Periodic quantum graphs with predefined spectral gaps

Let Γ be an arbitrary Z n -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian H ε on Γ with the action −ɛ −1d2/dx 2 on its edges; here ɛ > 0 is a small parameter. Let m ∈ N . We show that under a proper choice of vertex conditions the spectrum σ ( H ε ) of H ε has at least m gaps as ɛ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of σ ( H ε ) as ɛ → 0 can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) ɛ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.


INTRODUCTION
Traditionally the name quantum graph refers to a pair (Γ, H ), where Γ is a network-shaped structure of vertices connected by edges of certain positive lengths (metric graph) and H is a second order self-adjoint differential operator on Γ (Hamiltonian). Hamiltonians are determined by differential operations on the edges and certain interface conditions at the vertices. We refer to the monograph [5] for a broad overview and an extensive bibliography on this topic.
Quantum graphs arise naturally in mathematics, physics, chemistry and engineering as simplified models of wave propagation in quasi-one-dimensional systems looking like narrow neighborhoods of graphs. Typical applications include quantum wires [23,24], photonic crystals [29,30], graphene and carbon nanostructures [21,31], quantum chaos [25,26] and many other areas. For more details concerning origins of quantum graphs see [27] and [5,Chapter 7].
In various applications (for example, to aforementioned graphene and carbon nano-structures, and photonic crystals) periodic infinite graphs are studied. In what follows in order to simplify the presentation (but without any loss of generality) we assume that our graphs are embedded into R d for some d ∈ N. An infinite metric graph Γ ⊂ R d is said to be Z n -periodic (n ≤ d) if it invariant under translations through some linearly independent vectors ν 1 , . . . , ν n ∈ R d . The Hamiltonian H on a Z n -periodic metric graph Γ is said to be periodic if it commutes with these translations.
It is well-known that the spectrum of a periodic Hamiltonian on a periodic metric graph can be represented as a locally finite union of compact intervals (spectral bands). The bounded open interval is called a gap if it has an empty intersection with the spectrum, but its ends belong to it. The band structure of the spectrum suggests that gaps may exist in principle. In general, however, the presence of gaps is not guaranteed: two spectral bands may overlap, and then the corresponding gap disappears. For instance, if Γ is a rectangular lattice, H is defined by the operation −d 2 /dx 2 on the edges and the standard Kirchhoff conditions at the vertices, then σ (H ) has no gaps -it coincides with [0, ∞).
Existence and locations of spectral gaps are of primary interest because of various applications, for example in physics of photonic crystals -periodic nanostructures, whose characteristic property is that the light waves at certain optical frequencies fail to propagate in them, which is caused by gaps in the spectrum of the Maxwell operator or related scalar operators. For more details we refer to [29,30], where periodic high contrast photonic and acoustic media are studied in high contrast regimes leading to appearance of Dirichlet-to-Neumann type operators on periodic graphs.
To create spectral gaps one can use geometrical means. For example, given a fixed graph we "decorate" it changing its geometrical structure at each vertex: either one attaches to each vertex a copy of certain fixed compact graph [28] (see also [39] where similar idea was used for discrete graphs) or in each vertex one disconnects the edges emerging from it and then connects their loose endpoints by a certain additional graph ("spider") [8,37].
Another way to open spectral gaps is to use "advanced" vertex conditions. For example, as we already noted the spectrum of the Kirchhoff Laplacian on a rectangular lattice has no gaps, however (see [9]) if we replace Kirchhoff conditions by the so-called δ -conditions of the strength α = 0 one immediately gets infinitely many gaps provided the lattice-spacing ratio is a rational number.
