Periodic quantum graphs with predefined spectral gaps

Let $n\in \mathbb{N}$ and let Γ be an arbitrary connected ${\mathbb{Z}}^{n}$-periodic locally finite metric graph. The only assumption we impose on the geometry of Γ is that it does not coincide with a line (see the footnote ² explaining the role of this assumptions) and its fundamental domain is compact (see below). W.l.o.g. (cf the discussion after definition 4.1.1 in [5]) one can assume that Γ is embedded into ${\mathbb{R}}^{d}$ with d = n as n ⩾ 3 and d = 3 as n = 1, 2. We also assume that Γ has no loops—otherwise one can break them into pieces by introducing a new intermediate vertex.

By ${\mathcal{E}}_{{\Gamma}}$ and ${\mathcal{V}}_{{\Gamma}}$ we denote the sets of edges and vertices of Γ, respectively. By l = l(e) we denote the function assigning to the edge e its length l(e). We assume that l(e) < ∞ for each $e{\in \mathcal{E}}_{{\Gamma}}$. In a natural way we introduce on each edge $e{\in \mathcal{E}}_{{\Gamma}}$ the local coordinate x_e ∈ [0, l(e)], so that x_e = 0 and x_e = l(e) correspond to the endpoints of e. For $v\in {\mathcal{V}}_{{\Gamma}}$ we denote by $\mathcal{E}\left(v\right)$ the set of edges emanating from v.

The ${\mathbb{Z}}^{n}$-periodicity of Γ means that

$$\begin{equation*}{\Gamma}+{\nu }_{k}={\Gamma},\quad k=1,\dots ,n\end{equation*}$$

for some linearly independent vectors ${\nu }_{1},\dots ,{\nu }_{n}\in {\mathbb{R}}^{d}$. Let us introduce for $i=\left({i}_{1},\dots ,{i}_{n}\right)\in {\mathbb{Z}}^{n}$ the mapping i⋅ :  Γ → Γ defined by

$$\begin{equation}i\cdot x=x+\sum _{k=1}^{n}{i}_{k}{\nu }_{k},\quad x\in {\Gamma}.\end{equation} \tag{ 1.1 }$$

We denote by Y a fundamental domain of Γ, i.e. a compact set (see the assumption above) satisfying

$$\begin{equation*}\bigcup _{i\in {\mathbb{Z}}^{n}}i\cdot Y={\Gamma},\quad \text{the}\;\text{sets}\;Y\;\text{and}\;i\cdot Y\;\text{may}\;\text{have}\;\text{only}\;\text{finitely}\;\text{many}\;\text{common}\;\text{points}\;\text{as}\;i\ne 0.\end{equation*}$$

In particular, the above condition implies that the vertices on the boundary of the fundamental domain cannot have any common edges. Evidently a fundamental domain in not uniquely defined. Note that for any

$$\begin{equation*}r\in {\mathbb{N}}_{0}^{n}{:=}\left\{r=\left({r}_{1},\dots ,{r}_{n}\right):\enspace {r}_{k}\in \mathbb{N}\cup \left\{0\right\},\quad k=1,\dots ,n\right\}\end{equation*}$$

the graph Γ is also invariant under translations through vectors ${\nu }_{1}^{r},\dots ,{\nu }_{n}^{r}$ defined by ${\nu }_{k}^{r}={r}_{k}{\nu }_{k}$ The corresponding fundamental domain is the set Y^r given by

$$\begin{equation}{Y}^{r}=\bigcup _{i{\in \mathcal{I}}^{r}}i\cdot Y,\quad {\;\text{where}\;\enspace \mathcal{I}}^{r}=\left\{i\in {\mathbb{Z}}^{n}:\enspace {i}_{k}\in \left[0,{r}_{k}\right],\quad k=1,\dots ,n\right\}.\end{equation} \tag{ 1.2 }$$

Finally, we denote by ${\mathcal{U}}_{Y}$ the set of points of a fundamental domain Y that simultaneously belong to 'neighboring' fundamental domains, i.e.

$$\begin{equation*}{\mathcal{U}}_{Y}=\left\{v\in Y:\enspace \exists i\in {\mathbb{Z}}^{n}{\backslash}\left\{0\right\}\quad \;\text{such}\;\text{that}\;\enspace v\in i\cdot Y\right\}.\end{equation*}$$

An example of a ${\mathbb{Z}}^{2}$-periodic graph is presented on figure 2(a). This is an equilateral hexagonal lattice in ${\mathbb{R}}^{2}$, which is invariant under translations through vectors ${ \overrightarrow {\nu }}_{1}=\left(\sqrt{3},0\right)$, ${ \overrightarrow {\nu }}_{2}=\left(-\frac{\sqrt{3}}{2},\frac{3}{2}\right)$. Its fundamental domain Y is highlighted in bold lines. On figure 2(b) one sees the fundamental domain Y^r (1.2) for r = (2, 2). On these figures the bold dots are vertices belonging to ${\mathcal{U}}_{Y}$ and ${\mathcal{U}}_{{Y}^{r}}$, respectively.

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It is easy to see that for any $m\in \mathbb{N}$ there exists such $r=\left({r}_{1},\dots ,{r}_{n}\right)\in {\mathbb{N}}_{0}^{n}$ that the fundamental domain Y^r (1.2) can be represented as a union

$$\begin{equation}{Y}^{r}=\bigcup _{j=0}^{m}{Y}_{j}\end{equation} \tag{ 1.3 }$$

of non-empty compact sets Y_j , j = 0, ..., m satisfying the following conditions:

$$\begin{align}\hfill & \left(\text{i}\right)\hfill & \hfill & {Y}_{j}\;\text{are}\;\text{connected},\quad j=0,\dots ,m,\hfill \\ \hfill & \left(\text{ii}\right)\hfill & \hfill & {\mathcal{U}}_{Y}\subset {Y}_{0}{,\quad \mathcal{U}}_{Y}\cap {Y}_{j}=\varnothing ,\quad j=1,\dots m,\hfill \\ \hfill & \left(\text{iii}\right)\hfill & \hfill & \forall \left(j\ne 0,\enspace k\ne 0,\enspace j\ne k\right):\enspace {Y}_{j}\cap {Y}_{k}=\varnothing ,\hfill \\ \hfill & \left(\text{iv}\right)\hfill & \hfill & \text{the}\;\text{sets}\;{\mathcal{V}}_{j}{:=}{Y}_{j}\cap {Y}_{0},j=1,\dots ,m\;\text{are}\;\text{non}-\text{empty}\;\text{and}\;\text{consist}\;\text{of}\;\text{finitely}\;\text{many}\;\text{points},\hfill \\ \hfill & \left(\text{v}\right)\hfill & \hfill & {Y}_{0}\;\text{has}\;\text{a}\;\text{vertex}\;\tilde {v}{\;\text{belonging}\;\text{neither}\;\text{to}\;\mathcal{U}}_{Y}\;\text{nor}\;\text{to}\;{\cup }_{j=1}^{m}{Y}_{j}.\hfill \end{align} \tag{ 1.4 }$$

W.l.o.g. we may assume that the points belonging to ${\mathcal{V}}_{j}$ are the vertices of Γ (if $v\in {\mathcal{V}}_{j}$ lies on the interior of an edge of Γ we can regard it as a vertex with two outgoing edges). It is easy to see that such a decomposition is always possible for large enough r₁, r₂, ..., r_n ² . Of course such a decomposition is not unique. For example on figure 2(c) the domain Y^r is decomposed in such a way that (1.3) and (1.4) with m = 3 holds: Y₀ consists of bold solid lines, while Y₁, Y₂, Y₃ consist of one dashed edge, the black square is $\tilde {v}$, the white circles indicate vertices belonging to ${\mathcal{V}}_{j}$ (on the figure each ${\mathcal{V}}_{j}$, j = 1, 2, 3 consists of two vertices denoted by v_j1 and v_j2).

Now, let $m\in \mathbb{N}$ be given and let us fix such $r=\left({r}_{1},\dots ,{r}_{n}\right)\in {\mathbb{N}}_{0}^{n}$ that the fundamental domain Y^r admits representation (1.3) and (1.4). We set for $i\in {\mathbb{Z}}^{n}$:

$$\begin{equation*}{\mathcal{V}}_{ij}{:=}{i}^{r}\cdot {\mathcal{V}}_{j},\quad j=1,\dots ,m,\quad {Y}_{ij}{:=}{i}^{r}\cdot {Y}_{j},\quad j=0,\dots ,m,\quad {\tilde {v}}_{i}{:=}{i}^{r}\cdot \tilde {v},\end{equation*}$$

where the mapping i^r ⋅ :  Γ → Γ is defined by ${i}^{r}\cdot x=x+\sum _{k=1}^{n}{i}_{k}{r}_{k}{\nu }_{k},\quad x\in {\Gamma}.$

The vertices belonging to ${\mathcal{V}}_{ij}$ will support δ'-type conditions, the vertices ${\tilde {v}}_{i}$ will support δ-conditions, in the remaining vertices the Kirchhoff conditions will be posed.

