\Vert}{\mathcal{J}}_{\varepsilon }f-{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathcal{H}}_{\varepsilon }}^{2}={\Vert}{\mathcal{J}}_{\varepsilon }f-{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}^{2}+{\Vert}{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}.\end{equation} \tag{ 4.17 }$$

To estimate the first term in (4.17) we need the following standard Poincaré inequality following from the variational characterisation of the first Dirichlet eigenvalue:

$$\begin{equation}\forall \enspace f\in {\mathsf{H}}_{0}^{1}(\mathcal{I}):\quad {\Vert}f{{\Vert}}_{{\mathsf{L}}^{2}(\mathcal{I})}^{2}\leqslant {\pi }^{-2}\vert \mathcal{I}{\vert }^{2}{\Vert}{f}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}(\mathcal{I})}^{2},\end{equation} \tag{ 4.18 }$$

where $\mathcal{I}$ is a bounded interval and $\vert \mathcal{I}\vert$ denotes its length. Using (4.18) and the fact that the function f − f ◦ Φ_ɛ belongs to ${\mathsf{H}}_{0}^{1}(-\frac{\varepsilon }{2},\frac{\varepsilon }{2})$ we get

$$\begin{equation}{\Vert}{\mathcal{J}}_{\varepsilon }f-{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}^{2}={\Vert}{f}_{1}-{f}_{1}{\circ}{{\Phi}}_{\varepsilon }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}\leqslant {\pi }^{-2}{\varepsilon }^{2}{\Vert}{({f}_{1}-{f}_{1}{\circ}{{\Phi}}_{\varepsilon })}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}.\end{equation} \tag{ 4.19 }$$

Taking into account that 2d_ɛ ⩽ ɛ (see (2.4)), we get

$$\begin{equation}1\leqslant {{\Phi}}_{\varepsilon }^{\prime }(x)\leqslant 2\quad \text{as}\enspace \vert x\vert \in \left(\frac{{d}_{\varepsilon }}{2},\frac{\varepsilon }{2}\right).\end{equation} \tag{ 4.20 }$$

Using (4.20), the fact that Φ_ɛ (x) = 0 as $\vert x\vert \leqslant \frac{{d}_{\varepsilon }}{2}$ and the chain rule, we can further estimate (4.19) as

$$\begin{align}{\Vert}{\mathcal{J}}_{\varepsilon }f-{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}^{2}& \leqslant 2{\pi }^{-2}{\varepsilon }^{2}\left({\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}+{\Vert}({f}_{1}^{\prime }{\circ}{{\Phi}}_{\varepsilon })\cdot {{\Phi}}_{\varepsilon }^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right){\backslash}\left[-\frac{{d}_{\varepsilon }}{2},\frac{{d}_{\varepsilon }}{2}\right]\right)}^{2}\right)\\ & \leqslant 2{\pi }^{-2}{\varepsilon }^{2}\left({\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}+2{\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}\right)\\ & \leqslant 6{\pi }^{-2}{\varepsilon }^{2}{\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}\leqslant 6{\pi }^{-2}{\varepsilon }^{2}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}^{2}.\end{align} \tag{ 4.21 }$$

Finally, we estimate the second term in (4.17). Recall that ${\tilde{{\Omega}}}_{0}={{\Omega}}_{0}\cap \left[-\frac{1}{2},\frac{1}{2}\right]$. Using (2.2), (4.13) and lemma 4.1, we conclude

$$\begin{align}{\Vert}{\mathcal{J}}_{\varepsilon }^{1}f{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}& \leqslant \mathrm{max}\left\{{\varepsilon }^{-1}\vert {f}_{1}(0){\vert }^{2},\enspace {({b}_{\varepsilon })}^{-2}\vert {f}_{2}{\vert }^{2}\right\}{h}_{\varepsilon }{d}_{\varepsilon }\\ & \leqslant \mathrm{max}\left\{{\varepsilon }^{-1}{\ell }_{{\tilde{{\Omega}}}_{0}}{\Vert}{f}_{1}{{\Vert}}_{{\mathsf{H}}^{1}({\tilde{{\Omega}}}_{0})}^{2},\enspace {({b}_{\varepsilon })}^{-2}\vert {f}_{2}{\vert }^{2}\right\}{h}_{\varepsilon }{d}_{\varepsilon }\\ & \leqslant {\varepsilon }^{-1}{h}_{\varepsilon }{d}_{\varepsilon }\mathrm{max}\left\{{\ell }_{{\tilde{{\Omega}}}_{0}},\varepsilon {({b}_{\varepsilon })}^{-2}\right\}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}^{2}\\ & \leqslant \gamma {\varepsilon }^{2\alpha }{\ell }_{{\tilde{{\Omega}}}_{0}}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}^{2}.\end{align} \tag{ 4.22 }$$

In the last estimate we use β < 1/2 as then $\varepsilon {({b}_{\varepsilon })}^{-2}={\varepsilon }^{1-2\beta }\leqslant 1< {\ell }_{{\tilde{{\Omega}}}_{0}}$. Combining (4.17), (4.21) and (4.22), we arrive at the desired estimate (4.16) with the constant explicitly given by ${C}_{10}=\sqrt{6{\pi }^{-2}+\gamma {\ell }_{{\tilde{{\Omega}}}_{0}}}$. □

Now we come to the key lemma of this work; the following estimate on the two forms associated with ${\mathcal{A}}_{\varepsilon }$ and ${\mathcal{A}}_{0}$:

Lemma 4.5. One has

$$\begin{equation}\forall \enspace f\in {\mathcal{H}}_{0}^{2},\enspace u\in {\mathcal{H}}_{\varepsilon }^{2}:\quad \left\vert {\mathfrak{a}}_{\varepsilon }[{\mathcal{J}}_{\varepsilon }^{1}f,u]-{\mathfrak{a}}_{0}[f,{\tilde{\mathcal{J}}}_{\varepsilon }^{1}u]\right\vert \leqslant {C}_{11}{\varepsilon }^{\mathrm{min}\left\{\alpha ,1/2-\beta \right\}}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}.\end{equation} \tag{ 4.23 }$$

Proof. Let $f=({f}_{1},{f}_{2})\in \mathrm{dom}({\mathcal{A}}_{0})={\mathcal{H}}_{0}^{2}$ and $u\in \mathrm{dom}({\mathcal{A}}_{\varepsilon })={\mathcal{H}}_{\varepsilon }^{2}$. We denote

$$\begin{equation*}{f}_{\varepsilon }{:=}{\mathcal{J}}_{\varepsilon }^{1}f,\qquad {u}_{\varepsilon ,1}{:=}{({\tilde{\mathcal{J}}}_{\varepsilon }^{1}u)}_{1},\qquad {u}_{\varepsilon ,2}{:=}{({\tilde{\mathcal{J}}}_{\varepsilon }^{1}u)}_{2}\end{equation*}$$

(i.e. ${u}_{\varepsilon ,1}\in {\mathsf{H}}^{1}({{\Omega}}_{0})$, ${u}_{\varepsilon ,2}\in \mathbb{C}$ are the first and the second components of ${\tilde{\mathcal{J}}}_{\varepsilon }^{1}u\in {\mathsf{H}}^{1}({{\Omega}}_{0})\times \mathbb{C}$, respectively). Taking into account that f_ɛ is constant on R_ɛ , and V_ɛ = 0 on P_ɛ ∪ R_ɛ , one has

$$\begin{equation*}{\mathfrak{a}}_{\varepsilon }[{\mathcal{J}}_{\varepsilon }^{1}f,u]-{\mathfrak{a}}_{0}[f,{\tilde{\mathcal{J}}}_{\varepsilon }^{1}u]={I}_{\varepsilon }^{1}+{I}_{\varepsilon }^{2}+{I}_{\varepsilon }^{3},\end{equation*}$$

where

$$\begin{align*}{I}_{\varepsilon }^{1}& {:=}{\left(\nabla {f}_{\varepsilon },\nabla u\right)}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}-{\left({f}_{1}^{\prime },{u}_{\varepsilon ,1}^{\prime }\right)}_{{\mathsf{L}}^{2}({{\Omega}}_{0})},\qquad (\text{the part on the strip})\\ {I}_{\varepsilon }^{2}& {:=}{\left(\nabla {f}_{\varepsilon },\nabla u\right)}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}-\gamma {f}_{1}(0)\bar{{u}_{\varepsilon ,1}(0)},\qquad (\text{the part on the passage})\\ {I}_{\varepsilon }^{3}& {:=}{({V}_{\varepsilon }{f}_{\varepsilon },u)}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}-{({V}_{0}{f}_{1},{u}_{\varepsilon ,1})}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}\qquad (\text{the potential term})\text{.}\end{align*}$$

Estimate of ${I}_{\varepsilon }^{1}$ (on the strip). It is easy to see that ${I}_{\varepsilon }^{1}={\left({({f}_{1}{\circ}{{\Phi}}_{\varepsilon })}^{\prime }-{f}_{1}^{\prime },{u}_{\varepsilon ,1}^{\prime }\right)}_{{\mathsf{L}}^{2}(-\frac{\varepsilon }{2},\frac{\varepsilon }{2})}$, therefore we have

$$\begin{equation}\vert {I}_{\varepsilon }^{1}\vert \leqslant {\Vert}{({f}_{1}{\circ}{{\Phi}}_{\varepsilon })}^{\prime }-{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}{\Vert}{u}_{\varepsilon ,1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}.\end{equation} \tag{ 4.24 }$$

Using the chain rule and (4.20), we obtain

$$\begin{align}& {\Vert}{({f}_{1}{\circ}{{\Phi}}_{\varepsilon })}^{\prime }-{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}\\ & \quad \leqslant {\Vert}({f}_{1}^{\prime }{\circ}{{\Phi}}_{\varepsilon })\cdot {{\Phi}}_{\varepsilon }^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right){\backslash}\left[-\frac{{d}_{\varepsilon }}{2},\frac{{d}_{\varepsilon }}{2}\right]\right)}+{\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}\leqslant (\sqrt{2}+1){\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}\end{align} \tag{ 4.25 }$$

Since $f\in \mathrm{dom}({\mathcal{A}}_{0})$, one has

$$\begin{equation}{f}_{1}^{\prime }\in {\mathsf{H}}^{1}({{\Omega}}_{0}^{+})\;\text{and}\;{f}_{1}^{\prime }\in {\mathsf{H}}^{1}({{\Omega}}_{0}^{-}).\end{equation} \tag{ 4.26 }$$

