Periodic discrete graphs with prescribed spectrum
January 01, 1970
Abstract
We construct a periodic weighted graph whose discrete Laplacian has a spectrum with precisely \(n\) gaps. Moreover, we show that by an appropriate choice of the weights, the endpoints of these gaps, as well as the upper edge of the spectrum, attain the prescribed values. The underlying graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to \(n\) additional pendant vertices by extra edges. Semi-explicit formulae for the weight coefficients are provided: some of the coefficients are determined explicitly, while others are given as roots of an explicitly determined polynomial.
1 Introduction↩︎
The topic of this paper lies within the spectral theory of Laplacians on periodic discrete graphs. These operators act on functions defined on the vertices as difference operators, while the edges play a secondary role, serving merely as labels that encode the connectivity between vertices. This is an active area of research that brings together several branches of mathematics and has a wide range of applications, e.g. in nano-technology, crystallography, optics and chemistry. We refer the reader to the monographs [1]–[6] and references therein.
In this paper, we study Laplacians on periodic discrete graphs. For such operators, Floquet–Bloch theory (see, e.g., [1] for its version for discrete graphs) implies that the spectrum exhibits the so-called band-gap structure, that is, it consists of a union of finitely many closed finite intervals (bands) possibly separated by open intervals (gaps). In general, however, gaps are not guaranteed to occur, since the bands may overlap, causing the corresponding gaps to close. For example, the spectrum of the combinatorial discrete Laplacian \[\Delta:\ell^2(\mathbb{Z})\to\ell^2(\mathbb{Z}), \qquad (\Delta f)(i) = 2f(i)-f(i-1)-f(i+1),\quad i\in\mathbb{Z}\] on the infinite chain of vertices \(\{\dots,-2,-1,0,1,2,\dots\}\) has no gaps, namely \(\sigma(\Delta)=[0,4].\)
The existence of spectral gaps is important in various applications, since the spectrum encodes the transport properties of the underlying medium, and spectral gaps correspond to ranges of wavelengths for which wave propagation through the medium is prohibited. We refer the reader to [7] for applications to dielectric materials and to [8]–[11] for connections between the spectral properties of graph operators and the behavior of nanomaterials, especially with regard to photonic band gaps (see, e.g., [12]).
A method for opening spectral gaps was proposed by Schenker and Aizenman in [13]. The idea is to start with a fixed graph and “decorate” it by “gluing” to each vertex a copy of a given finite graph. Note that this method is originally not restricted to periodic graphs. See also [14], where a similar idea was applied to differential operators on metric graphs (the so-called quantum graphs).
In the current paper we investigate spectral properties of some specific class of periodic graphs with weights, i.e. additionally to the combinatorial structure (vertices linked by edges) we equip a graph \(\Gamma\) with two positive functions \(\mu\) and \(m\) defined on the set of edges and on the set of vertices, respectively. The graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to \(n\) additional pendant vertices by extra edges – see Figure 1. The Laplacian \(\mathscr{L}\) on the weighted graph \(\Gamma\) acts on \(f:\mathscr{V}\to\mathbb{C}\), where \(\mathscr{V}\) is the vertex set of \(\Gamma\), as follows: \[\begin{gather} (\mathscr{L}f)(v)=(m(v))^{-1}\sum\mu(e)(f(v) - f(u)) , \quad v\in\mathscr{V}. \end{gather}\] where the sum above is taken over all vertices \(u\) adjacent to \(v\), with \(e\) denoting the edge connecting \(u\) and \(v\).
The main peculiarity of these graphs is that their spectral gaps can be effectively controlled through an appropriate choice of the weight functions. Firstly, we show (Theorem 1) that, under rather mild assumptions on the weights (see 5 ), the spectrum of \(\mathscr{L}\) possesses exactly \(n\) gaps. Our main result (Theorem 4) addresses the inverse problem: we prove that the gap edges, as well as the upper edge of the spectrum, can be prescribed arbitrarily and realized by a suitable choice of the weights.
Semi-explicit formulae for the weights realizing the prescribed spectral structure are obtained. By this we mean that some of the weight coefficients are not given in closed form, but as roots of a certain polynomial whose coefficients are explicitly expressed in terms of the prescribed gap edges.
For periodic quantum graphs, this type of inverse problem was studied by Barseghyan and the first author in [15]. The result obtained there is asymptotic in nature: a sequence of operators was constructed whose first \(n\) spectral gaps converge to prescribed intervals. At the end of the paper, in Remark 6, we compare the results of [15] with those of the present paper.
The paper is organized as follows. In the next section, we introduce the weighted graph and the associated Laplacian, and describe its spectrum. In Section 3 we prove some auxiliary technical lemmas. In Section 4 we investigate several properties of the spectrum. The inverse problem is then solved in Section 5.
2 Graph \(\Gamma\), operator \(\mathscr{H}\) and its spectrum↩︎
Let \(\Gamma=(\mathscr{V},\mathscr{E})\) be a graph with vertex set \(\mathscr{V}=\{v_{i,j}\,:\;i\in\mathbb{Z},\;j=0,\dots,n\}\) and edge set \(\mathscr{E}=\{e_{i,j}\,:\;i\in\mathbb{Z},\;j=0,\dots,n\}.\) The edges connect the vertices as follows: \[\begin{gather} \label{graph:structure} v_{i,0}\overset{e_{i,0}}\sim v_{i+1,0}, \quad i\in\mathbb{Z}, \qquad v_{i,0}\overset{e_{i,j}}\sim v_{i,j}, \quad i\in\mathbb{Z},\;j=1,\dots,n. \end{gather}\tag{1}\] Here, the notation \(u\overset{e}\sim v\) means that the vertices \(u\) and \(v\) are connected by the edge \(e\). The graph is depicted in Figure 1. Note that we consider the edges as relations between vertices, rather then physical or geometrical links.
We equip the graph \(\Gamma\) with two weight functions \(\mu:\mathscr{E}\to \mathbb{R}\) and \(m:\mathscr{V}\to \mathbb{R}\) defined by \[\begin{gather} \label{mu:m} \mu(e_{i,j})=\mu_j,\quad i\in\mathbb{Z},\;j=0,\dots,n, \qquad m(v_{i,j})=m_j,\quad i\in\mathbb{Z},\;j=0,\dots,n, \end{gather}\tag{2}\] where \(\mu_j,\, m_j\), \(j=0,\dots,n\), are positive numbers.