In what follows if $u:{\Gamma}\to \mathbb{C}$ and $e{\in \mathcal{E}}_{{\Gamma}}$ then by u_e we denote the restriction of u onto the interior of e. Via a local coordinate x_e we identify u_e with a function on (0, l(e)).

The space L ²(Γ) consists of functions $u:{\Gamma}\to \mathbb{C}$ such that u_e ∈ L ²(0, l(e)) for each edge e and

$$\begin{equation*}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left({\Gamma}\right)}^{2}{:=}\sum _{e{\in \mathcal{E}}_{{\Gamma}}}{\Vert}{u}_{e}{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}{< }\infty .\end{equation*}$$

The space ${\tilde {\mathsf{H}}}^{k}\left({\Gamma}\right)$, $k\in \mathbb{N}$ consists of functions $u:{\Gamma}\to \mathbb{C}$ such that u_e belongs to the Sobolev space H ^k (0, l(e)) for each edge e and

$$\begin{equation*}{\Vert}u{{\Vert}}_{{\tilde {\mathsf{H}}}^{k}\left({\Gamma}\right)}^{2}{:=}\sum _{e{\in \mathcal{E}}_{{\Gamma}}}{\Vert}{u}_{e}{{\Vert}}_{{\mathsf{H}}^{k}\left(0,l\left(e\right)\right)}^{2}{< }\infty .\end{equation*}$$

By ${\mathsf{H}}_{\mathfrak{h}}^{1}\left({\Gamma}\right)$ we denote a subspace of ${\tilde {\mathsf{H}}}^{1}\left({\Gamma}\right)$ consisting of such function $u\in {\tilde {\mathsf{H}}}^{1}\left({\Gamma}\right)$ that

• if $v\in {\mathcal{V}}_{{\Gamma}}{\backslash}\left({\cup }_{i\in {\mathbb{Z}}^{n}}{\cup }_{j=1}^{m}{\mathcal{V}}_{ij}\right)$ then u is continuous at v, i.e. the limiting value of u(x) when x approaches v along $e\in \mathcal{E}\left(v\right)$ is the same for each $e\in \mathcal{E}\left(v\right)$. We denote this value by u(v);
• if $v\in {\mathcal{V}}_{ij}={Y}_{ij}\cap {Y}_{i0}$ for some $i=\left({i}_{1},\dots ,{i}_{n}\right)\in {\mathbb{Z}}^{n}$, j ∈ {1, ..., m} then
  • 1  
    the limiting value of u(x) when x approaches v along $e\in \mathcal{E}\left(v\right)\cap {Y}_{i0}$ is the same for each $e\in \mathcal{E}\left(v\right)\cap {Y}_{i0}$. We denote this value by u₀(v).
  • 2  
    the limiting value of u(x) when x approaches v along $e\in \mathcal{E}\left(v\right)\cap {Y}_{ij}$ is the same for each $e\in \mathcal{E}\left(v\right)\cap {Y}_{ij}$. We denote this value by u_j (v).

Let ɛ > 0 be a small parameter. In L ²(Γ) we introduce the quadratic form ${\mathfrak{h}}_{\varepsilon }$,

$$\begin{align}\hfill {\mathfrak{h}}_{\varepsilon }\langle u,u\rangle & ={\varepsilon }^{-1}\sum _{e{\in \mathcal{E}}_{{\Gamma}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}+\sum _{i\in {\mathbb{Z}}^{n}}\sum _{j=1}^{m}\sum _{v\in {\mathcal{V}}_{ij}}{\alpha }_{j}\vert {u}_{0}\left(v\right)-{\beta }_{j}{u}_{j}\left(v\right){\vert }^{2}\hfill \\ \hfill & \quad +\enspace \sum _{i\in {\mathbb{Z}}^{n}}\gamma \vert u\left({\tilde {v}}_{i}\right){\vert }^{2}\hfill \end{align} \tag{ 1.5 }$$

on the domain $\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }\right)={\mathsf{H}}_{\mathfrak{h}}^{1}\left({\Gamma}\right)$. Here α_j , β_j , γ are real constants, moreover α_j ≠ 0, β_j ≠ 0 [this assumption is needed to avoid the decoupling at the vertex v, cf (1.9)]. These constants are at our disposal and they will be specified later in section 3. The second and third terms in the right-hand side of (1.5) are indeed finite on $u\in {\tilde {\mathsf{H}}}^{1}\left({\Gamma}\right)$, this follows easily from the trace inequality [5, lemma 1.3.8]

$$\begin{equation*}\vert u\left(0\right){\vert }^{2}{\leqslant}2{l}^{-1}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\right)}^{2}+l{\Vert}{u}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\right)}^{2},\quad \forall \enspace u\in {\mathsf{H}}^{1}\left(0,l\right)\end{equation*}$$

and periodicity of Γ. It is also straightforward to verify that the form ${\mathfrak{h}}_{\varepsilon }$ is densely defined in L ²(Γ), lower semibounded and closed. By the first representation theorem [16, theorem 6.2.1] there exists the unique self-adjoint operator ${\mathcal{H}}_{\varepsilon }$ associated with the form ${\mathfrak{h}}_{\varepsilon }$, i.e.

$$\begin{equation}{\left({\mathcal{H}}_{\varepsilon }u,w\right)}_{{\mathsf{L}}^{2}\left({\Gamma}\right)}={\mathfrak{h}}_{\varepsilon }\langle u,w\rangle ,\quad \forall \enspace u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathcal{H}}_{\varepsilon }\right)\subset \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }\right),\quad \forall \enspace w\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }\right),\end{equation} \tag{ 1.6 }$$

where ${\mathfrak{h}}_{\varepsilon }\langle u,w\rangle$ is the sesquilinear form, which corresponds to the quadratic form (1.5).

The domain of ${\mathcal{H}}_{\varepsilon }$ consists of functions $u\in {\mathsf{H}}_{\mathfrak{h}}^{1}\left({\Gamma}\right)\cap {\tilde {\mathsf{H}}}^{2}\left({\Gamma}\right)$ satisfying

$$\begin{equation}\sum _{e\in \mathcal{E}\left(v\right)}{\left.\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\right\vert }_{{\mathbf{x}}_{e}=0}=0\quad \text{at}\enspace v\in {\mathcal{V}}_{{\Gamma}}{\backslash}\bigcup _{i\in {\mathbb{Z}}^{n}}\left(\left\{{\tilde {v}}_{i}\right\}\cup \left(\bigcup _{j=1}^{m}{\mathcal{V}}_{ij}\right)\right),\end{equation} \tag{ 1.7 }$$

$$\begin{equation}\sum _{e\in \mathcal{E}\left(v\right)}{\left.\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\right\vert }_{{\mathbf{x}}_{e}=0}=\gamma \varepsilon \enspace u\left(v\right)\quad \text{at}\enspace v={\tilde {v}}_{i},\end{equation} \tag{ 1.8 }$$

$$\begin{equation}\left.\begin{matrix}\hfill \sum _{e\in \mathcal{E}\left(v\right)\cap {Y}_{i0}}{\left.\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\right\vert }_{{\mathbf{x}}_{e}=0}={\alpha }_{j}\varepsilon \left({u}_{0}\left(v\right)-{\beta }_{j}{u}_{j}\left(v\right)\right),\hfill \\ \hfill \sum _{e\in \mathcal{E}\left(v\right)\cap {Y}_{ij}}{\left.\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\right\vert }_{{\mathbf{x}}_{e}=0}=-{\alpha }_{j}\varepsilon \left({u}_{0}\left(v\right)-{\beta }_{j}{u}_{j}\left(v\right)\right)\hfill \end{matrix}\right\}\quad \text{at}\enspace v\in {\mathcal{V}}_{ij},\end{equation} \tag{ 1.9 }$$

where x_e ∈ [0, l(e)] is a natural coordinate on $e\in \mathcal{E}\left(v\right)$ such that x_e = 0 at v. The action of ${\mathcal{H}}_{\varepsilon }$ is

$$\begin{equation}{\left({\mathcal{H}}_{\varepsilon }u\right)}_{e}=-{\varepsilon }^{-1}\frac{{\mathrm{d}}^{2}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}^{2}},\quad e\;{\in \mathcal{E}}_{{\Gamma}}.\end{equation} \tag{ 1.10 }$$

Condition (1.7) is usually referred as Kirchhoff coupling (sometimes the name Neumann coupling is used), condition (1.8) is known as δ-coupling of the strength γɛ. We may refer to conditions (1.9) as δ'-type coupling. The reason for this is as follows. Suppose that $v\in {\mathcal{V}}_{ij}$ has only two outgoing edges $e\in \mathcal{E}\left(v\right)\cap {Y}_{i0}$ and $\tilde {e}\in \mathcal{E}\left(v\right)\cap {Y}_{ij}$. Also let β_j = 1. Then conditions (1.9) are equivalent to

$$\begin{equation*}{\left.\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\right\vert }_{{\mathbf{x}}_{e}=0}+{\left.\frac{\mathrm{d}{u}_{\tilde {e}}}{\mathrm{d}{\mathbf{x}}_{\tilde {e}}}\right\vert }_{{\mathbf{x}}_{\tilde {e}}=0}=0,\qquad {\left({\alpha }_{j}\varepsilon \right)}^{-1}{\left.\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\right\vert }_{{\mathbf{x}}_{e}=0}={u}_{e}{\vert }_{{\mathbf{x}}_{e}=0}-{u}_{\tilde {e}}{\vert }_{{\mathbf{x}}_{\tilde {e}}=0}.\end{equation*}$$

Taking into account the definition of coordinates x_e and ${\mathbf{x}}_{\tilde {e}}$ we conclude that (1.9) coincides with the usual δ'-conditions of the strength ${\left({\alpha }_{j}\varepsilon \right)}^{-1}$ at a point on the line [2, section 1.4].