Then, by virtue of (4.2) and (4.26) we can continue the (square of) estimate (4.25) as follows:

$$\begin{align}& {\Vert}{({f}_{1}{\circ}{{\Phi}}_{\varepsilon })}^{\prime }-{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}^{2}& \\ & \qquad \leqslant \frac{{(\sqrt{2}+1)}^{2}\varepsilon }{2}\left({\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{\infty }\left(-\frac{\varepsilon }{2},0\right)}^{2}+{\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{\infty }\left(0,\frac{\varepsilon }{2}\right)}^{2}\right)& \\ & \qquad \leqslant \frac{{(\sqrt{2}+1)}^{2}\varepsilon }{2}\mathrm{max}\left\{{\ell }_{{\tilde{{\Omega}}}_{0}^{-}};\enspace {\ell }_{{\tilde{{\Omega}}}_{0}^{+}}\right\}\left({\Vert}{f}_{1}^{{\prime\prime}}{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0}^{-}\cup {{\Omega}}_{0}^{+})}^{2}+{\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}^{2}\right).\end{align} \tag{ 4.27 }$$

Finally, we need the estimates

$$\begin{equation}{\Vert}{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}\leqslant {\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}\leqslant {\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}},\end{equation} \tag{ 4.28 }$$

and

$$\begin{align}{\Vert}{f}_{1}^{{\prime\prime}}{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0}^{-}\cup {{\Omega}}_{0}^{+})}& \leqslant {\Vert}{\hat{\mathcal{A}}}_{0}{f}_{1}+{f}_{1}{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0}^{-}\cup {{\Omega}}_{0}^{+})}+{\Vert}{V}_{0}{f}_{1}+{f}_{1}{{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0}^{-}\cup {{\Omega}}_{0}^{+})}\\ & \leqslant {\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}}+({{\Vert}{V}_{0}{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}+1){\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}}\leqslant ({{\Vert}{V}_{0}{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}+2){\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}}\end{align} \tag{ 4.29 }$$

(to derive these estimates we have used, in particular, (3.3)). Inequalities (4.27)–(4.29) yield

$$\begin{equation}{\Vert}{({f}_{1}{\circ}{{\Phi}}_{\varepsilon })}^{\prime }-{f}_{1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}\leqslant {C}_{12}{\varepsilon }^{1/2}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}}\end{equation} \tag{ 4.30 }$$

with ${C}_{12}=\sqrt{\frac{{(\sqrt{2}+1)}^{2}}{2}\mathrm{max}\left\{{\ell }_{{\tilde{{\Omega}}}_{0}^{-}};\enspace {\ell }_{{\tilde{{\Omega}}}_{0}^{+}}\right\}\left({({{\Vert}{V}_{0}{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}+2)}^{2}+1\right)}$. Also, using (4.15), we get

$$\begin{equation}{\Vert}{u}_{\varepsilon ,1}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(-\frac{\varepsilon }{2},\frac{\varepsilon }{2}\right)}\leqslant {\Vert}{\partial }_{1}u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}\leqslant {\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}\leqslant {\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}.\end{equation} \tag{ 4.31 }$$

Combining (4.24), (4.30) and (4.31) we arrive at the estimate

$$\begin{equation}\vert {I}_{\varepsilon }^{1}\vert \leqslant {C}_{12}{\varepsilon }^{1/2}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}.\end{equation} \tag{ 4.32 }$$

Estimate of ${I}_{\varepsilon }^{2}$ (on the passage). Now we come to the key part of the form estimate, where the δ-potential in the form ${\mathfrak{a}}_{0}$ appears. We have

$$\begin{equation}{\Delta}{f}_{\varepsilon }=0,\qquad {\partial }_{1}{f}_{\varepsilon }=0,\qquad {\partial }_{2}{f}_{\varepsilon }=\frac{1}{{h}_{\varepsilon }}\left({({b}_{\varepsilon })}^{-1}{f}_{2}-{\varepsilon }^{-1/2}{f}_{1}(0)\right)\quad \text{on}\enspace {P}_{\varepsilon }.\end{equation} \tag{ 4.33 }$$

Using (4.33) and integrating by parts, we obtain

$$\begin{align}{I}_{\varepsilon }^{2}& ={\int }_{\partial {P}_{\varepsilon }}({\partial }_{\text{n}}{f}_{\varepsilon }(\mathbf{x}))\cdot \bar{u(\mathbf{x})}\enspace \mathrm{d}\mathbf{x}-\gamma {f}_{1}(0)\bar{{u}_{\varepsilon ,1}(0)}\\ & ={\partial }_{2}{f}_{\varepsilon }\cdot \left({\int }_{{D}_{\varepsilon }^{+}}\bar{u(\mathbf{x})}\enspace \mathrm{d}\mathbf{x}-{\int }_{{D}_{\varepsilon }^{-}}\bar{u(\mathbf{x})}\enspace \mathrm{d}\mathbf{x}\right)-\gamma {f}_{1}(0)\bar{{u}_{\varepsilon ,1}(0)}\\ & =\frac{1}{{h}_{\varepsilon }}\left({({b}_{\varepsilon })}^{-1}{f}_{2}-{\varepsilon }^{-1/2}{f}_{1}(0)\right)\cdot {d}_{\varepsilon }\left({\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{+}}-{\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{-}}\right)-\gamma {f}_{1}(0)\bar{{u}_{\varepsilon ,1}(0)}\\ & =\gamma \left({\varepsilon }^{1/2}{({b}_{\varepsilon })}^{-1}{f}_{2}-{f}_{1}(0)\right)\cdot {\varepsilon }^{1/2}\left({\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{+}}-{\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{-}}\right)-\gamma {f}_{1}(0)\bar{{u}_{\varepsilon ,1}(0)},\end{align} \tag{ 4.34 }$$