When designing materials with prescribed properties it is desirable not only to open up spectral gaps, but also be able to control their location and length -via a suitable choice of operator coefficients or/and geometry of the medium. We addressed this problem for various classes of periodic operators in a series of papers [4,11,[17][18][19]. In particular, periodic quantum graphs were treated in [4]. In this paper the required structure for the spectrum is achieved via the combination of two approaches described above: taking a fixed periodic graph Γ 0 we decorate it attaching to each period cell m compact graphs Y i j ; here j = 1, . . . , m, while the subscript i ∈ Z n indicates to which period cell we attach Y i j (see Figure 1, here m = 2). On Γ we considered the Hamiltonian H ε defined by the operation −ε −1 d 2 /dx 2 on the edges and the Kirchhoff conditions in all its vertices except the points of attachment of Y i j to Γ 0 -in these points we pose (a kind of) δ -conditions 1 . Note, that the vertex conditions we dealt with in [4] "generate" only Hamiltonians with inf(σ (H ε )) = 0. It was proven that σ (H ε ) has at least m gaps for small enough ε, these gaps converge (as ε → 0) to some intervals (A j , B j ) ⊂ [0, ∞) whose location and lengths can be nicely controlled by a suitable choice of coupling constants standing in those δ -conditions and a suitable choice the "sizes" of attached graphs Y i j .
Example of a periodic graph utilized in [4] In the current paper we continue the research started in [4]. We will prove that the required structure of the spectrum can be achieved solely by an appropriate choice of vertex conditions without any assumptions on the graph geometry. Namely, let Γ be a Z n -periodic metric graph. The only assumption we impose on it is that Γ does not coincide with a line. On Γ we consider the Hamiltonian H ε defined by the operation −ε −1 d 2 /dx 2 on edges and either Kirchhoff, δ or δ -type (different from those treated in [4]) conditions at vertices -see (1.7)-(1.9). We prove that σ (H ε ) has at least m gaps; when ε → 0 the first m gaps (respectively, the infimum of σ (H ε )) converge to some intervals (A j , B j ) ⊂ R, j = 1, . . . , m (respectively, to some number B 0 ∈ R); the location of A j , j = 1, . . . , m and B j , j = 0, . . . , m depends in explicit way from couplings constants standing in δ and δ -type vertex 1 For the definition of δ and δ -conditions in the graph context see, e.g., [9].
conditions; see Theorem 1.1. Moreover, choosing these coupling constants in a proper way one can completely control A j and B j making them coincident with predefined numbers; see Theorem 3.1. Note, that in contrast to [4], the limiting intervals and the bottom of the spectrum do not necessary lie on the positive semi-axis. Finally we show that for fixed (small enough) ε one can guarantee the precise coincidence of the left endpoints of the first m gaps with prescribed numbers; see Theorem 3.2.
The method we use to prove the convergence of spectra is different from the one used in [4], where we utilized Simon's result [40] about monotonic sequences of forms. In the current work we apply the abstract lemma from [12] serving to compare eigenvalues of two self-adjoint operators acting in different Hilbert spaces. The advantage of this approach is that we are able not only to prove the convergence of spectra, but also to estimate the rate of convergence.
The structure of the paper is as follows. In Section 1 we introduce the Hamiltonian H ε and formulate the main convergence result. Its proof is given in Section 2. In Section 3 we demonstrate how to control the location of spectral gaps.

SETTING OF THE PROBLEM AND MAIN RESULT
1.1. Metric graph Γ. Let n ∈ N and let Γ be an arbitrary connected 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 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 E Γ and V Γ 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 ∈ E Γ . In a natural way we introduce on each edge e ∈ E Γ 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 ∈ V Γ we denote by E (v) the set of edges emanating from v.
The Z n -periodicity of Γ means that Γ + ν k = Γ, k = 1, . . . , n for some linearly independent vectors ν 1 , . . . , ν n ∈ R d . Let us introduce for i = (i 1 , . . . , i n ) ∈ Z n the mapping i· : Γ → Γ defined by We denote by Y a fundamental domain of Γ, i.e. a compact set (see the assumption above) satisfying i∈Z n i ·Y = Γ, the sets Y and i ·Y may have only finitely many common points as i = 0.
Evidently a fundamental domain in not uniquely defined. Note that for any r = (r 1 , . . . , r n ) ∈ N n 0 the graph Γ is also invariant invariant under translations through vectors ν r 1 , . . . , ν r n defined by ν r k = r k ν k The corresponding fundamental domain is the set Y r given by (1.2) Finally, we denote by U Y the set of points of a fundamental domain Y that simultaneously belong to "neighboring" fundamental domains, i.e.