We denote

$$\begin{equation}{l}_{j}{:=}\sum _{e{\in \mathcal{E}}_{{Y}_{j}}}l\left(e\right),\enspace j=0,\dots ,m,\quad {N}_{j}{:=}\enspace \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\;\;\mathrm{o}\mathrm{f}\;\;{\mathcal{V}}_{j},\quad j=1,\dots ,m.\end{equation} \tag{ 1.11 }$$

Then for j = 1, ..., m we set

$$\begin{equation}{A}_{j}{:=}{\alpha }_{j}{\beta }_{j}^{2}{N}_{j}{l}_{j}^{-1},\end{equation} \tag{ 1.12 }$$

where α_j , β_j are real non-zero constants from (1.5). We assume that A_j are pairwise distinct; in this case we can renumber them in such a way that

$$\begin{equation}\forall \enspace j=1,\dots ,m-1:\quad {A}_{j}{< }{A}_{j+1}.\end{equation} \tag{ 1.13 }$$

Finally, we consider the following equation (for unknown $\lambda \in \mathbb{C}{\backslash}\left\{{A}_{1},\dots ,{A}_{m}\right\}$):

$$\begin{equation}\lambda \left({l}_{0}+\sum _{j=1}^{m}\frac{{A}_{j}{l}_{j}}{{\beta }_{j}^{2}\left({A}_{j}-\lambda \right)}\right)=\gamma \end{equation} \tag{ 1.14 }$$

where γ is a real constant from (1.5). It is easy to show that this equation has exactly m + 1 roots B_j , j = 0, ..., m, they are real, moreover (after an appropriate renumeration) these roots satisfy

$$\begin{equation}{B}_{0}{< }{A}_{1}{< }{B}_{1}{< }{A}_{2}{< }{B}_{2}{< }\dots {< }{A}_{m}{< }{B}_{m}.\end{equation} \tag{ 1.15 }$$

We are now in position to formulate the first main result of this work.

Theorem 1.1. There exist such positive constants Λ₀ (depending on Y) and C_A , C_B , ɛ₀ (depending on α_j , β_j , γ and Y) that for all ɛ < ɛ₀ the spectrum of ${\mathcal{H}}_{\varepsilon }$ has the following structure within (−∞, Λ₀ ɛ⁻¹]:

$$\begin{equation}\sigma \left({\mathcal{H}}_{\varepsilon }\right)\cap \left(-\infty ,{{\Lambda}}_{0}{\varepsilon }^{-1}\right]=\left[{B}_{0,\varepsilon },{{\Lambda}}_{0}{\varepsilon }^{-1}\right]{\backslash}\bigcup _{j=1}^{m}\left({A}_{j,\varepsilon },{B}_{j,\varepsilon }\right),\end{equation} \tag{ 1.16 }$$

where the numbers A_j,ɛ , j = 1, ..., m and B_j,ɛ , j = 0, ..., m satisfy

$$\begin{equation}{B}_{0,\varepsilon }{< }{A}_{1,\varepsilon }{< }{B}_{1,\varepsilon }{< }{A}_{2,\varepsilon }{< }{B}_{2,\varepsilon }{< }\dots {< }{A}_{m,\varepsilon }{< }{B}_{m,\varepsilon }{< }{{\Lambda}}_{0}{\varepsilon }^{-1},\end{equation} \tag{ 1.17 }$$

moreover

$$\begin{equation}0{\leqslant}{A}_{j}-{A}_{j,\varepsilon }{\leqslant}{C}_{A}{\varepsilon }^{1/2},\quad j=1,\dots ,m,\quad 0{\leqslant}{B}_{j}-{B}_{j,\varepsilon }{\leqslant}{C}_{B}{\varepsilon }^{1/2},\quad j=0,\dots ,m,\end{equation} \tag{ 1.18 }$$

the numbers A_j , B_j are specified by (1.12)–(1.14).

To simplify the notations we assume that the fundamental domain Y^r admits representation (1.3) and (1.4) for r = 0, i.e., already the initial fundamental domain Y admits such a representation. In the general case one should simply change the notations accordingly.

In the following if $\mathcal{H}$ is a self-adjoint lower semi-bounded operator with purely discrete spectrum, we denote by ${\left\{{\lambda }_{k}\left(\mathcal{H}\right)\right\}}_{k\in \mathbb{N}}$ the sequence of its eigenvalues arranged in the ascending order and repeated according to their multiplicity.

The Floquet–Bloch theory [5, chapter 4] establishes a relationship between the spectrum of the operator ${\mathcal{H}}_{\varepsilon }$ and the spectra of certain operators ${\mathcal{H}}_{\varepsilon }^{\theta }$ in L ²(Y). Namely, let

$$\begin{equation*}\theta \in {\mathbb{T}}^{n}=\left\{\theta =\left({\theta }_{1},\dots ,{\theta }_{n}\right)\in {\mathbb{C}}^{n},\quad \vert {\theta }_{k}\vert =1\quad \;\text{for}\;\text{all}\;\enspace k=1,\dots ,n\right\}.\end{equation*}$$

We denote by ${\mathsf{H}}_{\mathfrak{h}}^{1,\theta }\left({\Gamma}\right)$ the set of such functions $u:{\Gamma}\to \mathbb{C}$ that u_e ∈ H ¹(0, l(e)) for each $e\;{\in \mathcal{E}}_{{\Gamma}}$, u satisfy the same conditions at vertices of Γ as functions from ${\mathsf{H}}_{\mathfrak{h}}^{1}\left({\Gamma}\right)$, and

$$\begin{equation*}\forall \enspace x\in {\Gamma},\quad \forall \enspace i=\left({i}_{1},\dots ,{i}_{n}\right)\in {\mathbb{Z}}^{n}:\enspace u\left(i\cdot x\right)={\theta }^{i}u\left(x\right),\quad \;\text{where}\;\enspace {\theta }^{i}{:=}\left(\prod _{k=1}^{n}{\left({\theta }_{k}\right)}^{{i}_{k}}\right)\end{equation*}$$

[recall, that the mapping i⋅ : Γ → Γ is defined by (1.1)]. We introduce the quadratic form ${\mathfrak{h}}_{\varepsilon }^{\theta }$ by

$$\begin{equation}\begin{aligned}\hfill {\mathfrak{h}}_{\varepsilon }^{\theta }\langle u,u\rangle & ={\varepsilon }^{-1}\sum _{e{\in \mathcal{E}}_{Y}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}+\sum _{j=1}^{m}\sum _{v\in {\mathcal{V}}_{j}}{\alpha }_{j}{\left\vert {u}_{0}\left(v\right)-{\beta }_{j}{u}_{j}\left(v\right)\right\vert }^{2}+\gamma {\left\vert u\left(\tilde {v}\right)\right\vert }^{2},\hfill \\ \hfill \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{\theta }\right)& =\left\{u=f{\vert }_{Y},\enspace f\in {\mathsf{H}}_{\mathfrak{h}}^{1,\theta }\left({\Gamma}\right)\right\}.\hfill \end{aligned}\end{equation} \tag{ 2.1 }$$

Hereinafter by ${\mathcal{E}}_{Y}$ and ${\mathcal{V}}_{Y}$ we denote the set of edges and vertices of Y, respectively; similar notations will be used for Y_j . The form ${\mathfrak{h}}_{\varepsilon }^{\theta }$ is densely defined in L ²(Y), lower semibounded and closed. We denote by ${\mathcal{H}}_{\varepsilon }^{\theta }$ the operator associated with ${\mathfrak{h}}_{\varepsilon }^{\theta }$. The spectrum of ${\mathcal{H}}_{\varepsilon }^{\theta }$ is purely discrete, moreover for each $k\in \mathbb{N}$ the function $\theta {\mapsto}{\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{\theta }\right)$ is continuous. Consequently, the set

$$\begin{equation}{L}_{k,\varepsilon }=\bigcup _{\theta \in {\mathbb{T}}^{n}}\left\{{\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{\theta }\right)\right\}\quad \;\text{is}\;\text{a}\;\text{compact}\;\text{interval}.\end{equation} \tag{ 2.2 }$$

According to the Floquet–Bloch theory we have the following representation:

$$\begin{equation}\sigma \left({\mathcal{H}}_{\varepsilon }\right)=\bigcup _{k=1}^{\infty }{L}_{k,\varepsilon }.\end{equation} \tag{ 2.3 }$$