where ∂_n stands for the normal derivative on ∂P_ɛ , and where we used d_ɛ /h_ɛ = ɛγ in the last line. Taking into account that

$$\begin{equation*}{\varepsilon }^{1/2}{\langle u\rangle }_{{D}_{\varepsilon }^{0}}={u}_{\varepsilon ,1}(0)\quad \text{and}\quad {b}_{\varepsilon }{\langle u\rangle }_{{R}_{\varepsilon }}={u}_{\varepsilon ,2},\end{equation*}$$

one can rewrite (4.34) as follows,

$$\begin{equation*}{I}_{\varepsilon }^{2}={I}_{\varepsilon }^{2,1}+{I}_{\varepsilon }^{2,2}+{I}_{\varepsilon }^{2,3}+{I}_{\varepsilon }^{2,4},\end{equation*}$$

where

$$\begin{align*}{I}_{\varepsilon }^{2,1}& =\gamma \left({\varepsilon }^{1/2}{({b}_{\varepsilon })}^{-1}{f}_{\hspace{-2pt}2}-{f}_{1}(0)\right)\cdot {\varepsilon }^{1/2}\left({\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{+}}-{\langle \bar{u}\rangle }_{{R}_{\varepsilon }}\right),\\ {I}_{\varepsilon }^{2,2}& =\gamma \left({\varepsilon }^{1/2}{({b}_{\varepsilon })}^{-1}{f}_{\hspace{-2pt}2}-{f}_{1}(0)\right)\cdot {\varepsilon }^{1/2}{({b}_{\varepsilon })}^{-1}{\bar{u}}_{\varepsilon ,2},\\ {I}_{\varepsilon }^{2,3}& =-\gamma \varepsilon {({b}_{\varepsilon })}^{-1}{f}_{\hspace{-2pt}2}{\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{-}},\\ {I}_{\varepsilon }^{2,4}& =\gamma {f}_{1}(0)\cdot {\varepsilon }^{1/2}\left({\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{-}}-{\langle \bar{u}\rangle }_{{D}_{\varepsilon }^{0}}\right).\end{align*}$$

We first estimate the common first factor in ${I}_{\varepsilon }^{2,1}$ and ${I}_{\varepsilon }^{2,2}$ by

$$\begin{equation*}{\left\vert {\varepsilon }^{1/2}{({b}_{\varepsilon })}^{-1}{f}_{\hspace{-2pt}2}-{f}_{1}(0)\right\vert }^{2}\leqslant \frac{2\varepsilon }{{b}_{\varepsilon }^{2}}{\left\vert {f}_{2}\right\vert }^{2}+2{\ell }_{{\tilde{{\Omega}}}_{0}}{{\Vert}{f}_{1}{\Vert}}_{{\mathsf{H}}^{1}({{\Omega}}_{0})}^{2}\leqslant 2{\ell }_{{\tilde{{\Omega}}}_{0}}{{\Vert}f{\Vert}}_{{\mathcal{H}}_{0}^{1}}^{2}\end{equation*}$$

using (4.2) and the fact that $\varepsilon {({b}_{\varepsilon })}^{-2}={\varepsilon }^{1-2\beta }\leqslant 1< {\ell }_{{\tilde{{\Omega}}}_{0}}$ provided β < 1/2 (as usual ɛ ⩽ ɛ₀ < 1). In particular, using (4.3) and the fact that the quantity ${\varepsilon }^{\beta }{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}$ is uniformly bounded as ɛ ∈ (0, 1], namely

$$\begin{equation}{\varepsilon }^{\beta }{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}\leqslant {(2\beta \mathrm{e})}^{-1/2}\quad \text{as}\enspace \varepsilon \in (0,1],\end{equation} \tag{ 4.35 }$$

we have

$$\begin{align*}\left\vert {I}_{\varepsilon }^{2,1}\right\vert & \leqslant \gamma {(2{\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}{C}_{3}{\varepsilon }^{1/2}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{{\Vert}f{\Vert}}_{{\mathcal{H}}_{0}^{1}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}\\ & \leqslant \gamma {(2{\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}{C}_{3}{\varepsilon }^{\beta }{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}\cdot {\varepsilon }^{1/2-\beta }{{\Vert}f{\Vert}}_{{\mathcal{H}}_{0}^{1}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}\\ & \leqslant \underset{=:{C}_{13}}{\underbrace{\gamma {(2{\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}{C}_{3}{(2\beta \mathrm{e})}^{-1/2}}}\cdot {\varepsilon }^{1/2-\beta }{{\Vert}f{\Vert}}_{{\mathcal{H}}_{0}^{1}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}.\end{align*}$$