Now, we introduce the Laplacian \(\mathscr{L}\) on the weighted graph \(\Gamma\) (cf. [1] or [2]). We define the Hilbert space \(\mathscr{H}\) by \[\mathscr{H}=\Big\{f:\mathscr{V}\to\mathbb{C}:\quad\sum_{i\in\mathbb{Z}}\sum_{j=0}^n m_j|f(v_{i,j})|^2<\infty\Big\}\] equipped with the inner product \((f,g)_{\mathscr{H}}= \sum_{i\in\mathbb{Z}}\sum_{j=0}^n m_j f(v_{i,j})\overline{g(v_{i,j})}.\) Then we introduce the operator \(\mathscr{L}:\mathscr{H}\to \mathscr{H}\) by \[\begin{gather} \label{L:action} (\mathscr{L}f)(v)=\frac{1}{m(v)}\sum_{u\overset{e}\sim v}\mu(e)(f(v) - f(u)),\quad v\in\mathscr{V}. \end{gather}\tag{3}\] The operator \(\mathscr{L}\) is bounded, non-negative, and self-adjoint.
We denote: \[\begin{gather} \label{ab:notations} a_j\mathrel{\vcenter{:}}=\mu_j/m_j,\;j=1,\dots,n,\qquad b_j\mathrel{\vcenter{:}}=\mu_j/m_0,\;j=0,\dots,n. \end{gather}\tag{4}\] We assume that the numbers \(a_j\) are pairwise distinct. Without loss of generality, we may therefore order them so that \[\begin{gather} \label{aa} a_j<a_{j+1},\quad j=1,\dots,n-1. \end{gather}\tag{5}\]
To describe the spectrum of the operator \(\mathscr{L}\) we introduce a rational function \[\begin{gather} \label{F} F(x)=x\left(\sum_{j=1}^n\frac{b_j}{a_j-x}+1\right). \end{gather}\tag{6}\] The function \(F(x)\) is strictly increasing on each interval \((a_j,a_{j+1})\), as well as on \((-\infty,a_1)\) and on \((a_n,\infty)\). Moreover, one has \[\lim_{x\to\pm\infty}F(x)=\pm\infty,\quad \lim_{x\to a_j\pm 0}F(x)=\mp\infty,\quad F(0)=0.\] Consequently, \(F(x)\) has exactly \(n+1\) roots \(s_j\in[0,\infty)\), \(j=0,\dots,n\), which we enumerate in ascending order. Similarly, the equation \[\begin{gather} \label{t:eq} F(x)=4b_0 \end{gather}\tag{7}\] has exactly \(n+1\) positive roots \(t_j\), \(j=1,\dots,n+1\), which we also enumerate in ascending order. Clearly (see Figure 2), \[\begin{gather} \label{interlace} 0=s_0,\qquad s_{j-1}<t_j<a_j<s_j,\;j=1,\dots,n,\qquad s_n<t_{n+1}. \end{gather}\tag{8}\]
We are now in the position to formulate the first result.
Theorem 1. One has: \[\begin{gather} \label{th1:eq} \displaystyle\sigma(\mathscr{L})= \bigcup_{j=1}^{n+1} (s_{j-1},t_j)= [0,t_{n+1}]\setminus\Big( \bigcup_{j=1}^n (t_j,s_j)\Big). \end{gather}\tag{9}\]
Proof. Since the graph \(\Gamma\) is \(\mathbb{Z}\)-periodic and the operator \(\mathscr{L}\) commutes with the \(\mathbb{Z}\)-action, the standard Floquet–Bloch theory applies (see [1] for more details). In particular, the spectrum \(\sigma(\mathscr{L})\) has a band-gap structure: it is a union of closed finite intervals \(I_j\) (spectral bands) determined by the eigenvalues of the fiber operators \(\mathscr{L}(k)\), \(k\in[0,2\pi)\), acting in the space \(\mathscr{H}(k)\) of \(k\)-quasi-periodic functions \(f:\Gamma\to\mathbb{C}\), that is, \[f(v_{i,j})=e^{\mathrm{i} k}f(v_{i-1,j}),\quad i\in\mathbb{Z},\;j=0,\dots,n,\] with the action of \(\mathscr{L}(k)\) given by 3 . The quasiperiodicity condition implies that the spectral problem \[\begin{gather} \label{Lf61lambdaf} \mathscr{L}(k)f=\lambda f,\quad 0\not=f\in\mathscr{H}(k) \end{gather}\tag{10}\] reduces to the finite-dimensional problem \[\begin{gather} \label{Lf61lambdaf:matrix} L(k)f=\lambda f,\quad 0\not=f=(f_0,\dots,f_n)^T\in\mathbb{C}^{n+1}, \end{gather}\tag{11}\] where \(L(k)\) is the \((n+1)\times(n+1)\) matrix given by \[L(k)= \begin{pmatrix} b_0(1-e^{\mathrm{i}k})+b_0(1-e^{-\mathrm{i}k}) + \displaystyle\sum_{j=1}^n b_j &\quad -b_1 & -b_2 & \cdots & -b_n \\ -a_1 &\quad a_1 & 0 & \cdots & 0 \\ -a_2 &\quad 0 & a_2 & \cdots & 0 \\ \vdots &\quad \vdots & \vdots & \ddots & \vdots \\ -a_n &\quad 0 & 0 & \cdots & a_n \end{pmatrix}\] (recall that \(a_j\), \(j=1,\dots,n\) and \(b_j\), \(j=0,\dots,n\) are defined by 4 ). To obtain 11 one writes the equation 10 at the vertices \(v_{0,j}\), \(j=0,\dots,n\), which form a fundamental domain of \(\Gamma\), and use the quasiperiodicity conditions \[f(v_{1,0})=f(v_{0,0})e^{\mathrm{i}k},\qquad f(v_{-1,0})=f(v_{0,0})e^{-\mathrm{i}k}.\] The matrix \(L(k)\) is symmetric in \(\mathbb{C}^{n+1}\) equipped with the weighted inner product \[(f,g)_{\mathbb{C}^{n+1}}=\sum_{j=0}^n m_j f_j\overline{g_j}= m_0 \Big(f_0\overline{g_0}+\sum_{j=1}^n b_j a_j^{-1}\overline{g_j}\Big).\] We denote its eigenvalues, ordered increasingly and counted with multiplicities, by \[\lambda_1(k)\leq \lambda_2(k)\le\dots\le \lambda_{n+1}(k).\] For each fixed \(j\), \(\lambda_j(k)\) depends continuously on \(k\in [0,2\pi]\). Consequently, the set \[I_j\mathrel{\vcenter{:}}=\bigcup_{k\in [0,2\pi]}\lambda_j(k),\] is a compact interval. The Floquet-Bloch theory yields the representation \[\begin{gather} \label{Floquet-Bloch} \sigma(\mathscr{L})=\bigcup_{j=1}^{n+1} I_j. \end{gather}\tag{12}\] Therefore, the description of the spectrum of \(\mathscr{L}\) reduces to the analysis of the eigenvalues of the matrices \(L(k)\).