Along with ${\mathfrak{h}}_{\varepsilon }^{\theta }$ we also introduce the forms ${\mathfrak{h}}_{\varepsilon }^{N}$ and ${\mathfrak{h}}_{\varepsilon }^{D}$ acting on the domains

$$\begin{equation*}\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{N}\right)=\left\{u=f{\vert }_{Y},\enspace f\in {\mathsf{H}}_{\mathfrak{h}}^{1}\left({\Gamma}\right)\right\}\text{and}\;\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{D}\right)=\left\{u=f{\vert }_{Y},\enspace f\in {\mathsf{H}}_{\mathfrak{h}}^{1}\left({\Gamma}\right),\enspace \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(v\right)\subset Y\right\}\end{equation*}$$

and with the action being again specified by (2.1). By ${\mathcal{H}}_{\varepsilon }^{N}$ and ${\mathcal{H}}_{\varepsilon }^{D}$ we denote the associated operators. The spectra of these operators are purely discrete. It is easy to see that

$$\begin{equation*}\forall \enspace \theta \in {\mathbb{T}}^{n}:\quad \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{N}\right)\supset \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{\theta }\right)\supset \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{D}\right),\end{equation*}$$

whence, using the min–max principle [7, section 4.5], we conclude

$$\begin{equation}\forall \enspace k\in \mathbb{N},\quad \forall \enspace \theta \in {\mathbb{T}}^{n}:\quad {\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{N}\right){\leqslant}{\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{\theta }\right){\leqslant}{\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{D}\right).\end{equation} \tag{ 2.4 }$$

In the following we mostly use two distinguished points of ${\mathbb{T}}^{n}$,

$$\begin{equation}{\theta }_{\mathrm{p}}{:=}\left(1,1,\dots ,1\right)\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {\theta }_{\mathrm{a}}{:=}-\left(1,1,\dots ,1\right).\end{equation} \tag{ 2.5 }$$

The subscripts p and a means periodic and antiperiodic, respectively.

Remark 2.1. The main ingredients for the proof of theorem 1.1 are two-side estimates for the eigenvalues ${\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{\theta }\right)$, ${\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{N}\right)$, ${\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{D}\right)$, see lemmata 2.1, 2.3–2.6 below. The proof of these estimates is based on the standard min–max principle [7, theorem 4.5.3] and the result [12, lemma 2.1] both requiring no information on the structure of the domains of the operators ${\mathcal{H}}_{\varepsilon }^{\theta }$, ${\mathcal{H}}_{\varepsilon }^{N}$, ${\mathcal{H}}_{\varepsilon }^{D}$ (all calculations are conducted on the level of the associated quadratic forms). However it is interesting to take a more close look on these operators. Let $u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathcal{H}}_{\varepsilon }^{{\ast}}\right)$ with ∗ ∈ {θ, N, D}. Then

• for each $e\;{\in \mathcal{E}}_{Y}$ one has u_e ∈ H ²(0, l(e)),
• at the vertices from ${\mathcal{V}}_{Y}{{\backslash}\mathcal{U}}_{Y}\enspace u$ satisfies the same conditions as functions belonging to $\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathcal{H}}_{\varepsilon }\right)$.

To describe the behaviour of u on ${\mathcal{U}}_{Y}$ we assume for simplicity that points of ${\mathcal{U}}_{Y}$ lie on the interior of edges of Γ (in fact, one can always choose a period cell in such a way that this assumption fulfills). This assumption on a period cell implies, in particular, that for any $v{\in \mathcal{U}}_{Y}$ there is only one edge of ${\mathcal{E}}_{Y}$ (we denote it e_v ) emanating from v. Then we get the following boundary conditions at ${\mathcal{U}}_{Y}$:

• $u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathcal{H}}_{\varepsilon }^{\theta }\right)$ satisfies θ-periodic conditions at $v\;{\in \mathcal{U}}_{Y}$:
  $$\begin{equation*}{u}_{{e}_{w}}\left(w\right)={\theta }^{i}{u}_{{e}_{v}}\left(v\right),\qquad \frac{\mathrm{d}{u}_{{e}_{w}}}{\mathrm{d}{\mathbf{x}}_{{e}_{w}}}\left(w\right)=-{\theta }^{i}\frac{\mathrm{d}{u}_{{e}_{v}}}{\mathrm{d}{\mathbf{x}}_{{e}_{v}}}\left(v\right)\end{equation*}$$
  where $w{\in \mathcal{U}}_{Y}$ is such that w = i ⋅ v for some $i\in {\mathbb{Z}}^{n}$ (one can show that for each $v{\in \mathcal{U}}_{Y}$ there exists a unique w with w = i ⋅ v for some $i\in {\mathbb{Z}}^{n}$ provided the period cell is chosen as above),
• $u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathcal{H}}_{\varepsilon }^{N}\right)$ satisfies Neumann conditions $\frac{\mathrm{d}{u}_{e}}{\mathrm{d}{\mathbf{x}}_{e}}\left(v\right)=0$ at $v\;{\in \mathcal{U}}_{Y}$,
• $u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathcal{H}}_{\varepsilon }^{D}\right)$ satisfies Dirichlet conditions u_e (v) = 0 at $v\;{\in \mathcal{U}}_{Y}$.Above ${\mathbf{x}}_{{e}_{v}}\in \left[0,l\left({e}_{v}\right)\right]$ is a natural coordinate on e_v such that ${\mathbf{x}}_{{e}_{v}}=0$ at v; in the same way ${\mathbf{x}}_{{e}_{w}}$ is defined. The action of all operators above is given by (1.10).

Recall, that ${\theta }_{a}\in {\mathbb{T}}^{n}$ is given in (2.5).

Lemma 2.1. There exist Λ₀ > 0 and ɛ_Λ > 0 such that

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{{\Lambda}}:\quad {{\Lambda}}_{0}{\varepsilon }^{-1}{< }{\lambda }_{m+1}\left({\mathcal{H}}_{\varepsilon }^{{\theta }_{a}}\right).\end{equation} \tag{ 2.6 }$$

Proof. For $\theta \in {\mathbb{T}}^{n}$ and ɛ ⩾ 0 we introduce in L ²(Y) the form ${\mathbf{h}}_{\varepsilon }^{\theta }$,

$$\begin{align*}\hfill {\mathbf{h}}_{\varepsilon }^{\theta }\langle u,u\rangle & =\sum _{e{\in \mathcal{E}}_{Y}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}+\varepsilon \sum _{j=1}^{m}\sum _{v\in {\mathcal{V}}_{j}}{\alpha }_{j}{\left\vert {u}_{0}\left(v\right)-{\beta }_{j}{u}_{j}\left(v\right)\right\vert }^{2}+\varepsilon \gamma {\left\vert u\left(\tilde {v}\right)\right\vert }^{2},\hfill \\ \hfill \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathbf{h}}_{\varepsilon }^{\theta }\right)& =\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{\theta }\right).\hfill \end{align*}$$

We denote by ${\mathbf{H}}_{\varepsilon }^{\theta }$ the self-adjoint operator associated with this form. Obviously,

$$\begin{equation}\forall \enspace \varepsilon { >}0\quad \forall \enspace k\in \mathbb{N}:\quad {\varepsilon }^{-1}{\lambda }_{k}\left({\mathbf{H}}_{\varepsilon }^{\theta }\right)={\lambda }_{k}\left({\mathcal{H}}_{\varepsilon }^{\theta }\right).\end{equation} \tag{ 2.7 }$$

Also we observe that with respect to the space decomposition ${\mathsf{L}}^{2}\left(Y\right)={\oplus }_{j=0}^{m}{\mathsf{L}}^{2}\left({Y}_{j}\right)$ the operator ${\mathbf{H}}_{0}^{\theta }$ can be decomposed in a sum

$$\begin{equation}{\mathbf{H}}_{0}^{\theta }={\mathbf{H}}_{0,0}^{\theta }\oplus {\mathbf{H}}_{0,1}^{N}\oplus {\mathbf{H}}_{0,2}^{N}\oplus \cdots \oplus {\mathbf{H}}_{0,m}^{N},\end{equation} \tag{ 2.8 }$$

where the operators ${\mathbf{H}}_{0,0}^{\theta }$, ${\mathbf{H}}_{0,j}^{N}$ are associated with the forms ${\mathbf{h}}_{0,0}^{\theta }$, ${\mathbf{h}}_{0,j}^{N}$ defined as follows,

$$\begin{equation}{\mathbf{h}}_{0,0}^{\theta }\langle u,u\rangle =\sum _{e{\in \mathcal{E}}_{{Y}_{0}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2},\qquad \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathbf{h}}_{0,0}^{\theta }\right)=\left\{u=v{\vert }_{{Y}_{0}},\enspace v\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{\theta }\right)\right\},\end{equation} \tag{ 2.9 }$$

$$\begin{equation}{\mathbf{h}}_{0,j}^{N}\langle u,u\rangle =\sum _{e{\in \mathcal{E}}_{{Y}_{j}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2},\qquad \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathbf{h}}_{0,j}^{N}\right)=\left\{u=v{\vert }_{{Y}_{j}},\enspace v\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{\theta }\right)\right\}.\end{equation} \tag{ 2.10 }$$