Moreover,

$$\begin{equation*}\left\vert {I}_{\varepsilon }^{2,2}\right\vert \leqslant \underset{=:{C}_{14}}{\underbrace{\gamma {(2{\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}}}\cdot {\varepsilon }^{1/2-\beta }{{\Vert}f{\Vert}}_{{\mathcal{H}}_{0}^{1}}{{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}\end{equation*}$$

using the fact that $\left\vert {u}_{\varepsilon ,2}\right\vert \leqslant {{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}$. For the third term ${I}_{\varepsilon }^{2,3}$ we need the estimate

$$\begin{align*}\left\vert {\langle u\rangle }_{{D}_{\varepsilon }^{-}}\right\vert & \leqslant \left\vert {\langle u\rangle }_{{D}_{\varepsilon }^{-}}-{\langle u\rangle }_{{Y}_{\varepsilon }}\right\vert +\left\vert {\langle u\rangle }_{{Y}_{\varepsilon }}\right\vert \\ & \leqslant {C}_{4}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}+{\varepsilon }^{-1}{{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}\\ & \leqslant {C}_{4}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}+{\varepsilon }^{-1/2}{({\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}{\Vert}u{{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })}\\ & \leqslant \left({C}_{4}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}+{\varepsilon }^{-1/2}{({\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}\right){{\Vert}u{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })},\end{align*}$$

following by (4.4) and (4.11). Hence, we have

$$\begin{align*}\left\vert {I}_{\varepsilon }^{2,3}\right\vert \leqslant \gamma \varepsilon {({b}_{\varepsilon })}^{-1}\left\vert {f}_{2}\right\vert \left\vert {\langle u\rangle }_{{D}_{\varepsilon }^{-}}\right\vert & \leqslant \gamma \left({C}_{4}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}+{\varepsilon }^{-1/2}{({\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}\right){\varepsilon }^{1-\beta }\left\vert {f}_{2}\right\vert {{\Vert}u{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })}\\ & \leqslant \underset{=:{C}_{15}}{\underbrace{\gamma \left({C}_{4}+{({\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}\right)}}{\varepsilon }^{1/2-\beta }\left\vert {f}_{2}\right\vert {{\Vert}u{\Vert}}_{{\mathsf{H}}^{1}({S}_{\varepsilon })}\end{align*}$$

(in the last estimate we use the fact that ɛ ⩽ ɛ₀ < 1, whence ɛ|ln ɛ| < 1). Finally, we have

$$\begin{align*}\left\vert {I}_{\varepsilon }^{2,4}\right\vert & \leqslant \gamma {({\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}{C}_{5}\cdot {\varepsilon }^{1/2}{\left\vert \mathrm{ln}\enspace \varepsilon \right\vert }^{1/2}{{\Vert}{f}_{1}{\Vert}}_{{\mathsf{H}}^{1}({{\Omega}}_{0})}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })}\\ & \leqslant \underset{=:{C}_{16}}{\underbrace{\gamma {({\ell }_{{\tilde{{\Omega}}}_{0}})}^{1/2}{C}_{5}{(2\beta \mathrm{e})}^{-1/2}}}\cdot {\varepsilon }^{1/2-\beta }{{\Vert}{f}_{1}{\Vert}}_{{\mathsf{H}}^{1}({{\Omega}}_{0})}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({Y}_{\varepsilon })},\end{align*}$$

using (4.2), (4.5), and (4.35). As a result, we arrive at the estimate

$$\begin{equation}\vert {I}_{\varepsilon }^{2}\vert \leqslant ({C}_{13}+{C}_{14}+{C}_{15}+{C}_{16}){\varepsilon }^{1/2-\beta }{\Vert}f{{\Vert}}_{{\mathcal{H}}_{1}}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{1}}.\end{equation} \tag{ 4.36 }$$

Estimate of ${I}_{\varepsilon }^{3}$ (the potential term). By virtue of (2.8), (2.11), (2.12) and (4.14) one gets the equality

$$\begin{equation*}{I}_{\varepsilon }^{3}={({V}_{\varepsilon }({\mathcal{J}}_{\varepsilon }^{1}-{\mathcal{J}}_{\varepsilon })f,u)}_{{\mathsf{L}}^{2}({S}_{\varepsilon })},\end{equation*}$$

hence, using lemma 4.4, we obtain the estimate

$$\begin{equation}\vert {I}_{\varepsilon }^{3}\vert \leqslant {{\Vert}{V}_{0}{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}{C}_{10}{\varepsilon }^{\mathrm{min}\left\{1,\alpha \right\}}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }};\end{equation} \tag{ 4.37 }$$

note that ${\Vert}{V}_{\varepsilon }{{\Vert}}_{{\mathsf{L}}^{\infty }({S}_{\varepsilon })}={\Vert}{V}_{0}{{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}$.

Finally, combining (4.32), (4.36) and (4.37) and taking into account (3.3), we arrive at the desired estimate (4.23) with

$$\begin{equation*}{C}_{11}={C}_{12}+{C}_{13}+{C}_{14}+{C}_{15}+{C}_{16}+{{\Vert}{V}_{0}{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}{C}_{10}\end{equation*}$$

□

It follows from (2.11), (4.14), (4.16) and (4.23) that the conditions of theorem 3.1 (applied to the spaces and operators as in (4.1)) hold with

$$\begin{equation*}\delta =\mathrm{max}\left\{{C}_{10},{C}_{11}\right\}{\varepsilon }^{\mathrm{min}\left\{\alpha ,1/2-\beta \right\}}.\end{equation*}$$

Hence, applying theorem 3.1, we immediately arrive at the desired first estimate in (2.15); the second follows from ${\tilde{\mathcal{J}}}_{\varepsilon }{\mathcal{R}}_{\varepsilon }-{\mathcal{R}}_{0}{\tilde{\mathcal{J}}}_{\varepsilon }={({\mathcal{R}}_{\varepsilon }{\mathcal{J}}_{\varepsilon }-{\mathcal{J}}_{\varepsilon }{\mathcal{R}}_{0})}^{{\ast}}={\mathcal{L}}_{\varepsilon }^{{\ast}}$ and ${\Vert}{\mathcal{L}}_{\varepsilon }{\Vert}={\Vert}{\mathcal{L}}_{\varepsilon }^{{\ast}}{\Vert}$. In particular theorem 2.2 is proven with C₁ = 4 max{C₁₀, C₁₁}.