An example of a Z 2 -periodic graph is presented on Figure 2(a). This is an equilateral hexagonal lattice in R 2 , which is invariant under translations through vectors ν 1 = ( √ 3, 0), ν 2 = (− √ 3 2 , 3 2 ). 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 this figures the bold dots are vertices belonging to U Y and U Y r , respectively.

Decomposition of a fundamental domain.
It is easy to see that for any m ∈ N there exists such r = (r 1 , . . . , r n ) ∈ N n 0 that the fundamental domain Y r (1.2) can be represented as a union of non-empty compact sets Y j , j = 0, . . . , m satisfying the following conditions: (iv) the sets V j := Y j ∩Y 0 are non-empty and consist of vertices, j = 1, . . . , m, (v) Y 0 has a vertex v belonging neither to U Y nor to ∪ m j=1 Y j .

(1.4)
It is easy to see that such a decomposition is always possible for large enough r 1 , r 2 , . . . , 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)-(1.4) with m = 3 holds: Y 0 consists of bold solid lines, while Y 1 ,Y 2 ,Y 3 consist of one dashed edge, the black square is v. Now, let m ∈ N be given and let us fix such r = (r 1 , . . . , r n ) ∈ N n 0 that the fundamental domain Y r admits representation (1.3)-(1.4). We set for i ∈ Z n : The vertices belonging to V i j will support δ -type conditions, the vertices v i will support δ -conditions, in the remaining vertices the Kirchhoff conditions will be posed. 2 In order to achieve decomposition (1.3)-(1.4) we require our initial assumption on Γ that it does not coincide with a line. If Γ is a line, its fundamental domain Y r would be a compact interval; one can decompose it in such a way that properties (ii)-(v) hold, but then the set Y 0 will be always disconnected.

Functional spaces.
In what follows if u : Γ → C and e ∈ E Γ 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 : Γ → C such that u e ∈ L 2 (0, l(e)) for each edge e and The space H k (Γ), k ∈ N consists of functions u : Γ → C such that u e belongs to the Sobolev space H k (0, l(e)) for each edge e and 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 on 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 ∈ H 1 (Γ), this follows easily from the trace inequality [5, Lemma 1.3.8] , ∀u ∈ H 1 (0, l) and periodicity of Γ. It is also straightforward to verify that the form h ε is densely defined in L 2 (Γ), lower semibounded and closed. By the first representation theorem [16, Theorem VI.2.1] there exists the unique self-adjoint operator H ε associated with the form h ε , i.e.
where h ε u, w is the sesquilinear form, which corresponds to the quadratic form (1.5).
Condition (1.7) is usually referred as Kirchhoff coupling, condition (1.8) is known as δ -coupling of the strength γε. We may refer to conditions (1.9) as δ -type coupling. The reason for this is as follows. Suppose that v ∈ V i j has only two outgoing edges e ∈ E (v) ∩Y i0 and e ∈ E (v) ∩Y i j . Also let β j = 1. Then conditions (1.9) are equivalent to Taking into account the definition of coordinates x e and x e we conclude that (1.9) coincides with the usual δ -conditions of the strength (α j ε) −1 at a point on the line [2, Section I.4].
1.5. Main results. We denote Then for j = 1, . . . , m we set We assume that A j are pairwise distinct; in this case we can renumber them in such a way that Finally, we consider the following equation (for unknown λ ∈ C \ {A 1 , . . . , A m }) : (1.14) 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 We are now in position to formulate the first main result of this work.