It is easy to see that

$$\begin{equation}{\lambda }_{1}\left({\mathbf{H}}_{0,j}^{N}\right)=0,\quad j=1,\dots ,m,\end{equation} \tag{ 2.11 }$$

and the corresponding eigenspace consists of constant functions. Due to the connectivity of Y_j one has

$$\begin{equation}{\lambda }_{2}\left({\mathbf{H}}_{0,j}^{N}\right){ >}0,\quad j=1,\dots ,m.\end{equation} \tag{ 2.12 }$$

If ${\lambda }_{1}\left({\mathbf{H}}_{0,0}^{\theta }\right)=0$, the corresponding eigenfunction would be constant which is possible iff θ = θ_p. Thus

$$\begin{equation}{\lambda }_{1}\left({\mathbf{H}}_{0,0}^{\theta }\right){ >}0,\quad \theta \ne {\theta }_{\mathrm{p}}.\end{equation} \tag{ 2.13 }$$

It follows from (2.8), (2.11)–(2.13) that ${\lambda }_{k}\left({\mathbf{H}}_{0}^{\theta }\right)=0$ for k = 1, ..., m, while

$$\begin{equation}{\lambda }_{m+1}\left({\mathbf{H}}_{0}^{\theta }\right)=\mathrm{min}\left\{{\lambda }_{1}\left({\mathbf{H}}_{0,0}^{\theta }\right);{\lambda }_{2}\left({\mathbf{H}}_{0,1}^{N}\right);\enspace {\lambda }_{2}\left({\mathbf{H}}_{0,2}^{N}\right);\enspace \dots ;\enspace {\lambda }_{2}\left({\mathbf{H}}_{0,m}^{N}\right)\right\}{ >}0,\quad \theta \ne {\theta }_{\mathrm{p}}.\end{equation} \tag{ 2.14 }$$

Using the fact that the sequence of forms ${\mathbf{h}}_{\varepsilon }^{\theta }$ increases monotonically as ɛ decreases, and moreover ${\mathrm{lim}}_{\varepsilon \to 0}\enspace {\mathbf{h}}_{\varepsilon }^{\theta }\langle u,u\rangle ={\mathbf{h}}_{0}^{\theta }\langle u,u\rangle$, $\forall \enspace u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathbf{h}}_{\varepsilon }^{\theta }\right)=\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathbf{h}}_{0}^{\theta }\right)$ we conclude [40, theorem 4.1]:

$$\begin{equation}\forall \enspace f\in {\mathsf{L}}^{2}\left(Y\right):\enspace {\Vert}{\left({\mathbf{H}}_{\varepsilon }^{\theta }+\mathrm{I}\right)}^{-1}f-{\left({\mathbf{H}}_{0}^{\theta }+\mathrm{I}\right)}^{-1}f{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}\to 0\quad \;\text{as}\;\enspace \varepsilon \to 0.\end{equation} \tag{ 2.15 }$$

Moreover, since the sequence of resolvents ${\left({\mathbf{H}}_{\varepsilon }^{\theta }+\mathrm{I}\right)}^{-1}$ decrease monotonically as ɛ decreases, and both resolvents ${\left({\mathbf{H}}_{\varepsilon }^{\theta }+\mathrm{I}\right)}^{-1}$ and ${\left({\mathbf{H}}_{0}^{\theta }+\mathrm{I}\right)}^{-1}$ are compact one can upgrade (2.15) to the norm resolvent convergence [16, theorem 8.3.5]. As a consequence we get the convergence of spectra, namely

$$\begin{equation}\forall \enspace k\in \mathbb{N}:\enspace {\lambda }_{k}\left({\mathbf{H}}_{\varepsilon }^{\theta }\right)\to {\lambda }_{k}\left({\mathbf{H}}_{0}^{\theta }\right)\quad \;\text{as}\;\enspace \varepsilon \to 0.\end{equation} \tag{ 2.16 }$$

We set

$$\begin{equation}{{\Lambda}}_{0}{:=}\frac{{\lambda }_{m+1}\left({\mathbf{H}}_{0}^{{\theta }_{\mathrm{a}}}\right)}{2}.\end{equation} \tag{ 2.17 }$$

Since θ_a ≠ θ_p, then Λ₀ > 0. It follows from (2.16) that there exists ɛ_Λ > 0 such that

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{{\Lambda}}:\quad {{\Lambda}}_{0}{< }{\lambda }_{m+1}\left({\mathbf{H}}_{\varepsilon }^{{\theta }_{\mathrm{a}}}\right).\end{equation} \tag{ 2.18 }$$

Combining (2.7) and (2.18) we arrive at the desired estimate (2.6). The lemma is proven.□

Here we recall a result from [12] serving to compare eigenvalues of two operators acting in different Hilbert spaces. Let H and H' be separable Hilbert spaces, $\mathcal{H}$ and ${\mathcal{H}}^{\prime }$ be non-negative self-adjoint operators in these spaces, and $\mathfrak{h}$ and ${\mathfrak{h}}^{\prime }$ be the associated quadratic forms. We assume that both operators $\mathcal{H}$ and ${\mathcal{H}}^{\prime }$ have purely discrete spectra.

Lemma 2.2. [12, lemma 2.1] Suppose that ${\Phi}:\mathrm{d}\mathrm{o}\mathrm{m}\left(\mathfrak{h}\right)\to \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}^{\prime }\right)$ is a linear map such that

$$\begin{align*}\hfill {\Vert}u{{\Vert}}_{H}^{2}& {\leqslant}{\Vert}{\Phi}u{{\Vert}}_{{H}^{\prime }}^{2}+{\delta }_{1}\left({\Vert}u{{\Vert}}_{H}^{2}+\mathfrak{h}\langle u,u\rangle \right),\hfill \\ \hfill {\mathfrak{h}}^{\prime }\langle {\Phi}u,{\Phi}u\rangle & {\leqslant}\mathfrak{h}\langle u,u\rangle +{\delta }_{2}\left({\Vert}u{{\Vert}}_{H}^{2}+\mathfrak{h}\langle u,u\rangle \right)\hfill \end{align*}$$

for all $u\in \mathrm{d}\mathrm{o}\mathrm{m}\left(\mathfrak{h}\right)$. Here δ₁, δ₂ are some positive constants. Then for each $j\in \mathbb{N}$ we have

$$\begin{equation}{\lambda }_{j}\left({\mathcal{H}}^{\prime }\right){\leqslant}{\lambda }_{j}\left(\mathcal{H}\right)+\frac{{\lambda }_{j}\left(\mathcal{H}\right)\left(1+{\lambda }_{j}\left(\mathcal{H}\right)\right){\delta }_{1}+\left(1+{\lambda }_{j}\left(\mathcal{H}\right)\right){\delta }_{2}}{1-\left(1+{\lambda }_{j}\left(\mathcal{H}\right)\right){\delta }_{1}}\end{equation} \tag{ 2.19 }$$

provided the denominator $1-\left(1+{\lambda }_{k}\left(\mathcal{H}\right)\right){\delta }_{1}$ is positive.

Remark 2.2. The above result was established in [12] under the assumption that dim H = dim H' = ∞, however, it is easy to see from its proof that the result remains valid for dim H' < ∞ as well. In that case (2.19) holds for j ∈ {1, ..., dim H'}.

In this subsection we denote by bold letters (e.g., u) the elements of ${\mathbb{C}}^{m+1}$. Their entries will be enumerated starting from zero, i.e.

$$\begin{equation*}\mathbf{u}\in {\mathbb{C}}^{m+1}\enspace {\Rightarrow}\enspace \mathbf{u}=\left({u}_{0},\dots ,{u}_{m}\right)\quad \;\text{with}\;\enspace {u}_{j}\in \mathbb{C}.\end{equation*}$$

Let ${\mathbb{C}}_{l}^{m+1}$ be the same space ${\mathbb{C}}^{m+1}$ equipped with the weighted scalar product

$$\begin{equation}{\left(\mathbf{u},\mathbf{v}\right)}_{{\mathbb{C}}_{l}^{m+1}}=\sum _{j=0}^{m}{u}_{j}\overline{{v}_{j}}{l}_{j}\end{equation} \tag{ 2.20 }$$

[recall that l_j and N_j are defined by (1.11)]. Note that ${\mathbb{C}}_{l}^{m+1}$ is isomorphic to a subspace of L ²(Y) consisting of functions being constant on each Y_j , j = 0, ..., m. In ${\mathbb{C}}_{l}^{m+1}$ we introduce the form ³

$$\begin{equation}{\mathfrak{h}}_{0}^{N}\langle \mathbf{u},\mathbf{u}\rangle =\sum _{j=1}^{m}{\alpha }_{j}{N}_{j}{\left\vert {u}_{0}-{\beta }_{j}{u}_{j}\right\vert }^{2}+\gamma {\left\vert {u}_{0}\right\vert }^{2},\qquad \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{0}^{N}\right)={\mathbb{C}}_{l}^{m+1}.\end{equation} \tag{ 2.21 }$$