Remark 4.6 (On the error estimates in theorem 2.2). Tracing the model parameters, we see that the constant C₁ = 4 max{C₁₀, C₁₁} depends on upper bounds of the parameters ${{\Vert}V{\Vert}}_{{\mathsf{L}}^{\infty }({{\Omega}}_{0})}$, ${\ell }_{\tilde{{\Omega}}}^{1/2}$, ${\ell }_{{\tilde{{\Omega}}}_{\pm }}^{1/2}$, β^−1/2, α^1/2 and γ entering in our model. Note that the (worst) error ɛ^1/2−β appears in ${I}_{\varepsilon }^{2,2}$ and ${I}_{\varepsilon }^{2,3}$. All other error terms are of better order, namely

$$\begin{equation*}{I}_{\varepsilon }^{1}=\mathcal{O}({\varepsilon }^{1/2}),\quad {I}_{\varepsilon }^{2,1},{I}_{\varepsilon }^{2,4}=\mathcal{O}({(\varepsilon \left\vert \mathrm{ln}\enspace \varepsilon \right\vert )}^{1/2}),\quad \text{and}\quad {I}_{\varepsilon }^{3}=\mathcal{O}({\varepsilon }^{\mathrm{min}\left\{1,\alpha \right\}})\end{equation*}$$

are better; and the ɛ^α -term comes from ${\tilde{\mathcal{J}}}_{\varepsilon }^{1}u$ on the second component, i.e. from the ${\mathsf{L}}^{2}$-estimate of u on P_ɛ , see (4.22).

Remark 4.7. 

• (a)  
  Let ${V}_{0}^{\prime }\in {\mathsf{L}}^{\infty }({{\Omega}}_{0})$ with V₀' ⩾ 0, and κ > 0. We define ${V}_{\varepsilon }^{\kappa }\in {\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$ by

  $$\begin{equation*}{V}_{\varepsilon }^{\kappa }(\mathbf{x})=\begin{cases}{V}_{0}^{\prime }({x}_{1}),\quad & \mathbf{x}=({x}_{1},{x}_{2})\in {S}_{\varepsilon },\\ \kappa ,\quad & \mathbf{x}\in {P}_{\varepsilon }\cup {R}_{\varepsilon }.\end{cases}\end{equation*}$$

  We introduce the operators ${\mathcal{A}}_{\varepsilon }^{\kappa }=-{{\Delta}}_{{{\Omega}}_{\varepsilon }}+{V}_{\varepsilon }^{\kappa }$ and ${\mathcal{A}}_{0}^{\kappa }={\hat{\mathcal{A}}}_{0}\oplus (\kappa \mathrm{I})$ acting in ${\mathsf{L}}^{2}({{\Omega}}_{\varepsilon })$ and ${\mathsf{L}}^{2}({{\Omega}}_{0})\oplus \mathbb{C}$, respectively. Then one has the following counterpart of theorem 2.2:

  $$\begin{equation}{\Vert}{\mathcal{R}}_{\varepsilon }^{\kappa }{\mathcal{J}}_{\varepsilon }-{\mathcal{J}}_{\varepsilon }{\mathcal{R}}_{0}^{\kappa }{{\Vert}}_{{\mathcal{H}}_{0}\to {\mathcal{H}}_{\varepsilon }}={\Vert}{\tilde{\mathcal{J}}}_{\varepsilon }{\mathcal{R}}_{\varepsilon }^{\kappa }-{\mathcal{R}}_{0}^{\kappa }{\tilde{\mathcal{J}}}_{\varepsilon }{{\Vert}}_{{\mathcal{H}}_{\varepsilon }\to {\mathcal{H}}_{0}}\leqslant C{\varepsilon }^{\mathrm{min}\left\{\alpha ,1/2-\beta \right\}},\end{equation} \tag{ 4.38 }$$

  where ${\mathcal{R}}_{\varepsilon }^{\kappa }={({\mathcal{A}}_{\varepsilon }^{\kappa }+\mathrm{I})}^{-1}$ and ${\mathcal{R}}_{0}^{\kappa }={({\mathcal{A}}_{0}^{\kappa }+\mathrm{I})}^{-1}$. The proof is similar to the one of theorem 2.2. The only difference pops up in the proof of the estimate

  $$\begin{equation*}\forall \enspace f\in {\mathcal{H}}_{0}^{2},\enspace u\in {\mathcal{H}}_{\varepsilon }^{2}:\quad \left\vert {\mathfrak{a}}_{\varepsilon }^{\kappa }[{\mathcal{J}}_{\varepsilon }^{1}f,u]-{\mathfrak{a}}_{0}^{\kappa }[f,{\tilde{\mathcal{J}}}_{\varepsilon }^{1}u]\right\vert \leqslant C{\varepsilon }^{\mathrm{min}\left\{\alpha ,1/2-\beta \right\}}{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{2}}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}\end{equation*}$$