Theorem 1.1. There exist such positive constants Λ 0 (depending on Y ) and C A , C B , ε 0 (depending on α j , β j , γ and Y ) that the spectrum of H ε has the following structure within (−∞, Λ 0 ε −1 ] : where the numbers A j,ε , j = 1, . . . , m and B j,ε , j = 0, . . . , m satisfy In the general case one should simply change the notations accordingly. In the following if H is a self-adjoint lower semi-bounded operator with purely discrete spectrum, we denote by {λ k (H )} k∈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 H ε and the spectra of certain operators H θ ε in L 2 (Y ). Namely, let We denote by H 1,θ h (Γ) the set of such functions u : Γ → C that u e ∈ H 1 (0, l(e)) for each e ∈ E Γ , u satisfy the same conditions at vertices of Γ as functions from H 1 h (Γ), and Hereinafter by E Y and V Y we denote the set of edges and vertices of Y , respectively; similar notations will be used for Y j . The form h θ ε is densely defined in L 2 (Y ), lower semibounded and closed. We denote by H θ ε the operator associated with h θ ε . The spectrum of H θ ε is purely discrete, moreover for each k ∈ N the function θ → λ k (H θ ε ) is continuous. Consequently, the set According to the Floquet-Bloch theory we have the following representation: Along with h θ ε we also introduce the forms h N ε and h D ε acting on the domains with the action being again specified by (2.1). By H N ε and H D ε we denote the associated operators. The spectra of these operators are purely discrete. It is easy to see that , whence, using the min-max principle [7, Section 4.5], we conclude In the following we mostly use two distinguished points of T n , The subscripts p and a means periodic and antiperiodic, respectively.
Remark 2.1. Despite in the following we work only on the level of forms one is curious to take a more close look on the operators H θ ε , H N ε , H D ε . Let u ∈ dom(H * ε ) with * ∈ {θ , N, D}. Then • for each e ∈ E Y one has u e ∈ H 2 (0, l(e)), • at the vertices from V Y \U Y u satisfies the same conditions as functions belonging to dom(H ε ).
To describe the behaviour of u on U Y we assume for simplicity that points of U Y lye on the interiors 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 ∈ U Y there is only one edges of E Y (we denote it e v ) emanating from v. Then we get the following boundary conditions at U Y : where w ∈ U Y is such that w = i · v for some i ∈ Z n (one can show that for each v ∈ U Y there exists a unique w with w = i · v for some i ∈ Z n provided the period cell is chosen as above), is a natural coordinate on e v such that x e v = 0 at v; in the same way x e w is defined. The action of all operators above is given by (1.10).
Lemma 2.1. There exist Λ 0 > 0 and ε Λ > 0 such that Proof. For θ ∈ T n and ε ≥ 0 we introduce in L 2 (Y ) the form h θ ε , We denote by H θ ε the self-adjoint operator associated with this form. Obviously, Also we observe that with respect to the space decomposition L 2 (Y ) = ⊕ m j=0 L 2 (Y j ) the operator H θ 0 can be decomposed in a sum where the operators H θ 0,0 , H N 0, j are associated with the forms h θ 0,0 , h N 0, j defined as follows, It is easy to see that If λ 1 (H θ 0,0 ) = 0, the corresponding eigenfunction would be constant which is possible iff θ = θ p . Thus It follows from (2.8), (2.11)-(2.13) that λ k (H θ 0 ) = 0 for k = 1, . . . , m, while (2.14) Using the fact that sequence of forms h θ ε increases monotonically as ε decreases, and moreover Moreover, since the sequence of resolvents (H θ ε + I) −1 decrease monotonically as ε decreases, and both resolvents (H θ ε + I) −1 and (H θ 0 + I) −1 are compact one can upgrade (2.15) to the norm resolvent convergence [16, Theorem VIII-3.5]. As a consequence we get the convergence of spectra, namely We set Since θ a = θ p , then Λ 0 > 0. It follows from (2.16) that there exists ε Λ > 0 such that provided the denominator 1 − (1 + λ k (H ))δ 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 }.

2.4.
Estimates on λ k (H N ε ) and λ k (H θ p ε ). In this subsection we denote by bold letters (e.g., u) the elements of C m+1 . Their entries will be enumerated starting from zero, i.e. u ∈ C m+1 ⇒ u = (u 0 , . . . , u m ) with u j ∈ C.