This form is associated with the operator ${\mathcal{H}}_{0}^{N}$ in ${\mathbb{C}}_{l}^{m+1}$ being given by the symmetric [with respect to the scalar product (2.20)] matrix

$$\begin{equation*}{\mathcal{H}}_{0}^{N}=\left(\begin{pmatrix}\hfill \gamma {l}_{0}^{-1}+\sum _{j=1}^{m}{\alpha }_{j}{N}_{j}{l}_{0}^{-1}\hfill & \hfill -{\alpha }_{1}{\beta }_{1}{N}_{1}{l}_{0}^{-1}\hfill & \hfill -{\alpha }_{2}{\beta }_{2}{N}_{2}{l}_{0}^{-1}\hfill & \hfill \dots \hfill & \hfill -{\alpha }_{m}{\beta }_{m}{N}_{m}{l}_{0}^{-1}\hfill \\ \hfill -{\alpha }_{1}{\beta }_{1}{N}_{1}{l}_{1}^{-1}\hfill & \hfill {\alpha }_{1}{\beta }_{1}^{2}{N}_{1}{l}_{1}^{-1}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill -{\alpha }_{2}{\beta }_{2}{N}_{2}{l}_{2}^{-1}\hfill & \hfill 0\hfill & \hfill {\alpha }_{2}{\beta }_{2}^{2}{N}_{2}{l}_{2}^{-1}\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill \ddots \hfill & \hfill {\vdots}\hfill \\ \hfill -{\alpha }_{m}{\beta }_{m}{N}_{m}{l}_{m}^{-1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill {\alpha }_{m}{\beta }_{m}^{2}{N}_{m}{l}_{m}^{-1}\hfill \end{pmatrix}\right).\end{equation*}$$

We denote by ${\lambda }_{1}\left({\mathcal{H}}_{0}^{N}\right){\leqslant}{\lambda }_{2}\left({\mathcal{H}}_{0}^{N}\right){\leqslant}\dots {\leqslant}{\lambda }_{m+1}\left({\mathcal{H}}_{0}^{N}\right)$ its eigenvalues. It turns out that

$$\begin{equation}{\lambda }_{j}\left({\mathcal{H}}_{0}^{N}\right)={B}_{j-1},\quad j=1,\dots ,m+1.\end{equation} \tag{ 2.22 }$$

Indeed, let λ be the eigenvalue of ${\mathcal{H}}_{0}^{N}$ such that λ ∉ {A₁, A₂, ..., A_m }, and let 0 ≠ u = (u₀, ..., u_m ) be the corresponding eigenfunction. The equation ${\mathcal{H}}_{0}^{N}\mathbf{u}=\lambda \mathbf{u}$ is a linear algebraic system for u₀, ..., u_m . From the last m equations of this system we infer

$$\begin{equation}{u}_{j}=\frac{{\alpha }_{j}{\beta }_{j}{N}_{j}{l}_{j}^{-1}}{{\alpha }_{j}{\beta }_{j}^{2}{N}_{j}{l}_{j}^{-1}-\lambda }{u}_{0},\quad j=1,\dots ,m.\end{equation} \tag{ 2.23 }$$

Note, that the denominator in (2.23) is non-zero since $\lambda \ne {A}_{j}={\alpha }_{j}{\beta }_{j}^{2}{N}_{j}{l}_{j}^{-1}$. Inserting (2.23) into the first equation of the system we arrive at

$$\begin{equation*}{u}_{0}\left[\lambda \left({l}_{0}+\sum _{j=1}^{m}\frac{{A}_{j}{l}_{j}}{{\beta }_{j}^{2}\left({A}_{j}-\lambda \right)}\right)-\gamma \right]=0.\end{equation*}$$

Moreover, u₀ ≠ 0 [otherwise, due to (2.23), u would vanish]. Hence λ is a root of equation (1.14). Evidently, the converse assertion also holds, that is

$$\begin{equation}\lambda \in \sigma \left({\mathcal{H}}_{0}^{N}\right){\backslash}\left\{{A}_{1},{A}_{2},\dots ,{A}_{m}\right\}\enspace \Longleftrightarrow\enspace \lambda \quad \;\text{is}\;\text{a}\;\text{root}\;\text{of}\;\left(1.14\right).\end{equation} \tag{ 2.24 }$$

Then (2.22) follows immediately from (1.15) and (2.24).

Lemma 2.3. There exist constants C_B > 0 and ɛ_B > 0 such that

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{B}:\quad {B}_{j-1}{\leqslant}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)+{C}_{B}{\varepsilon }^{1/2},\quad j=1,\dots ,m+1.\end{equation} \tag{ 2.25 }$$

Proof. W.l.o.g. we may assume that α_j and γ are non-negative. Evidently, under this assumption the operators ${\mathcal{H}}_{\varepsilon }^{N}$ are non-negative. Consequently, the operator ${\mathcal{H}}_{0}^{N}$ is also non-negative, see footnote ³ . Thus we are in the framework of lemma 2.2. In the general case we have to consider the shifted operators ${\mathcal{H}}_{\varepsilon }^{N}-\mu \mathrm{I}$ and ${\mathcal{H}}_{0}^{N}-\mu \mathrm{I}$, where μ is the smallest eigenvalues of ${\mathcal{H}}_{\varepsilon }^{N}{\vert }_{\varepsilon =1}$ (this eigenvalue could be indeed negative if one of the numbers α_j and γ is negative). The operator ${\mathcal{H}}_{\varepsilon }^{N}-\mu \mathrm{I}$ is non-negative for each ɛ ∈ (0, 1] due to the fact that the sequence of forms ${\mathfrak{h}}_{\varepsilon }^{N}$ increases monotonically as ɛ decreases; the non-negativity of ${\mathcal{H}}_{0}^{N}-\mu \mathrm{I}$ is again due to footnote ³ .

We introduce the operator ${\Phi}:\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{N}\right)\to {\mathbb{C}}_{l}^{m+1}$ by

$$\begin{equation}{\left({\Phi}u\right)}_{j}={l}_{j}^{-1}\sum _{e{\in \mathcal{E}}_{{Y}_{j}}}{\int }_{0}^{l\left(e\right)}{u}_{e}\left(x\right)\mathrm{d}x,\quad j=0,\dots ,m.\end{equation} \tag{ 2.26 }$$

Our goal is to show that the following estimates hold for each $u\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{N}\right)$:

$$\begin{equation}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}{\leqslant}{\Vert}{\Phi}u{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}}^{2}+{C}_{1}\varepsilon \left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right),\end{equation} \tag{ 2.27 }$$

$$\begin{equation}{\mathfrak{h}}_{0}^{N}\langle {\Phi}u,{\Phi}u\rangle {\leqslant}{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle +{C}_{2}{\varepsilon }^{1/2}\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right).\end{equation} \tag{ 2.28 }$$

with some C₁, C₂ > 0. By lemma 2.2 (see also remark 2.2 after it) we infer from (2.27) and (2.28) that

$$\begin{equation}{B}_{j-1}={\lambda }_{j}\left({\mathcal{H}}_{0}^{N}\right){\leqslant}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)+\frac{{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)\left(1+{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)\right){C}_{1}\varepsilon +\left(1+{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)\right){C}_{2}{\varepsilon }^{1/2}}{1-\left(1+{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)\right){C}_{1}\varepsilon }\end{equation} \tag{ 2.29 }$$

provided $\left(1+{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)\right){C}_{1}\varepsilon {< }1$. Set

$$\begin{equation*}{\varepsilon }_{B}{:=}\mathrm{min}\left\{\frac{1}{2{C}_{1}\left(1+{B}_{m}\right)};\enspace 1\right\}.\end{equation*}$$

Since $0{\leqslant}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right){\leqslant}{B}_{j-1}$ [the last estimate follows from (2.4) and lemma 2.4 below] and B_j−1 ⩽ B_m for j = 1, ..., m + 1, the denominator in (2.29) is larger than 1/2 for ɛ < ɛ_B . Moreover, since ɛ_B ⩽ 1, one has ɛ < ɛ^1/2 for ɛ < ɛ_B . Taking all above into account we deduce from (2.29):

$$\begin{equation*}\forall \enspace \varepsilon {< }{\varepsilon }_{B}:\quad {B}_{j-1}{\leqslant}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right)+2\left({B}_{m}\left(1+{B}_{m}\right){C}_{1}+\left(1+{B}_{m}\right){C}_{2}\right){\varepsilon }^{1/2}.\end{equation*}$$

Thus estimate (2.25) holds for ɛ < ɛ_B with ${C}_{B}=2\left({B}_{m}\left(1+{B}_{m}\right){C}_{1}+\left(1+{B}_{m}\right){C}_{2}\right)$.