Taking into account 5 , one can easily show that if \(\lambda= a_j\) for some \(j\in\{1,\dots,n\}\), then the problem 11 possesses only a zero solution \(f\). Thus, \[\begin{gather} \label{lambdanotin} \lambda\notin \{a_j,\;j=1,\dots,n\}. \end{gather}\tag{13}\] The matrix equality 11 yields the system of \(n+1\) equations for unknowns \(f_j\), \(j=0,\dots,n\). From the last \(n\) equations of this systems we deduce, taking into account 13 : \[\begin{gather} \label{fj} f_j=\frac{a_j f_0}{a_j-\lambda},\quad j=1,\dots,n. \end{gather}\tag{14}\] Substituting 14 into the first equation and taking into account that \(f_0\not=0\) (since otherwise 14 would imply \(f_j=0\) for all \(j=1,\dots,n\)), we arrive, after straightforward computations, at the following equation for \(\lambda\): \[\begin{gather} 4b_0\sin^2{k\over 2}=F(\lambda), \end{gather}\] where the function \(F\) is given by 6 . Thus, the eigenvalues \(\lambda_j(k)\) correspond to the intersection points of the graph of \(F(x)\) with the horizontal line \(y=4b_0\sin^2{k\over 2}\). The properties of the function \(F(x)\) (see Figure 2) imply that \[\forall k\in [0,2\pi]:\quad \lambda_{j}(0)\le \lambda_{j}(k)\le \lambda_j(\pi),\quad j=1,\dots,n.\] By definition of the numbers \(s_j\) and \(t_j\), we get \(\lambda_j(0)=s_{j-1}\) and \(\lambda_j(\pi)=t_{j}\). Hence, \(I_j=[s_{j-1},t_j]\), which together with 12 implies the first equality in 9 . The second equality follows from 8 . The theorem is proven. ◻
Theorem 1 demonstrates that the operator \(\mathscr{L}\) has \(n\) gaps \((t_j,s_j)\), \(j=1,\dots,n\). Our main objective is to solve the inverse problem: to prove that, through a suitable choice of the coefficients \(\mu_j\) and \(m_j\), the positions of the gap endpoints, as well as the upper spectral edge \(t_{n+1}\), can be made to coincide with prescribed values.
The rest of the paper is organized as follows. In Section 3, we prove two auxiliary technical lemmas concerning a polynomial whose roots are the numbers \(a_j\). In Section 4, we derive several useful relations linking \(a_j\), \(b_j\), and \(s_j\), \(t_j\). Finally, in Section 4, we formulate and prove the main result of the paper, Theorem 4, which solves the inverse problem stated above.
3 Auxiliary lemmas↩︎
To prove the main result of this work, we will need two technical lemmas stated below. Since their proof is rather lengthy and technical, we devote a separate section to them.
Lemma 1. Let \(B>0\). Let \(s_j\), \(j=0,\dots,n\) and \(t_j\), \(j =1,\dots, n+1\) be given numbers satisfying \[\begin{gather} \label{st} 0=s_0< t_1,\quad t_j< s_j< t_{j+1},\;j=1,\dots,n. \end{gather}\tag{15}\] Define the polynomial \(Q(x)\) of degree \(n\) by \[\begin{gather} \label{Qdef} Q(x)=\prod_{i=1}^n(t_i-x)-\dfrac 1B\sum_{j=1}^n \left( \prod_{m=0}^n (s_m-t_j) \prod_{\substack{i=1\\i\ne j}}^n\dfrac{x-t_i}{t_j-t_i}\right). \end{gather}\tag{16}\] Then \[\label{eq::Qslstatement} \forall \ell\in\{0,\dots,n\}:\quad Q(s_\ell)=\left(\prod_{i=1}^n (t_i-s_\ell)\right)\left(1+\dfrac{1}{B}\left(\sum_{\substack{i=1\\i\ne \ell}}^n s_i-\sum_{\substack{i=1}}^n t_i\right)\right).\tag{17}\]
Remark 2. The role of the above lemma, which at first glance may appear somewhat unrelated to our problem, is the following. We will show later (see Lemma 3(i)) that if \(s_j\) and \(t_j\) are defined as in Theorem 1, then the numbers \(a_j\) are precisely the roots of the polynomial \(Q(x)\) with \(B=4b_0\). This observation will allow us to solve the inverse problem by choosing \(a_j\) as the roots of \(Q(x)\). However, the numbers \(a_j\) chosen in this way must satisfy 8 , and the proof of this fact relies essentially on Lemma 1.
Remark 3. It is easy to see that the leading coefficient of the polynomial \(Q(x)\) is equal to \((-1)^n\) and that \(Q(t_j)= {-P(t_j)}/{B}\) for \(j=1,\dots,n,\) where \(P(x)\) is a polynomial of degree \(n+1\) given by \[\begin{gather} \label{Pdef} P(x)\mathrel{\vcenter{:}}=\prod_{m=0}^n (s_m-x). \end{gather}\tag{18}\] This follows from the fact that the second term in the right-hand-side of 16 is precisely the Lagrange interpolation polynomial of degree \(n-1\) passing through the points \((t_j,{-P(t_j)}/{B})\), \(j=1,\dots,n\)). These two properties (the prescribed leading coefficient and the interpolation data at \(n\) distinct points) uniquely determine the polynomial \(Q\). We will use this alternative characterization of \(Q\) later on.