Let C m+1 l be the same space C m+1 equipped with the weighted scalar product (recall that l j and N j are defined by (1.11)). Note that C m+1 l is isomorphic to a subspace of L 2 (Y ) consisting of functions being constant on each Y j , j = 0, . . . , m. In C m+1 l we introduce the form This form is associated with the operator H N 0 in C m+1 l being given by the symmetric (with respect to the scalar product (2.20)) matrix Indeed, let λ be the eigenvalue of H N 0 such that λ / ∈ {A 1 , A 2 , . . . , A m }, and let 0 = u = (u 0 , . . . , u m ) be the corresponding eigenfunction. The equation H N 0 u = λ u is a linear algebraic system for u 0 , . . . , u m . From the last m equations of this system we infer Note, that the denominator in (2.23) is non-zero since λ = A j = α j β 2 j N j l −1 j . Inserting (2.23) into the first equation of the system we arrive at Moreover, u 0 = 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 Proof. W.l.o.g. we may assume that α j and γ are non-negative. Evidently, under this assumption the operators H N ε are non-negative. Moreover, the operator H N 0 is also non-negative, since (it is easy to see that the image of Ψ is contained in dom(h N ε )). Thus we are in the framework of Lemma 2.2. In the general case we have to consider the shifted operators H N ε − µI and where µ is the smallest eigenvalues of H N ε | ε=1 . The operator H N ε − µI is non-negative for each ε ∈ (0, 1] due to the fact that the sequence of forms h N ε increases monotonically as ε decreases; the non-negativity of H N 0 − µI follows from (2.26). We introduce the operator Φ : Our goal is to show that the following estimates hold for each u ∈ dom(h N ε ): with some C 1 ,C 2 > 0. By Lemma 2.2 (see also Remark 2.2 after it) we infer from (2.29)-(2.30) that Taking into account that 0 ≤ λ j (H N ε ) and λ j (H N ε ) ≤ B j−1 (this estimate follows from (2.4) and Lemma 2.4 below), we conclude from (2.31) that there exists such ε B > 0 that the required estimate (2.25) holds for ε < ε B .
To prove (2.29) we need a Poincaré-type inequality on each Y j . Namely, let the form h N 0, j be defined by (2.10), j = 1, . . . , m; in the same way we define h N 0, j for j = 0. By H N 0, j we denote the associated operators in L 2 (Y j ), j = 0, . . . , m. One has λ 1 (H N 0, j ) = 0 (the corresponding eigenspace consists of constants), while λ 2 (H N 0, j ) > 0. By the max-min principle [38] Using the above estimate for v := u − (Φu) j we get where C 1 = (λ 2 (H 0, j )) −1 . Using (2.32) we obtain (on the last step we use the fact that α j and γ are non-negative). Inequality (2.29) is checked. Now let us prove the estimate (2.30). One has: We estimate the remainder R ε as follows (below we use the estimate |a| 2 − |b| 2 ≤ |a − b| (|a| + |b|)): To proceed further we need a standard trace estimate where C > 0 depends on Y . Applying it for w := u Y j −(Φu) j and then using (2.32) we obtain Also, using the Cauchy-Schwarz inequality and (2.35) and taking into account that ε ≤ 1, one gets with some constant C 2 depending on α j , β j , γ, Y . The required estimate (2.30) follows from (2.33), (2.39); this ends the proof of Lemma 2.3.
Lemma 2.4. One has: . . , m + 1. Proof. By the min-max principle [7, Section 4.5] we have where H j is a set of all j-dimensional subspaces in dom(h It is easy to see that the image of Ψ is contained in dom(h θ p ε ), and (recall, that the form h N 0 is given by (2.21), by H N 0 we denote the associated operator). Let {e 1 , e 2 , . . . , e m+1 } be an orthonormal system of eigenvectors of H N 0 such that H N 0 e j = B j−1 e j (see (2.22)). For j = 1, . . . , m + 1 we set W j := span(e 1 , . . . , e j ). It is easy to see that Finally, we set V j := ΨW j , obviously V j ∈ H j . Then using (2.40)-(2.42) we obtain: The lemma is proven.
2.5. Estimates on λ k (H θ a ε ) and λ k (H D ε ). Let C m l be the subspace of C m+1 consisting of vectors of the form u = (0, u 1 , . . . , u m ) with u j ∈ C with the scalar product generated by (2.20), i.e.