To prove (2.27) we need a Poincaré-type inequality on each Y_j . Namely, let the form ${\mathbf{h}}_{0,j}^{N}$ be defined by (2.10), j = 1, ..., m; in the same way we define ${\mathbf{h}}_{0,j}^{N}$ for j = 0. By ${\mathbf{H}}_{0,j}^{N}$ we denote the associated operators in L ²(Y_j ), j = 0, ..., m. One has ${\lambda }_{1}\left({\mathbf{H}}_{0,j}^{N}\right)=0$ (the corresponding eigenspace consists of constants), while ${\lambda }_{2}\left({\mathbf{H}}_{0,j}^{N}\right){ >}0$. By the max–min principle [38] ${\lambda }_{2}\left({\mathbf{H}}_{0,j}^{N}\right){\leqslant}{\mathbf{h}}_{0,j}^{N}\langle v,v\rangle /{\Vert}v{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}$ for each $v\in \mathrm{d}\mathrm{o}\mathrm{m}\left({\mathbf{h}}_{0,j}^{N}\right)$ such that ${\left(v,1\right)}_{{\mathsf{L}}^{2}\left({Y}_{j}\right)}=0$. Using the above estimate for v := u − (Φu)_j we get

$$\begin{equation}\forall \enspace u\in {\mathbf{h}}_{0,j}^{N}:\quad {\Vert}u-{\left({\Phi}u\right)}_{j}{{\Vert}}_{{\mathsf{L}}^{2}\left({Y}_{j}\right)}^{2}{\leqslant}{C}_{1}\sum _{e{\in \mathcal{E}}_{{Y}_{j}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2},\quad j=0,\dots ,m,\end{equation} \tag{ 2.30 }$$

where ${C}_{1}={\left({\lambda }_{2}\left({\mathbf{H}}_{0,j}\right)\right)}^{-1}$. Using (2.30) we obtain

$$\begin{align*}\hfill {\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}& =\sum _{j=0}^{m}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left({Y}_{j}\right)}^{2}=\sum _{j=0}^{m}\left(\vert {\left({\Phi}u\right)}_{j}{\vert }^{2}{l}_{j}+{\Vert}u-{\left({\Phi}u\right)}_{j}{{\Vert}}_{{\mathsf{L}}^{2}\left({Y}_{j}\right)}^{2}\right)\hfill \\ \hfill & {\leqslant}{\Vert}{\Phi}u{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}}^{2}+{C}_{1}\sum _{e{\in \mathcal{E}}_{Y}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}{\leqslant}{\Vert}{\Phi}u{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}}^{2}+{C}_{1}\varepsilon {\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \hfill \\ \hfill & {\leqslant}{\Vert}{\Phi}u{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}}^{2}+{C}_{1}\varepsilon \left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right)\hfill \end{align*}$$

(on the penultimate step we use the fact that α_j and γ are non-negative). Inequality (2.27) is checked.

Now let us prove the estimate (2.28). One has:

$$\begin{align}\hfill {\mathfrak{h}}_{0}^{N}\langle {\Phi}u,{\Phi}u\rangle & ={\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle -{\varepsilon }^{-1}\sum _{e{\in \mathcal{E}}_{Y}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}\hfill \\ \hfill & \quad +\enspace \underset{{=:}{R}_{\varepsilon }}{\underbrace{\sum _{j=1}^{m}\sum _{v\in {\mathcal{V}}_{j}}{\alpha }_{j}\left[{\left\vert {\left({\Phi}u\right)}_{0}-{\beta }_{j}{\left({\Phi}u\right)}_{j}\right\vert }^{2}-{\left\vert {u}_{0}\left(v\right)-{\beta }_{j}{u}_{j}\left(v\right)\right\vert }^{2}\right]+\gamma \left[{\left\vert {\left({\Phi}u\right)}_{0}\right\vert }^{2}-{\left\vert u\left(\tilde {v}\right)\right\vert }^{2}\right]}}\hfill \\ \hfill & {\leqslant}{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle +{R}_{\varepsilon }.\hfill \end{align} \tag{ 2.31 }$$

We estimate the remainder R_ɛ as follows (below we use the estimate $\vert a{\vert }^{2}-\vert b{\vert }^{2}{\leqslant}\vert a-b\vert \left(\vert a\vert +\vert b\vert \right)$):

$$\begin{align}\hfill \vert {R}_{\varepsilon }\vert & {\leqslant}\sum _{j=1}^{m}\sum _{v\in {\mathcal{V}}_{j}}{\alpha }_{j}\left(\vert {\left({\Phi}u\right)}_{0}-{u}_{0}\left(v\right)\vert +\vert {\beta }_{j}\vert \cdot \vert {\left({\Phi}u\right)}_{j}-{u}_{j}\left(v\right)\vert \right)\hfill \\ \hfill & \quad {\times}\left(\vert {\left({\Phi}u\right)}_{0}\vert +\vert {u}_{0}\left(v\right)\vert +\vert {\beta }_{j}\vert \cdot \vert {\left({\Phi}u\right)}_{j}\vert +\vert {\beta }_{j}\vert \cdot \vert {u}_{j}\left(v\right)\vert \right)\hfill \\ \hfill & \quad +\enspace \gamma \enspace \vert {\left({\Phi}u\right)}_{0}-u\left(\tilde {v}\right)\vert \cdot \left(\vert {\left({\Phi}u\right)}_{0}\vert +\vert u\left(\tilde {v}\right)\vert \right).\hfill \end{align} \tag{ 2.32 }$$

To proceed further we need a standard trace estimate

$$\begin{equation}{\Vert}w{{\Vert}}_{{\mathsf{L}}^{\infty }\left({Y}_{j}\right)}{\leqslant}\tilde {C}{\Vert}w{{\Vert}}_{{\mathsf{H}}^{1}\left({Y}_{j}\right)},\quad w\in {\mathsf{H}}^{1}\left({Y}_{j}\right),\quad j=0,\dots ,m,\end{equation} \tag{ 2.33 }$$

where $\tilde {C}{ >}0$ depends on Y. Applying it for $w{:=}u{\upharpoonright }_{{Y}_{j}}-{\left({\Phi}u\right)}_{j}$ and then using (2.30) we obtain

$$\begin{align}\hfill {\Vert}u-{\left({\Phi}u\right)}_{j}{{\Vert}}_{{\mathsf{L}}^{\infty }\left({Y}_{j}\right)}& {\leqslant}\tilde {C}{\left({\Vert}u-{\left({\Phi}u\right)}_{j}{{\Vert}}_{{\mathsf{L}}^{2}\left({Y}_{j}\right)}+\sum _{e{\in \mathcal{E}}_{{Y}_{j}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}\right)}^{1/2}\hfill \\ \hfill & {\leqslant}\tilde {C}\left({C}_{1}+1\right){\left(\sum _{e{\in \mathcal{E}}_{{Y}_{j}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2}\right)}^{1/2}\hfill \\ \hfill & {\leqslant}\tilde {C}\left({C}_{1}+1\right){\varepsilon }^{1/2}\enspace {\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right)}^{1/2},\enspace j=0,\dots ,m.\hfill \end{align} \tag{ 2.34 }$$

Also, using the Cauchy–Schwarz inequality and (2.33) and taking into account that ɛ ⩽ 1, one gets

$$\begin{equation}\vert {\left({\Phi}u\right)}_{j}\vert {\leqslant}{l}_{j}^{-1/2}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left({Y}_{j}\right)}{\leqslant}\underset{j}{\mathrm{max}}{l}_{j}^{-1/2}{\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right)}^{1/2},\quad j=0,\dots ,m,\end{equation} \tag{ 2.35 }$$

$$\begin{equation}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{\infty }\left({Y}_{j}\right)}{\leqslant}\tilde {C}{\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right)}^{1/2},\quad j=0,\dots ,m.\end{equation} \tag{ 2.36 }$$

Combining (2.32), (2.34)–(2.36) we arrive at the estimate

$$\begin{equation}\vert {R}_{\varepsilon }\vert {\leqslant}{C}_{2}{\varepsilon }^{1/2}\left({\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}+{\mathfrak{h}}_{\varepsilon }^{N}\langle u,u\rangle \right)\end{equation} \tag{ 2.37 }$$

with some constant C₂ depending on α_j , β_j , γ, Y. The required estimate (2.28) follows from (2.31), (2.37); this ends the proof of lemma 2.3. □

Lemma 2.4. One has:

$$\begin{equation*}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\right){\leqslant}{B}_{j-1},\quad j=1,\dots ,m+1.\end{equation*}$$

Proof. By the min–max principle [7, section 4.5] we have

$$\begin{equation}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\right)=\underset{V\in {\mathfrak{H}}^{j}}{\mathrm{min}}\underset{u\in V{\backslash}\left\{0\right\}}{\mathrm{max}}\frac{{\mathfrak{h}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\langle u,u\rangle }{{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}},\end{equation} \tag{ 2.38 }$$

where ${\mathfrak{H}}^{j}$ is a set of all j-dimensional subspaces in $\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\right)$.

We introduce the operator ${\Psi}:{\mathbb{C}}_{l}^{m+1}\to {\mathsf{L}}^{2}\left(Y\right)$ by

$$\begin{equation}{\Psi}\mathbf{u}=\sum _{j=0}^{m}{u}_{j}{\chi }_{{Y}_{j}},\quad {\chi }_{{Y}_{j}}\;\text{is}\;\text{the}\;\text{indicator}\;\text{function}\;\text{of}\;{Y}_{j}.\end{equation} \tag{ 2.39 }$$

It is easy to see that the image of Ψ is contained in $\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\right)$, and

$$\begin{equation}{\Vert}{\Psi}\mathbf{u}{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}={\Vert}\mathbf{u}{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}},\qquad {\mathfrak{h}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\langle {\Psi}\mathbf{u},{\Psi}\mathbf{u}\rangle ={\mathfrak{h}}_{0}^{N}\langle \mathbf{u},\mathbf{u}\rangle \end{equation} \tag{ 2.40 }$$

[recall, that the form ${\mathfrak{h}}_{0}^{N}$ is given by (2.21), by ${\mathcal{H}}_{0}^{N}$ we denote the associated operator].