  (here ${\mathfrak{a}}_{\varepsilon }^{\kappa }$ and ${\mathfrak{a}}_{0}^{\kappa }$ are the forms associated with ${\mathcal{A}}_{\varepsilon }^{\kappa }$ and ${\mathcal{A}}_{0}^{\kappa }$), where one has two extra (comparing with (4.23)) terms:

  $$\begin{equation*}\kappa \left({({\mathcal{J}}_{\varepsilon }^{1}f,u)}_{{\mathsf{L}}^{2}({B}_{\varepsilon })}-{f}_{2}\bar{{({\tilde{\mathcal{J}}}_{\varepsilon }^{1}u)}_{2}}\enspace \right)\quad \text{and}\quad \kappa {({\mathcal{J}}_{\varepsilon }^{1}f,u)}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}.\end{equation*}$$

  The first term vanishes, which follows easily from the definition of the operators ${\mathcal{J}}_{\varepsilon }^{1}$ and ${\tilde{\mathcal{J}}}_{\varepsilon }^{1}$, while the second term is estimates by $C{\varepsilon }^{2\alpha }{\Vert}f{{\Vert}}_{{\mathcal{H}}_{0}^{1}}{\Vert}u{{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}$ for some constant C > 0, cf (4.6) and (4.22).
• (b)  
  Now, let ${V}_{0}\in {\mathsf{L}}^{\infty }({{\Omega}}_{0})$ be a real potential with μ := essinf(V₀) < 0 (in contrast to (2.13)). As before, we define V_ɛ via (2.12). In this case theorem 2.2 remains valid, but with ${\mathcal{R}}_{\varepsilon }$ and ${\mathcal{R}}_{0}$ (see (2.14)) replaced by

  $$\begin{equation*}{({\mathcal{A}}_{\varepsilon }+(1-\mu )\mathrm{I})}^{-1}\quad \text{and}\quad {({\mathcal{A}}_{0}+(1-\mu )\mathrm{I})}^{-1},\end{equation*}$$

  respectively. To prove this one should apply (4.38) for V₀' = V₀ − μ and κ = −μ.

To prove theorem 2.3 we will use theorem 3.4.

Lemma 4.8. One has

$$\begin{equation}\forall \enspace u\in \mathrm{dom}({\mathfrak{a}}_{\varepsilon }):\quad {{\Vert}u{\Vert}}_{{\mathcal{H}}_{\varepsilon }}^{2}\leqslant {\mu }_{\varepsilon }{{\Vert}{\tilde{\mathcal{J}}}_{\varepsilon }u{\Vert}}_{{\mathcal{H}}_{0}}^{2}+{\nu }_{\varepsilon }{\mathfrak{a}}_{\varepsilon }[u,u]\end{equation} \tag{ 4.39 }$$

with

$$\begin{equation*}{\mu }_{\varepsilon }=1+{C}_{6}{\varepsilon }^{2\alpha }\quad \text{and}\quad {\nu }_{\varepsilon }={C}_{17}{\varepsilon }^{2\mathrm{min}\left\{\alpha ,\beta \right\}}.\end{equation*}$$

Proof. For a bounded domain D we have the following (equivalent version of the) Poincaré inequality, namely

$$\begin{equation}{{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}(D)}^{2}\leqslant \left\vert D\right\vert {\left\vert {\langle u\rangle }_{D}\right\vert }^{2}+\frac{1}{{\lambda }_{2}(D)}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}(D)}^{2}\end{equation} \tag{ 4.40 }$$

for $u\in {\mathsf{H}}^{1}(D)$, where λ₂(D) denotes the second (first non-zero) Neumann eigenvalue of D. Applying this inequality with D = {x₁} × (−ɛ, 0), we obtain

$$\begin{equation*}{\int }_{-\varepsilon }^{0}\vert u({x}_{1},{x}_{2}){\vert }^{2}\enspace \mathrm{d}{x}_{2}\leqslant {\left\vert {\varepsilon }^{-1/2}{\int }_{-\varepsilon }^{0}u({x}_{1},{x}_{2})\enspace \mathrm{d}{x}_{2}\right\vert }^{2}+\frac{{\varepsilon }^{2}}{{\pi }^{2}}{\int }_{-\varepsilon }^{0}\vert {\partial }_{2}u({x}_{1},{x}_{2}){\vert }^{2}\enspace \mathrm{d}{x}_{2}\end{equation*}$$

for (almost) all x₁ ∈ Ω₀. Integrating this inequality over Ω₀ with respect to x₁ and taking into account the definition of ${\tilde{\mathcal{J}}}_{\varepsilon }$ in (2.10) and the fact that β < 1, we arrive at the estimate

$$\begin{align}\forall \enspace u\in {\mathsf{H}}^{1}({S}_{\varepsilon }):\quad {{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}& \leqslant {{\Vert}{({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{1}{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}^{2}+\frac{{\varepsilon }^{2}}{{\pi }^{2}}{{\Vert}{\partial }_{2}u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}\\ & \leqslant {{\Vert}{({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{1}{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}^{2}+\frac{{\varepsilon }^{2\beta }}{{\pi }^{2}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}.\end{align} \tag{ 4.41 }$$

Similarly, applying (4.40) with D = R_ɛ , we obtain

$$\begin{equation}{{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}^{2}\leqslant {\left\vert {({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{2}\right\vert }^{2}+\frac{{\varepsilon }^{2\beta }}{{\pi }^{2}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}^{2}\end{equation} \tag{ 4.42 }$$

using also b_ɛ = ɛ^β . Finally, by virtue of (4.6) and (4.41), we obtain

$$\begin{equation}{{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\leqslant {C}_{6}{\varepsilon }^{2\alpha }\left({{\Vert}{({\tilde{\mathcal{J}}}_{\varepsilon }u)}_{1}{\Vert}}_{{\mathsf{L}}^{2}({{\Omega}}_{0})}^{2}+\frac{{\varepsilon }^{2\beta }}{{\pi }^{2}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}+{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}+{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\right),\end{equation} \tag{ 4.43 }$$

where C₆ is given in (4.12). Summing up (4.41)–(4.43), and taking into account that ɛ^2β < 1, we arrive at the desired estimate (4.39) with C₁₇ := π⁻² + C₆(π⁻² + 1). □

