inverse spectral problem for the Schrödinger operator on the square lattice

Recently, there have been a lot of studies on quantum graphs, which are one-dimensional Schrödinger (Sturm-Liouville) operators $-\frac{d^{2}}{dz^{2}}+q_{e}(z)$ acting on the edges of a metric graph, while some matching conditions are imposed at the graph vertices. Such operators are used for modeling various processes on graph-like structures in physics, mechanics, chemistry, and other applications. Expositions of spectral theory results for quantum graphs can be found, e.g., in the monographs [13, 8, 9, 12] and references therein.

This paper is mostly focused on inverse spectral and scattering problems. Such problems consist in the reconstruction of unknown operator characteristics from spectral information. Till now, inverse problems have been studied for several types of quantum graphs. Reconstruction of differential operators on compact graphs has been investigated in [6, 7, 10, 16, 23, 24, 28, 29, 30] and other studies. Inverse spectral-scattering problems on non-compact graphs with finite and infinite edges were solved, e.g., in [11, 17, 27].

In recent years, Schrödinger operators on infinite periodic graphs have attracted considerable attention of scholars in connection with applications in material studies and nanotechnology (see [22, 20] and references therein). Spectral properties of such operators were studied in [22, 20, 21, 25, 26] and other papers. We also mention that in [14] some spectral theory issues were considered for infinite quantum graphs of general structure (not necessarily periodic). However, to the best of the authors’ knowledge, there were no studies on inverse problems for differential Schrödinger operators on periodic graphs except for the two preprints [4, 5] of Ando et al. Let us discuss their results in more detail.

The two preprints [4, 5] present the same results on the inverse scattering for the Schrödinger operator on the hexagonal lattice with finitely supported potential. The authors of [4, 5] have shown that, in this case, the scattering matrix (S-matrix) uniquely determines the Dirichlet-to-Neumann (D-N) map for a boundary value problem on a finite part of the graph. Furthermore, the differential (continuous) operator was reduced to a difference (discrete) one and the potentials on the graph edges were reconstructed from the D-N map. Thus, it has been shown that the S-matrix uniquely specifies a finitely supported potential on the hexagonal lattice. The preprints [4, 5] continue the previous studies of their authors [1, 2, 3, 18, 19], in which analogous ideas and methods were developed for the inverse scattering on discrete periodic graphs. The difference between [4] and [5] is that, in [5], the preliminary steps up to determining the D-N map by the S-matrix are implemented for the hexagonal lattice, while in [4] they are provided for a general periodic lattice. This opens a perspective of studying inverse spectral problems for different types of periodic graphs.

This paper is concerned with a family of one-dimensional Schrödinger operators $-\frac{d^{2}}{dz^{2}}+q_{e}(z)$ defined on the edges of the square lattice as in Figure 1, assuming the Kirchhoff conditions at the vertices (the details are given in Section 2). Here, $z$ varies over the interval $(0,1)$ and $e\in E$, where $E$ is the set of all edges of the square lattice.

Let us impose the following assumptions on the potentials.

(Q-1) $q_{e}(z)$ is real-valued, and $q_{e}\in L^{2}(0,1)$.

(Q-2) $q_{e}(z)=0$ on $(0,1)$ except for a finite number of edges.

(Q-3) $q_{e}(z)=q_{e}(1-z)$ for $z\in(0,1)$.

Since the support of the potential $q$ is finite, we can choose a sufficiently large square domain $D_{N}:=\{n_{1}+\mathrm{i}n_{2}\colon 0\leq n_{1},n_{2}\leq N\}$ such that $\mbox{supp}\,q$ is located inside of $D_{N}$ and consider the edge Dirichlet-to-Neumann map $\Lambda_{E}$ associated with this domain (see Section 2 for details). This paper is devoted to the following inverse spectral problem.

The main results of this paper are the uniqueness theorem (Theorem 2.1) for Inverse Problem 1.1 and the reconstruction procedure in Section 5. For solving Inverse Problem 1.1, we use the relation between the edge D-N map and the vertex D-N map, which is defined in Section 3. In other words, we reduce the continuous problem to the discrete one. Consequently, we develop a constructive algorithm for the recovery of the potentials $q_{e}$ from the vertex D-N map. Note that, although we follow the general idea of Ando et al, our reconstruction procedure is different from [4, 5]. First, our algorithm is based on specific properties of the square lattice. Second, the authors of [4, 5] apply the strategy of recovering the potentials along any zigzag line, which perfectly works for discrete operators with constant coefficients [3, 19] but causes difficulties for the reduced vertex Laplacian whose coefficients depend on the unknown potentials (see Remark 5.1 for details). In this paper, we step-by-step recover $q_{e}$ together with the Laplacian coefficients required at the next steps. For reconstruction of the potential on each fixed edge, we use the classical results of the inverse spectral theory for the Schrödinger operators on finite intervals (see, e.g., [15]). Finally, our results imply the uniqueness for solution of the inverse scattering problem by the S-matrix on the square lattice.