Let {e¹, e², ..., e^m+1} be an orthonormal system of eigenvectors of ${\mathcal{H}}_{0}^{N}$ such that ${\mathcal{H}}_{0}^{N}{\mathbf{e}}^{j}={B}_{j-1}{\mathbf{e}}^{j}$ [see (2.22)]. For j = 1, ..., m + 1 we set W^j := span(e¹, ..., e^j ). It is easy to see that

$$\begin{equation}\underset{\mathbf{u}\in {\mathbf{W}}^{j}{\backslash}\left\{0\right\}}{\mathrm{max}}\frac{{\mathfrak{h}}_{0}^{N}\langle \mathbf{u},\mathbf{u}\rangle }{{\Vert}\mathbf{u}{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}}^{2}}={B}_{j-1}.\end{equation} \tag{ 2.41 }$$

Finally, we set V^j := ΨW^j , obviously ${V}^{j}\in {\mathfrak{H}}^{j}$. Then using (2.38)–(2.41) we obtain:

$$\begin{equation*}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{N}\right){\leqslant}\underset{u\in {V}^{j}{\backslash}\left\{0\right\}}{\mathrm{max}}\frac{{\mathfrak{h}}_{\varepsilon }^{{\theta }_{\mathrm{p}}}\langle u,u\rangle }{{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left(Y\right)}^{2}}=\underset{\mathbf{u}\in {\mathbf{W}}^{j}{\backslash}\left\{0\right\}}{\mathrm{max}}\frac{{\mathfrak{h}}_{0}^{N}\langle \mathbf{u},\mathbf{u}\rangle }{{\Vert}\mathbf{u}{{\Vert}}_{{\mathbb{C}}_{l}^{m+1}}^{2}}={B}_{j-1}.\end{equation*}$$

The lemma is proven. □

Let ${\mathbb{C}}_{l}^{m}$ be the subspace of ${\mathbb{C}}^{m+1}$ consisting of vectors of the form u = (0, u₁, ..., u_m ) with ${u}_{j}\in \mathbb{C}$ with the scalar product generated by (2.20), i.e.

$$\begin{equation*}{\left(\mathbf{u},\mathbf{v}\right)}_{{\mathbb{C}}_{l}^{m}}=\sum _{j=1}^{m}{u}_{j}\overline{{v}_{j}}{l}_{j}.\end{equation*}$$

In this space we introduce the quadratic form

$$\begin{equation*}{\mathfrak{h}}_{0}^{{\theta }_{\mathrm{a}}}\langle \mathbf{u},\mathbf{u}\rangle =\sum _{j=1}^{m}{\alpha }_{j}{\beta }_{j}^{2}{N}_{j}{\left\vert {u}_{j}\right\vert }^{2}.\end{equation*}$$

It is easy to see that ${\mathfrak{h}}_{0}^{{\theta }_{\mathrm{a}}}={\mathfrak{h}}_{0}^{N}{\upharpoonright }_{{\mathbb{C}}_{l}^{m}}$. The operator associated with this form is given by the matrix

$$\begin{equation*}{\mathcal{H}}_{0}^{{\theta }_{\mathrm{a}}}=\mathrm{diag}\left({\alpha }_{1}{\beta }_{1}^{2}{N}_{1}{l}_{1}^{-1},\enspace {\alpha }_{2}{\beta }_{2}^{2}{N}_{2}{l}_{2}^{-1},\enspace \dots ,\enspace {\alpha }_{m}{\beta }_{m}^{2}{N}_{m}{l}_{m}^{-1}\right).\end{equation*}$$

Evidently, the eigenvalues of this matrix are the numbers A₁ < A₂ < ... < A_m .

Lemma 2.5. There exist such constants C_A > 0 and ɛ_A > 0 that

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{A}:\quad {A}_{j}{\leqslant}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{{\theta }_{\mathrm{a}}}\right)+{C}_{A}{\varepsilon }^{1/2},\quad j=1,\dots ,m.\end{equation} \tag{ 2.42 }$$

Proof. The proof is similar to the proof of lemma 2.3. There is only one essential difference: instead of the operator Φ (2.26) one should use the operator ${{\Phi}}_{0}:\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{{\theta }_{\mathrm{a}}}\right)\to {\mathbb{C}}_{l}^{m}$ defined by

$$\begin{equation*}{\left({{\Phi}}_{0}u\right)}_{0}=0,\quad {\left({{\Phi}}_{0}u\right)}_{j}={\left({\Phi}u\right)}_{j},\quad j=1,\dots ,m.\end{equation*}$$

and, as a consequence, instead of the Poincaré inequality (2.30) on Y₀ one should use the inequality

$$\begin{equation*}{\Vert}u{{\Vert}}_{{\mathsf{L}}^{2}\left({Y}_{0}\right)}^{2}{\leqslant}{C}_{1}\sum _{e{\in \mathcal{E}}_{{Y}_{0}}}{\Vert}{u}_{e}^{\prime }{{\Vert}}_{{\mathsf{L}}^{2}\left(0,l\left(e\right)\right)}^{2},\end{equation*}$$

where ${C}_{1}={\left({\lambda }_{1}\left({\mathbf{H}}_{0,0}^{{\theta }_{\mathrm{a}}}\right)\right)}^{-1}$ [recall, that the operator ${\mathbf{H}}_{0,0}^{\theta }$ is introduced in the proof of lemma 2.1, and its first eigenvalue is non-zero provided θ ≠ θ_p).□

Lemma 2.6. One has:

$$\begin{equation*}{\lambda }_{j}\left({\mathcal{H}}_{\varepsilon }^{D}\right){\leqslant}{A}_{j},\quad j=1,\dots ,m.\end{equation*}$$

Proof. The proof is similar to the proof of lemma 2.4. Namely, one has to replace everywhere in the proof of lemma 2.4 the superscript θ_p by D, the superscript N by θ_a, B_j−1 by A_j , and to use instead of the mapping Ψ (2.39) its restriction to ${\mathbb{C}}_{l}^{m}$ (the image of this restriction is contained in $\mathrm{d}\mathrm{o}\mathrm{m}\left({\mathfrak{h}}_{\varepsilon }^{D}\right)$). □

It follows from (2.2), (2.4) and lemmata 2.3 and 2.4 that

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{B}:\quad {B}_{j-1}-{C}_{B}{\varepsilon }^{1/2}{\leqslant}\mathrm{inf}\left({L}_{j,\varepsilon }\right){\leqslant}{B}_{j-1},\quad j=1,\dots ,m+1.\end{equation} \tag{ 2.43 }$$

Similarly, using (2.2), (2.4) and lemmata 2.5 and 2.6 we get

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{A}:\quad {A}_{j}-{C}_{A}{\varepsilon }^{1/2}{\leqslant}\mathrm{sup}\left({L}_{j,\varepsilon }\right){\leqslant}{A}_{j},\quad j=1,\dots ,m\end{equation} \tag{ 2.44 }$$

Finally, we infer from (2.2) and lemma 2.1 that

$$\begin{equation}\forall \enspace \varepsilon {< }{\varepsilon }_{{\Lambda}}:\quad {{\Lambda}}_{0}{\varepsilon }^{-1}{< }\mathrm{sup}\left({L}_{m+1,\varepsilon }\right).\end{equation} \tag{ 2.45 }$$

Set

$$\begin{equation}{A}_{j,\varepsilon }{:=}\mathrm{sup}\left({L}_{j,\varepsilon }\right),\quad j=1,\dots ,m,\quad {B}_{j,\varepsilon }{:=}\mathrm{inf}\left({L}_{j+1,\varepsilon }\right),\quad j=0,\dots ,m.\end{equation} \tag{ 2.46 }$$

Combining (1.15), (2.43)–(2.46) we conclude that there exists such ɛ₀ > 0 that properties (1.16)–(1.18) hold for ɛ < ɛ₀, Λ₀ being defined by (2.17), C_A being defined in lemma 2.5, C_B being defined in lemma 2.3. Evidently, Λ₀ depends only on Y, while ɛ₀, C_A , C_B depend also on α_j , β_j , γ. theorem 1.1 is proven.

Remark 2.3. The proof of theorem 1.1 relies, in particular, on some properties of the eigenvalues of the operator ${\mathcal{H}}_{\varepsilon }^{{\theta }_{\mathrm{a}}}$—see the estimates (2.6), (2.42). In fact, the only specific property of θ_a we use is that θ_a ≠ θ_p. Thus, instead of ${\mathcal{H}}_{\varepsilon }^{{\theta }_{\mathrm{a}}}$ one can utilize any other ${\mathcal{H}}_{\varepsilon }^{\theta }$ with θ ≠ θ_p—the above estimates are still valid for its eigenvalues (but, of course, with another constants Λ, ɛ_Λ, C_A , ɛ_A ).