In this space we introduce the quadratic form It is easy to see that h θ a 0 = h N 0 C m l . The operator associated with this form is given by the matrix Evidently, the eigenvalues of this matrix are the numbers A 1 < A 2 < . . . < A m .
Lemma 2.5. There exist such constants C A > 0 and ε A > 0 that Proof. The proof is similar to the proof of Lemma 2.3. There is only one essential difference: instead of the operator Φ (2.28) one should use the operator Φ 0 : dom(h θ a ε ) → C m l defined by (Φ 0 u) 0 = 0, (Φ 0 u) j = (Φu) j , j = 1, . . . , m. and, as a consequence, instead of the Poincaré inequality (2.32) on Y 0 one should use the inequality where C 1 = (λ 1 (H θ a 0,0 )) −1 (recall, that the operator H θ 0,0 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: 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 supscript θ p by D, the supscript N by θ a , B j−1 by A j , and to use instead of the mapping Ψ (2.27) its restriction to C m l (the image of this restriction is contained in dom(h D ε )). 2.6. End of the proof of Theorem 1.1. It follows from (2.2), (2.4) and Lemmata 2.3-2.4 that as ε < ε B . Similarly, using (2.2), (2.4) and Lemmata 2.5-2.6 we get as ε < ε A . Finally, we infer from (2.2) and Lemma 2.1 that Remark 2.3. The proof of Theorem 1.1 relies, in particular, on some properties of the eigenvalues of the operator H θ a ε -see the estimates (2.6), (2.43). In fact, the only specific property of θ a we use is that θ a = θ p . Thus, instead of H θ a ε one can utilize any other H θ ε with θ = θ p -the above estimates are still valid for its eigenvalues (but, of course, with another constants Λ, ε Λ , C A , ε A ).

CONTROL OVER THE ENDPOINTS OF SPECTRAL GAPS
Our first goal is to show that under a suitable choice of coupling constants α j , β j , γ the numbers A j , B j (cf. Theorem 1.1) coincide with prescribed ones.
Theorem 3.1. Let A j , j = 1, . . . , m and B j , j = 0, . . . , m be arbitrary numbers satisfying We set where r j , j = 1, . . . , m is defined by Proof of Theorem 3.1. The equality A j [ α j , β j , γ] = A j , j = 1, . . . , m is straightforward -one just needs to insert α j and β j defined by (3.2) into the definition of the numbers A j [ α j , β j , γ] (1.12). Now, let us prove that B j [ α j , β j , γ] = B j as j = 0, . . . , m. For this purpose, we consider the following system of linear algebraic equations (for unknown z = (z 1 , z 2 , , . . . , z m ) ∈ C m ): It was proven in [17] that z = ( r 1 , . . . , r m ) with r j being defined by (3.3) is the solution to this system.
Thus for j = 1, . . . , m one has It is straightforward to check that (3.4) is equivalent to Using A j [ α j , β j , γ] = A j we conclude from (3.5) that B j , j = 0, . . . , m are the roots of (1.14) in which . . , m. Theorem 3.1 is proven.
Applying this lemma we conclude that there exists such α ∈ D that F k (α) = A k , k = 1, . . . , m. Remark 3.2. The assumption A j = 0, j = 1, . . . , m in (3.1) is essential -one cannot avoid it when using the Hamiltonians H ε introduced in Subsection 1.4, since the numbers A j (1.12) are always nonzero. To overcome this restriction one can add to H ε a constant potential, which shift the spectrum accordingly. Another option is to to pick in each Y j , j = 0, . . . , m an internal point v j , and then to add at v j the δ -coupling of the strength γ l j , where l j is defined by (1.11) and γ ∈ R. Denote by H ε the modified Hamiltonian. Repeating verbatim the arguments we use in the proof of Theorem 1.1 one can show that the spectrum of H ε satisfies (1.16)-(1.18), but with A j + γ and B j + γ instead of A j and B j .

ACKNOWLEDGMENT
The author is supported by Austrian Science Fund (FWF) under the Project M 2310-N32.