It is worth mentioning that Inverse Problem 1.1 can be treated as the inverse spectral problem on a finite metric graph by the Weyl matrix associated to the boundary vertices. However, our results are novel in this direction. It is well-known the the Weyl matrix uniquely specifies the Schrödinger operator on a tree-graph (see [7, 28]). But, for graphs with cycles, this is not the case in general. Even for the simplest graphs with loops additional data are required (see [24, 30]). Our paper provides a new class of finite quantum graphs whose potentials are uniquely determined by the Weyl matrix. Here, the symmetry of the potentials (Q-3) is crucial.

The paper is organized as follows. In Section 2, we give the definitions related to the edge Schrödinger operator and the edge D-N map. In Section 3, we define the reduced vertex Laplacian and the vertex D-N map, and study the relation between the edge D-N map and the vertex D-N map. In Section 4, some auxiliary solutions of the vertex Schrödinger equation are obtained. In Section 5, we reconstruct the potentials from the D-N map. In Section 6, the inverse scattering by the S-matrix is discussed.

Let us define an infinite square lattice with the vertex set

and the edge set

In other words, we consider the vertices as points on the complex plane $\mathbb{C}$ and suppose that two vertices are joined by an edge if the distance between them equals $1$. For two vertices $w,v\in V$, the notation $w\sim v$ means that there exists an edge $e\in E$ such that $v,w$ are end points of $e$.

Let each edge $e$ be endowed with arclength metric and identified with the interval $(0,1):e=\{(1-z)e(0)+ze(1)\colon 0\leqslant z\leqslant 1\}$, where $e(0),e(1)\in V$. Put

Let $\mathcal{E}\subseteq E$. Then, we call $f=\{f_{e}\}_{e\in\mathcal{E}}$ a function on $\mathcal{E}$ if each $f_{e}$ ($e\in\mathcal{E}$) is a function on $[0,1]$. We will write that $f=\{f_{e}\}_{e\in\mathcal{E}}\in\mathcal{A}(E)$ if $f_{e}\in\mathcal{A}[0,1]$ for all $e\in\mathcal{E}$, where $\mathcal{A}$ is any functional class, e.g., $\mathcal{A}=C,L^{2}$, etc. For $v\in V$, $e\in E_{v}\cap\mathcal{E}$, and $f=\{f_{e}\}_{e\in\mathcal{E}}$, introduce the notations

In other words, $f(v,e)$ is the value of the function $f$ and $\partial_{e}f(v)$, of its derivative in the vertex $v$ along the edge $e$.

Let $v\in V$ be a vertex such that $E_{v}\subseteq\mathcal{E}$. Then, we say that $f=\{f_{e}\}_{e\in\mathcal{E}}$ satisfies the Kirchhoff conditions at $v$ if

(K-1) $f$ is continuous at $v$, that is, $f(v,e_{1})=f(v,e_{2})$ for any $e_{1},e_{2}\in E_{v}$. In this case, we denote $f(v)=f(v,e)$, $e\in E_{v}$.

(K-2) $f\in C^{1}(E_{v})$ and $\sum_{e\in E_{v}}\partial_{e}f(v)=0$.

We will also use the notation $f(v)=f(v,e)$ if $e$ is the only edge in $\mathcal{E}$ incident to the vertex $v$, that is, $\mathcal{E}\cap E_{v}=\{e\}$.

Let us define the Schrödinger operator on a finite subgraph of the square lattice. Suppose that the potential $q=\{q_{e}\}_{e\in E}$ satisfies the assumptions (Q-1), (Q-2), and (Q-3). Let $\mathring{\Omega}$ be the vertex set of some connected finite subgraph of the lattice $(V,E)$. Denote

Thus, $E_{\Omega}$ is the edge set of the subgraph with the vertices $\Omega$. The notations $\mathring{\Omega}$, $\partial\Omega$, $\mathring{E}_{\Omega}$, and $\partial E_{\Omega}$ are used for the interior vertices, the boundary vertices, the interior edges, and the boundary edges, respectively, of the subgraph $(\Omega,E_{\Omega})$.