Proof of Lemma 1. Instead of \(Q(x)\), we will work with the function \[\widehat{Q}(x)\mathrel{\vcenter{:}}= Q(x)\prod_{1\le i<k\le n}(t_k-t_i)\] This choice is motivated by the fact (see below) that it eliminates the factors \((t_j-t_i)\) appearing in the denominator of 16 .
It is easy to see that \[\forall j\in\{1,\dots,n\}:\quad \prod_{1\le i<k\le n}(t_k-t_i)=(-1)^{n-j} \prod_{\substack{i=1\\i\ne j}}^n {(t_j-t_i)} \prod_{\substack{1\le i<k\le n\\i\ne j,k\ne j}}(t_k-t_i)\] Using the above equality, we get for any fixed \(\ell\), \(0\le \ell\le n\): \[\begin{align} \widehat{Q}(s_\ell) &=\prod_{i=1}^n(t_i-s_\ell)\prod_{1\le i<k\le n}(t_k-t_i) \\ &+\sum_{j=1}^n \dfrac{(-1)^{n-j+1}}{B}\left(\prod_{m=0}^n(s_m-t_j)\prod_{\substack{i=1\\i\ne j}}^n {(s_\ell-t_i)}\prod_{\substack{1\le i<k\le n\\i\ne j, k\ne j}}(t_k-t_i)\right)\\ &=\prod_{i=1}^n(t_i-s_\ell)\left(\prod_{1\le i<k\le n}(t_k-t_i)+\sum_{j=1}^n \dfrac{(-1)^{j-1}}{B}\left({\prod_{\substack{m=0\\m\ne \ell}}^n(s_m-t_j)}\prod_{\substack{1\le i<k\le n\\i\ne j, k\ne j}}(t_k-t_i)\right)\right). \end{align}\] Thus \[\label{eq::wQsl} \widehat{Q}(s_\ell)=\prod_{i=1}^n(t_i-s_\ell)\left(\prod_{1\le i<k\le n}(t_k-t_i)+F^{(\ell)}(t_1,\dots,t_n)\right),\tag{19}\] where \(F^{(\ell)}(x_1,\dots, x_n)\) is a polynomial in \(n\) variables given by \[\begin{gather} \label{sumFj} F^{(\ell)}(x_1,\dots, x_n)\mathrel{\vcenter{:}}=\sum_{j=1}^n F_j^{(\ell)}(x_1,\dots x_n), \end{gather}\tag{20}\] with \[\label{Fj} F_j^{(\ell)}(x_1,\dots, x_n)\mathrel{\vcenter{:}}=\dfrac{(-1)^{j-1}}{B}{\prod_{\substack{m=0\\m\ne \ell}}^n(s_m-x_j)}\prod_{\substack{1\le i<k\le n\\i\ne j, k\ne j}}(x_k-x_i).\tag{21}\]
It follows easily from the definition 21 of \(F_j^{(\ell)}\) that for any fixed \(p,r\in\{1,\dots,n\}\) with \(p\ne r\), one has \[F_j^{(\ell)}(\sigma(x_1),\dots \sigma(x_n))=\begin{cases} -F_j^{(\ell)}(x_1,\dots x_n),& j\notin\{ p, r\}\\ -F_r^{(\ell)}(x_1,\dots x_n),& j=p,\\ -F_p^{(\ell)}(x_1,\dots x_n),& j=r, \end{cases}\] where \(\sigma(x_i)=x_i, i\notin\{ p, r\}\), \(\sigma(x_r)=x_p\), \(\sigma(x_p)=x_r\). Consequently, \(F^{(\ell)}(x_1,\dots, x_n)=\sum_{j=1}^n F_j^{(\ell)}(x_1,\dots x_n)\) is an alternating polynomial. It is well-known (see, e.g., [16]) that any alternating polynomial is a product of the Vandermonde polynomial \[W(x_1,\dots x_n)\mathrel{\vcenter{:}}=\prod_{1\le i<k\le n}(x_k-x_i)\] and a symmetric polynomial. Straightforward computations yield \[\deg F^{(\ell)}=\deg F_j^{(\ell)}= \dfrac{n(n-1)}{2}+1,\qquad \deg W =\dfrac{n(n-1)}{2}.\] Hence \[\begin{gather} \label{FeqC} F^{(\ell)}(x_1,\dots x_n) = S^{(\ell)}(x_1,\dots x_n) W(x_1,\dots x_n), \end{gather}\tag{22}\] where \(S^{(\ell)}(x_1,\dots x_n)\) is a symmetric polynomial with \(\deg S^{(\ell)} = 1\), i.e. \[S^{(\ell)}(x_1,\dots,x_n)=A^{(\ell)}\left(\sum_{i=1}^n x_i+C^{(\ell)}\right),\] with the coefficients \(A^{(\ell)},\,C^{(\ell)}\in \Bbb R\) to be determined.
To find \(A^{(\ell)}\), we regard the right-hand side of 22 as a polynomial of the single variable \(x_n\), treating all other variables as parameters. In this case, its coefficient at \(x_n^n\) equals \(A^{(\ell)}\prod_{1\le i<k\le n-1}(x_k-x_i).\) On the other hand, if we determine this coefficient from the left-hand side of 22 , i.e. directly from the definition 20 of \(F^{(\ell)}(x)\), we arrive at the expression \(-(1/B)\prod_{1\le i<k\le n-1}(x_k-x_i).\) By equating these two expressions, we infer \[\begin{gather} \label{A} A^{(\ell)}=-1/B. \end{gather}\tag{23}\]
To proceed further (in particular, to determine the coefficient \(C^{(\ell)}\)), we divide our analysis into two cases.