It follows from (2.15) and (4.39) that the conditions of theorem 3.4 (applied to the spaces and operators as in (4.1)) hold with

$$\begin{align*}& \eta =\tilde{\eta }=4{C}_{1}{\varepsilon }^{\mathrm{max}\left\{\alpha ,1/2-\beta \right\}},\qquad \mu =1,\qquad \tilde{\mu }={\mu }_{\varepsilon }=1+{C}_{6}{\varepsilon }^{2\alpha },\quad \\ & \nu =0,\qquad \tilde{\nu }={\nu }_{\varepsilon }={C}_{17}{\varepsilon }^{2\mathrm{min}\left\{\alpha ,\beta \right\}},\end{align*}$$

hence, applying theorem 3.4 with $\kappa =\tilde{\kappa }=1/2$ and taking into account (2.17), we immediately arrive at the desired estimate (2.18) with

$$\begin{equation}{C}_{2}=\mathrm{max}\left\{4{C}_{1}\sqrt{2(1+{C}_{6})},2{C}_{17}\right\}.\end{equation} \tag{ 4.44 }$$

The error ɛ^2β not yet appearing in the estimates of theorem 2.2 comes from the contribution of u on R_ɛ in (4.42).

In this subsection, we additionally show that the identification operators are also quasi-unitarily equivalent, see section 3.3. From this concept, also used in [25, 26], the convergence of operator functions as in proposition 3.7 follows. Note that we have given a more explicit (and better) estimate on the spectral convergence in theorem 3.4 here, although it was already shown in [25, 26]. We have commented on the differences in remark 3.9.

Lemma 4.9. We have ${\tilde{\mathcal{J}}}_{\varepsilon }{\mathcal{J}}_{\varepsilon }f=f$ for all $f\in {\mathcal{H}}_{0}$ and

$$\begin{equation*}\forall \enspace u\in \mathrm{dom}({\mathfrak{a}}_{\varepsilon }):\quad {{\Vert}u-{\mathcal{J}}_{\varepsilon }{\tilde{\mathcal{J}}}_{\varepsilon }u{\Vert}}_{{\mathcal{H}}_{\varepsilon }}\leqslant {C}_{18}{\varepsilon }^{\mathrm{min}\left\{\alpha ,\beta \right\}}{{\Vert}u{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}.\end{equation*}$$

Proof. For a domain D, the standard Poincaré inequality

$$\begin{equation}{{\Vert}u-{\langle u\rangle }_{D}{\Vert}}_{{\mathsf{L}}^{2}(D)}^{2}\leqslant \frac{1}{{\lambda }_{2}(D)}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}(D)}^{2}\end{equation} \tag{ 4.45 }$$

holds (in fact, (4.40) is equivalent to (4.45)). Using (4.45) with D = {x₁} × (−ɛ, 0) and D = R_ɛ , and the estimate (4.6), we obtain

$$\begin{align*}{{\Vert}u-{\mathcal{J}}_{\varepsilon }{\tilde{\mathcal{J}}}_{\varepsilon }u{\Vert}}_{{\mathcal{H}}_{\varepsilon }}^{2}& ={\int }_{{{\Omega}}_{0}}{{\Vert}u({x}_{1},\cdot )-{\langle u({x}_{1},\cdot )\rangle }_{(-\varepsilon ,0)}{\Vert}}_{{\mathsf{L}}^{2}((-\varepsilon ,0))}^{2}\enspace \mathrm{d}{x}_{1}\\ & \quad +{{\Vert}u{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}+{{\Vert}u-{\langle u\rangle }_{{R}_{\varepsilon }}{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}^{2}\\ & \leqslant \frac{{\varepsilon }^{2}}{{\pi }^{2}}{\int }_{{{\Omega}}_{0}}{{\Vert}{\partial }_{2}u({x}_{1},\cdot ){\Vert}}_{{\mathsf{L}}^{2}((-\varepsilon ,0))}^{2}\enspace \mathrm{d}{x}_{1}\\ & \quad +{C}_{6}{\varepsilon }^{2\alpha }\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}+{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({S}_{\varepsilon })}^{2}+{\Vert}\nabla u{{\Vert}}_{{\mathsf{L}}^{2}({P}_{\varepsilon })}^{2}\right)\\ & \quad +\frac{{\varepsilon }^{2\beta }}{{\pi }^{2}}{{\Vert}\nabla u{\Vert}}_{{\mathsf{L}}^{2}({R}_{\varepsilon })}^{2}\\ & \leqslant {C}_{18}^{2}{\varepsilon }^{2\mathrm{min}\left\{\alpha ,\beta \right\}}{{\Vert}u{\Vert}}_{{\mathcal{H}}_{\varepsilon }^{1}}^{2},\end{align*}$$

with ${C}_{18}{:=}{({C}_{6}+{\pi }^{-2})}^{1/2}$ (for the last step note that β < 1). The lemma is proven. □