Case (i): \(\ell\ne 0\). We have \[\forall \ell\not=0\;\forall j\in\{1,\dots,n\}:\quad F_j^{(\ell)}(s_1,\dots,s_{\ell-1},0,s_{\ell+1},\dots, s_n)=0.\] This follows from the fact (see 21 ) that \(F_\ell^{(\ell)}(x_1,\dots,x_n)\) is divisible by \(s_0-x_\ell\), and \(F_j^{(\ell)}(x_1,\dots,x_n)\) is divisible by \(s_j-x_j\) for every \(j\neq \ell\). Then, using 22 –23 , we get: \[0=F^{(\ell)}(s_1,\dots,s_{\ell-1},0,s_{\ell+1},\dots, s_n)=\frac{1}{B}\left(\sum_{\substack{i=1\\i\ne \ell}}^n s_i+C^{(\ell)}\right)(-1)^{\ell}\prod_{i=1}^n s_i\prod_{\substack{0\le i<k\le n\\i\ne \ell,k\ne \ell}}(s_k-s_i),\] and, since last two products above are non-zero, we conclude that \[C^{(\ell)}=-\sum_{\substack{i=1\\i\ne \ell}}^n s_i,\quad \forall \ell \in \{1,\dots,n\}.\] Hence, one has \[F^{(\ell)}(x_1,\dots, x_n)=-\frac{1}{B}\left(\sum_{i=1}^n x_i-\sum_{\substack{i=1\\i\ne \ell}}^ns_i\right)\prod_{1\le i<k\le n}(x_k-x_i), \quad \forall \ell \in \{1,\dots,n\}.\] Substituting the above equality into 19 , we obtain \[\begin{align} \widehat{Q}(s_\ell) =\prod_{i=1}^n(t_i-s_\ell)\left(\prod_{1\le i<k\le n}(t_k-t_i)+\frac{1}{B}\left(\sum_{\substack{i=1\\i\ne \ell}}^ns_i-\sum_{i=1}^n t_i\right)\prod_{1\le i<k\le n}(t_k-t_i)\right),\; \forall \ell \in \{1,\dots,n\}, \end{align}\] whence \[\begin{gather} \label{Qsl1} Q(s_\ell)=\dfrac{\widehat{Q}(s_\ell)}{\prod_{1\le i<k\le n}(t_k-t_i)}=\prod_{i=1}^n(t_i-s_\ell)\left(1+\dfrac1B\left(\sum_{\substack{i=1\\i\ne \ell}}^ns_i-\sum_{i=1}^n t_i\right)\right),\; \forall \ell \in \{1,\dots,n\}. \end{gather}\tag{24}\]
Case (ii): \(\ell=0\). Then we have \[\forall j\in\{1,\dots,n\}:\quad F_j^{(0)}(s_1,\dots, s_n)=0,\] which follows from the fact (see 21 ) that \(F_j^{(0)}(x_1,\dots, x_n)\) is divisible by \(s_j-x_j\). Then, using 22 –23 , we obtain: \[0=F^{(0)}(s_1\dots, s_n)=-\frac{1}{B}\left(\sum_{\substack{i=1}}^n s_i+C^{(0)}\right) \prod_{1\le i<k\le n}(s_k-s_i),\] and, since last product above is non-zero, we conclude that \[C^{(0)}= -\sum_{i=1}^n s_i,\] Hence, one has \[F^{(0)}(x_1,\dots, x_n)=-\frac{1}{B}\left(\sum_{i=1}^n x_i-\sum_{i=1}^ns_i\right)\prod_{1\le i<k\le n}(x_k-x_i).\] Substituting the above equality into 19 , we obtain \[\begin{align} \widehat{Q}(s_0)=\prod_{i=1}^n(t_i-s_0)\left(\prod_{1\le i<k\le n}(t_k-t_i)+\frac{1}{B}\left(\sum_{i=1}^ns_i-\sum_{i=1}^n t_i\right)\prod_{1\le i<k\le n}(t_k-t_i)\right), \end{align}\] whence, \[\begin{gather} \label{Qsl2} Q(s_0)=\dfrac{\widehat{Q}(s_0)}{\prod_{1\le i<k\le n}(t_k-t_i)}=\prod_{i=1}^n(t_i-s_0)\left(1+\dfrac1B\left(\sum_{i=1}^ns_i-\sum_{i=1}^n t_i\right)\right). \end{gather}\tag{25}\]
Lemma 2. Let \(B>0\). Let \(s_i\), \(j=0,\dots,n\) and \(t_j\), \(j =1,\dots, n+1\) be given numbers satisfying 15 . Define the polynomial \(Q(x)\) of degree \(n\) by 16 . Let \(a_j\), \(j=1,\dots,n\), be the roots of \(Q\), ordered according to increasing absolute value. Then the property \[\begin{gather} \label{a:in:ts} t_j<a_j<s_j\quad \text{for all }j =1,\dots, n, \end{gather}\tag{26}\] if fulfilled if and only if the inequality \[\begin{gather} \label{Bcond} \sum_{j=1}^n t_j<B+\sum_{j=1}^{n-1} s_j \end{gather}\tag{27}\] holds true.
Proof. Recall that the polynomial \(P(x)\) is given by 18 . Since \(Q(t_j)=- {P(t_j)}/{B}\) for \(j=1,\dots, n\) (see Remark 3) and \(\mathop{\mathrm{sgn}}P(t_j)=(-1)^{j}\), \(j=1,\dots, n\) (this follows easily from 15 ), then \[\label{eq::sgnQti} \mathop{\mathrm{sgn}}Q(t_j)= (-1)^{j-1},\quad j=1,\dots,n,\tag{28}\] whence, at the endpoints of each interval \((t_j,t_{j+1})\), \(j=1,\dots,n-1\), the function \(Q(x)\) is non-zero and takes opposite signs. Consequently, each interval \((t_j,t_{j+1})\), \(j=1,\dots,n\) contains at least one root of \(Q(x)\). Furthermore, since \[\mathop{\mathrm{sgn}}Q(x)=(-1)^n{ for sufficiently large }x,\] then the semibounded interval \((t_n,\infty)\) also contains a root of \(Q(x)\). From the above observations, together with the fact that \(a_j\), \(j=1,\dots,n\), are the roots of \(Q\) (recall that \(\deg Q = n\)) ordered by increasing absolute value, we conclude that \[t_j<a_j<t_{j+1}\quad \text{for all } j=1,\dots,n-1,\qquad t_n<a_n,\] and then the property 26 holds if and only if \[\begin{gather} \label{sgnQ} \mathop{\mathrm{sgn}}Q(t_j)=-\mathop{\mathrm{sgn}}Q(s_j)\ne 0\quad \text{for all } j=1,\dots,n. \end{gather}\tag{29}\] By 28 , the condition 29 is equivalent to \[\begin{gather} \label{sgnQ1}\mathop{\mathrm{sgn}}Q(s_j)=(-1)^j\quad \text{for all } j=1,\dots,n . \end{gather}\tag{30}\]
To investigate \(\mathop{\mathrm{sgn}}Q(s_j)\) we use Lemma 1. Namely, since \(\mathop{\mathrm{sgn}}\left(\prod_{i=1}^n(t_i-s_j)\right)=(-1)^{j}\) (this follows from 15 ), then by 17 , condition 30 holds if and only if \[1+\dfrac{1}{B}\left(\sum_{\substack{i=1\\i\ne j}}^n s_i-\sum_{\substack{i=1}}^n t_i\right)>0 \quad \text{for all } j=1,\dots,n ,\] which is, obviously, equivalent to \[\label{sllessB} \sum_{i=1}^n t_i<B+\sum_{\substack{i=1\\i\ne j}}^n s_i \quad \text{for all } j=1,\dots,n .\tag{31}\] Now, since \[\sum_{\substack{i=1\\i\ne j}}^n s_i\ge \sum_{i=1}^{n-1} s_i \quad \text{for all } j=1,\dots,n\] due to 15 , then the statement 31 is equivalent to 27 . The lemma is proven. ◻
4 Properties of \(\sigma(\mathscr{L})\)↩︎
In Theorem 1, we computed the spectrum of the operator \(\mathscr{L}\) 3 . In this section, we establish several additional useful relations between the parameters \(a_j, b_j\), which determine the coefficients of the operator, and the quantities \(s_j, t_j\), which represent the endpoints of the spectral bands. These results will be used later in the solution of the inverse problem.
Recall that the function \(F\) is given by 6 .
Lemma 3. Let \(a_j\), \(j=1,\dots,n\), and \(b_j\), \(j=0,\dots,n\), be positive numbers, moreover, let \(a_j\) satisfy 5 . Let \(s_j\), \(j=0,\dots,n\), and \(t_j\), \(j=1,\dots,n+1\), denote the roots of the equations \(F(x)=0\) and \(F(x)=4b_0\), respectively, enumerated in ascending order (that is, the interlacing property 8 is fulfilled). Then the following holds:
The numbers \(a_j,\;i=1,\dots, n,\) are the roots of the polynomial \(Q(x)\) defined by 16 , with \(B=4b_0\).
The upper edge of the spectrum \(t_{n+1}\) satisfies \[\begin{gather} \label{T} t_{n+1}= 4b_0+\sum_{i=1}^{n}(s_i-t_i). \end{gather}\tag{32}\]
Proof. (i) We define the function \(R(x)\) by \[R(x)\mathrel{\vcenter{:}}=\sum_{j=1}^n\frac{b_j}{a_j-x}+1.\] One has \(F(x)=x R(x)\), whence \[\begin{gather} \label{Rzeros} R(s_j)=0,\quad j=1,\dots,n. \end{gather}\tag{33}\] Reducing \(R(x)\) to a common denominator, we get \[R(x)=\dfrac{R_1(x)}{R_2(x)},\] where \[R_2(x)=\prod_{j=1}^n (a_j-x),\qquad R_1(x)=R(x)R_2(x)=\prod_{j=1}^n (a_j-x)+ \sum_{j=1}^n b_j \prod_{\substack{k=1\\k\ne j}}^n(a_k-x).\] Then, due to 33 , we have \[\begin{gather} \label{R1zeros} R_1(s_j)=0,\quad j=1,\dots,n. \end{gather}\tag{34}\] Using 34 and the facts that \(\deg R_1(x)=n\), the leading coefficient of \(R_1(x)\) is \((-1)^n\), the numbers \(s_j\) are pairwise distinct, and \(s_0=0\), we infer \[R_1(x) =\prod_{i=1}^n (s_i-x)=-x^{-1}P(x),\] where the polynomial \(P(x)\) is given in 18 . Then we have \(R_2(x)=-\dfrac{P(x)}{xR(x)}\), whence, taking into account that \(t_j\,\;j=1,\dots, n+1.\) are pairwise distinct solutions of \(x R(x)=4b_0\), we get \[\label{eq::R2ti} R_2(t_j)=-\dfrac{P(t_j)}{t_jR(t_j)}=-\dfrac{P(t_j)}{4b_0},\quad j=1,\dots, n+1.\tag{35}\] Thus \(R_2(x)\)
is a polynomial of degree \(n\),
its leading coefficient is \((-1)^n\),
its graph passes through the \(n\) distinct points 1 \[\left(t_j, \dfrac{-P(t_j)}{4b_0}\right),\quad j =1,\dots, n.\]
However, the polynomial \(Q(x)\) satisfies the same properties 1.-3. – see Remark 3. Since the properties 1.-3. determine the polynomial uniquely, we conclude that \[\label{eq::Q61R2} \prod_{j=1}^n (a_j-x)=R_2(x)=Q(x),\tag{36}\] which proves the statement (i).
(ii) We consider the following polynomial of degree \(n+1\): \[T(x)=Q(x)+\dfrac{P(x)}{4b_0}.\] The leading coefficient of \(T(x)\) is \((-1)^{n+1}/(4b_0)\), its free coefficient is \(Q(0)\), and the pairwise distinct numbers \(t_i\), \(i=1,\dots, n+1\) are the roots of \(T(x)\), which follows from 35 –36 . Then by Vieta’s theorem we obtain \(\prod_{i=1}^{n+1} t_i= 4b_0 Q(0),\) whence, \[\label{eq::TQ0} t_{n+1}=\dfrac{4b_0 Q(0)}{\prod_{i=1}^n t_i}.\tag{37}\] Finally, using Lemma 1 and taking into account that \(s_0=0\), one has \[\begin{align} Q(0) =\left(\prod_{i=1}^n t_i\right)\left(1+\dfrac1{4b_0}\left(\sum_{i=1}^n s_i-\sum_{i=1}^nt_i\right)\right)\label{Q0} \end{align}\tag{38}\] Substituting 38 into 37 we arrive at the formula 32 . The lemma is proven. ◻
5 Inverse problem↩︎
In this section we solve the inverse problem: for given \(s_j\) and \(t_j\), we construct an operator \(\mathscr{L}\) defined by 3 such that its spectrum admits the representation 9 with exactly these prescribed values of \(s_j\) and \(t_j\). Namely, we have the following result.
Theorem 4. Let \(s_j\), \(j=0,\dots,n\), and \(t_j\), \(j=1,\dots,n+1\), be given numbers satisfying \[\begin{gather} \label{st43} 0=s_0< t_1,\qquad t_j< s_j< t_{j+1},\;j=1,\dots,n. \end{gather}\tag{39}\] Using these numbers, we define \(b_0>0\) by \[\begin{gather} \label{b0} b_0\mathrel{\vcenter{:}}=\dfrac14\sum_{j=0}^{n}\left(t_{j+1}-s_j\right), \end{gather}\tag{40}\] and the polynomial \(Q(x)\) of degree \(n\) by \[\begin{gather} \label{Qdef:new} Q(x)=\prod_{i=1}^n(t_i-x)-{1\over 4 b_0}\sum_{j=1}^n \left( \prod_{m=0}^n (s_m-t_j) \prod_{\substack{i=1\\i\ne j}}^n\dfrac{x-t_i}{t_j-t_i}\right). \end{gather}\tag{41}\] Let \(a_j\), \(j=1,\dots,n\), be the roots of \(Q\), ordered according to increasing absolute value.
Then the following holds:
The numbers \(a_j\), \(j=1,\dots,n\) are positive and pairwise distinct, moreover, they satisfy the interlacing property \[\begin{gather} \label{a:in:ts43} t_j<a_j<s_j,\quad j=1,\dots,n. \end{gather}\tag{42}\]
Using the above given \(s_j\), \(j=1,\dots,n\), and the previously defined \(a_j\), \(j=1,\dots,n\), we define the numbers \(b_j\), \(j=1,\dots,n\), by \[\begin{gather} \label{bj} b_j\mathrel{\vcenter{:}}=(s_j-a_j)\prod_{\substack{i=1\\i\ne j}}^n\dfrac{s_i-a_j}{a_i-a_j}, \quad j=1,\dots,n. \end{gather}\tag{43}\] Then \(b_j\) are positive, moreover, one has \[\begin{gather} \label{SLAE} \sum_{j=1}^n\frac{b_j}{a_j-s_i}+1=0,\quad i=1,\dots,n. \end{gather}\tag{44}\]
Using the numbers \(a_j\), \(j=1,\dots,n\), and \(b_j\), \(j=0,\dots,n\), defined above, we construct the graph \(\Gamma\) by 1 –2 , with the functions \(\mu\) and \(m\) given by \[\mu_j=b_j,\;j=0,\dots,n, \qquad m_0=1,\qquad m_j=b_j/a_j,\;j=1,\dots,n.\] Then the spectrum of the operator \(\mathscr{L}\), defined by 3 , admits a representation \[\begin{gather} \sigma(\mathscr{L})=[0,t_{n+1}]\setminus\left( \bigcup_{j=1}^n (t_j,s_j)\right), \end{gather}\] where \(s_i\), \(j=1,\dots,n\), \(t_j\), \(j=1,\dots,n+1\), are exactly the numbers given above.
Proof. (i) By Lemma 2, the numbers \(a_j\) satisfy 42 provided 27 holds with \(B=4b_0=\sum_{j=0}^{n} (t_{j+1}-s_j),\) that is, \[\begin{gather} \sum_{j=1}^n t_j < \sum_{j=0}^{n} (t_{j+1}-s_j) + \sum_{j=1}^{n-1} s_j. \end{gather}\] Since \(s_0=0\), the above inequality is equivalent to \(s_n<t_{n+1}\), which indeed holds by 39 . Hence, the above choice of \(a_j\), \(j=1,\dots,n\), is admissible: all \(a_j\) have the property 42 , in particular, they are positive and satisfy 5 .
(ii) It follows from 39 and already established 42 that \[\mathop{\mathrm{sgn}}(s_i-a_j)=\mathop{\mathrm{sgn}}(a_i-a_j) \quad \text{for all } i\neq j, \qquad \mathop{\mathrm{sgn}}(s_j-a_j)=1.\] Hence, the numbers \(b_j\), \(j=1,\dots,n\), defined by 43 are positive.
Regarding 44 as a system of \(n\) linear algebraic equations for the unknowns \(b_j\), \(j=1,\dots,n\), it remains to show that its solution is given by 43 .
We prove the above statement by induction. For \(n=1\), the claim is obvious. Assume that it has been established for \(n=N-1\). We now prove it for \(n=N\). Multiplying the \(i\)-th equation in 44 (\(i=1,\dots,N\)) by \(a_N-s_i\) and then subtracting the \(N\)-th equation from the first \(N-1\) equations we arrive at the new system of \(N-1\) equations \[\begin{gather} \sum_{j=1}^{N-1}{\widehat b_j\over a_j-s_i}+1,\quad i=1,\dots, N-1, \end{gather}\] where the new unknowns \(\widehat b_j\), \(j=1,\dots, N-1\) are expressed in terms of \(b_j\) as follows, \[\begin{gather} \label{new} \widehat b_j=b _j\displaystyle{a_N-a_j\over s_N-a_j},\quad j=1,\dots, N-1 \end{gather}\tag{45}\] The system 45 coincides with the system 44 for \(n=N-1\). Therefore, by the induction hypothesis, we obtain \[\begin{gather} \label{syst95sol95ind} \widehat b_j= (s_j-a_j)\prod_{\substack{i=1\\i\ne j}}^{N-1}\dfrac{s_i-a_j}{a_i-a_j},\qquad j=1,\dots,N-1. \end{gather}\tag{46}\] Combining 45 and 46 , we conclude that \(b_j\), \(j=1,\dots,N-1\), satisfy 43 . The validity of 43 for \(b_N\) follows from the symmetry of the system 44 .
(iii) By Theorem 1 (taking into account 4 ) we have \[\sigma(\mathscr{L})=\displaystyle\bigcup_{j=1}^{n+1}[\widetilde{s}_{j-1},\widetilde{t}_{j}],\] where \(\widetilde{s}_0=0\), and \(\widetilde{s}_1< \dots<\widetilde{s}_{n}\) satisfy \[\label{wtsj:eq} \sum_{j=1}^n\dfrac{b_j}{a_j-\widetilde{s}_i}+1 =0,\quad i=1,\dots,n,\tag{47}\] while \(\widetilde{t}_1< \dots<\widetilde{t}_{n+1}\) satisfy \[\widetilde{t}_i\left(\sum_{j=1}^n\dfrac{b_j}{a_j-\widetilde{t}_i}+1\right)=4b_0,\quad i=1,\dots,n+1.\]
We define the polynomial \(\widetilde{Q}(x)\) of degree \(n\) by 41 , but with \(\widetilde{s}_j\), \(\widetilde{t}_j\) instead of \(s_j\), \(t_j\). respectively. We also introduce two polynomials of degree \(n+1\), \[T(x)\mathrel{\vcenter{:}}= Q(x)+\dfrac{P(x)}{4b_0}\text{\quad and\quad } \widetilde{T}(x)\mathrel{\vcenter{:}}=\widetilde{Q}(x)+\dfrac{\widetilde{P}(x)}{4b_0},\] where \(P(x)\mathrel{\vcenter{:}}=\prod_{j=0}^n (s_j-x)\) and \(\widetilde{P}(x)\mathrel{\vcenter{:}}=\prod_{j=0}^n (\widetilde{s}_j-x)\). The numbers \(t_j\), \(j=1,\dots,n\) (respectively, \(\widetilde{t}_j\), \(j=1,\dots,n\)) are the roots of \(T(x)\) (respectively, \(\widetilde{T}(x)\)); this follows immediately by substituting these numbers into the corresponding polynomials, see Remark 3. Furthermore, the number \(\widetilde{t}_{n+1}\) is also the root of \(\widetilde{T}(x)\) – see the beginning of the proof of Part (ii) of Lemma 3.
Using 44 (resp. 47 ), we conclude that \(s_j\) (resp. \(\widetilde{s}_j\)) are all \(n+1\) roots of the rational function \(F(x)\) 6 (see Figure 2). Hence, since \(s_j\) and \(\widetilde{s}_j\) are enumerated in ascending order, we conclude that \[\label{s:wts} s_j = \widetilde{s}_j,\quad j=1,\dots,n.\tag{48}\] Recall the numbers \(a_j\) are the roots of the polynomial \(Q(x)\) 41 and (see 42 ) they are pairwise distinct. By Lemma 3(i) \(a_j\) are also the roots of the polynomial \(\widetilde{Q}(x)\). Both \(Q\) and \(\widetilde{Q}\) are of degree \(n\) and their leading coefficient is \((-1)^{n}\). Hence the polynomials \(Q(x)\) and \(\widetilde{Q}(x)\) do coincide; consequently, by 48 , we have \[T(x)=\widetilde{T}(x),\quad \forall x,\] whence \[\begin{gather} \label{inclusion} \{t_1<\dots<t_n\}\subset \{\widetilde{t}_1<\dots<\widetilde{t}_n<\widetilde{t}_{n+1}\}. \end{gather}\tag{49}\] However, since \(\widetilde{t}_j<a_n\) for \(j=1,\dots,n\) and \(\widetilde{t}_{n+1}>a_n\) (in fact, 8 holds with \(\widetilde{t}_j\) in place of \(t_j\)), while, by 42 , we have \(t_j<a_n\) for \(j=1,\dots,n\), it follows from 49 that \[\label{t:wtt} t_j = \widetilde{t}_j,\quad j=1,\dots,n.\tag{50}\]
Finally, using Lemma 3(ii) and 48 , 50 , we get \[\begin{gather} \label{T:wtT} \widetilde{t}_{n+1}= 4b_0+\sum_{i=1}^{n}(\widetilde{s}_i-\widetilde{t}_i)= 4b_0+\sum_{i=1}^{n}( s_i- t_i)=t_{n+1}, \end{gather}\tag{51}\] where on the last step we substitute the expression 40 for \(b_0\) and use the fact that \(s_0=0\).
The desired result follows from 48 , 50 , 51 . The theorem is proven. ◻
Remark 5. It follows immediately from the theorem above and its proof that, given the graph \(\mathbb{Z}\) equipped with the constant positive edge weight \(\mu_0\) and vertex weight \(m_0\equiv 1\), one can “decorate” it by attaching periodically arranged “bristles” with suitably chosen weights so that the spectrum of the Laplacian on the resulting weighted graph possesses \(n\) gaps, which coincide with the prescribed intervals \((t_j,s_j)\). The upper edge of the spectrum of this operator will be given by \[t_{j+1}=4 {\mu_0}+\sum_{j=1}^{n}(s_j-t_j).\]
Remark 6. At the end of this section, we compare our results with the one obtained in [15] for quantum graphs. In that paper, a \(\mathbb{Z}^n\)-periodic metric graph \(\Gamma\) and a family of Hamiltonians \(\mathscr{A}_\varepsilon\) on \(\Gamma\) acting as \(-\frac{1}{\varepsilon}\frac{\mathrm{d}^2}{\mathrm{d} x^2}\) on the edges of \(\Gamma\), subject to appropriate (the so-called \(\delta'\)-type) conditions at the vertices is considered; where \(\varepsilon>0\) is a small parameter. It was shown that, as \(\varepsilon\to0\), the spectrum of \(\mathscr A_\varepsilon\) possesses at least \(n\) gaps (\(n\in\mathbb{N}\) being prescribed in advance). The first \(n\) gaps converge as \(\varepsilon\to 0\) to certain intervals whose location can be effectively controlled by an appropriate choice of the geometry of \(\Gamma\) (resembling the decorated graphs from [13], [14]) and of the coupling constants appearing in the interface conditions. Any remaining gaps escape to infinity as \(\varepsilon\to0\).
The inverse problem solved in [15] is considerably simpler than the one studied in the present work. The right endpoints of the gaps are given by the roots of an 6 -type function \(F(x)\), whereas the left endpoints are expressed explicitly in terms of those coupling constants. Thus, the left gap edges can be controlled very easily, while controlling the right edges requires solving an 44 -type system. This is substantially simpler than the situation in the present paper, where the right edges are determined by the equation 7 .
Finally, we note that the result of [15] was further refined in [17]. In particular, it was shown how to ensure, for fixed (sufficiently small) \(\varepsilon\), the exact coincidence of the left endpoints of the first \(n\) spectral gaps with prescribed values, however, the coupling constants appearing in the interface conditions are not determined explicitly in this approach.
6 Acknowledgements↩︎
A. K. is grateful to Excellence Project FoS UHK 2204/2025-2026 for the support.