Denote by $\Delta_{E}$ and call the edge Laplacian the second derivative operation along each edge:

For the subgraph $(\Omega,E_{\Omega})$, consider the Dirichlet boundary value problem for the edge Laplacian

where a function $u=\{u_{e}\}_{e\in E_{\Omega}}\in H^{2}(E_{\Omega})$ is assumed to fulfill the Kirchhoff conditions at every interior vertex $v\in\mathring{\Omega}$ and $f=\{f(v)\}_{v\in\partial\Omega}$ is a function given on the set of the boundary vertices.

Denote by $(-\Delta_{E,\Omega}+q_{E,\Omega})$ the operator acting by the rule $(-\Delta_{E}+q_{e})u_{e}$, whose domain $D(-\Delta_{E,\Omega}+q_{E,\Omega})$ is the set of all the functions $u=\{u_{e}\}_{e\in E_{\Omega}}\in H^{2}(E_{\Omega})$ satisfying the Dirichlet condition $u(v)=0$ at any boundary vertex $v\in\partial\Omega$ and the Kirchhoff conditions at any interior vertex $v\in\mathring{\Omega}$. By the standard argument, the operator $(-\Delta_{E,\Omega}+q_{E,\Omega})$ is self-adjoint and its spectrum $\sigma(-\Delta_{E,\Omega}+q_{E,\Omega})$ is a countable set of real eigenvalues. Below, we assume that

The edge Dirichlet-to-Neumann (D-N) map $\Lambda_{E,\Omega}(\lambda)$ is defined as follows:

where $u$ is the solution of the boundary value problem (2) for the boundary data $f$ and $e$ is the edge of $E_{\Omega}$ having $v$ as its end point. Below we assume that the vertex set $\Omega$ is fixed and use the short notation $\Lambda_{E}:=\Lambda_{E,\Omega}$. Note that $f=\{f(v)\}_{v\in\partial\Omega}$ can be treated as a vector of $\mathbb{C}^{M}$, $M:=|\partial\Omega|$, and $\Lambda_{E}(\lambda)$, as the $(M\times M)$ matrix function such that $\Lambda_{E}(\lambda)f=g$, $g=\left\{\frac{d}{dz}u_{e}(v)\right\}_{v\in\partial\Omega}\in\mathbb{C}^{M}$. The matrix function $\Lambda_{E}(\lambda)$ is analytic in $\lambda$ satisfying (3).

For simplicity, we consider the domain

where a natural number $N$ is chosen so large that $\mbox{supp}\,q\subseteq\mathring{E}_{\Omega}$. In other words, $q_{e}=0$ on all the edges except $e\in\mathring{E}_{\Omega}$. In Section 6, we show that the D-N map is uniquely specified by the S-matrix and vice versa, so the form of the domain is actually unimportant.

Consider Inverse Problem 1.1 for the domain $\Omega$. In fact, we have to find the potential on $\mathring{E}_{\Omega}$, since on all the other edges $q_{e}=0$. Our main result is the following uniqueness theorem.

The proof of Theorem 2.1 is based on the reduction of the edge Laplacian to the vertex one.

Consider the Schrödinger equation

on each fixed edge $e\in E$. Let $\phi_{e0}(z,\lambda),\phi_{e1}(z,\lambda)$ be the solutions of (5) with the initial data $\phi_{e0}(0,\lambda)=0,\,\phi_{e0}^{\prime}(0,\lambda)=1$ and $\phi_{e1}(1,\lambda)=0,\,\phi_{e1}^{\prime}(1,\lambda)=-1$, respectively.

Denote by $H_{e}$ the Schrödinger operator $\left(-\frac{d^{2}}{dz^{2}}+q_{e}\right)$ with the domain $D(H_{e})$ which consists of functions $u\in H^{2}[0,1]$ satisfying the Dirichlet conditions $u(0)=u(1)=0$. It is well-known that the operator $H_{e}$ is self-adjoint and its spectrum is a countable set of real eigenvalues. In the following, we assume that $\lambda\not\in\sigma(H_{e})$, $e\in E$. This guarantees that $\phi_{e0}(1,\lambda)\neq 0$ and $\phi_{e1}(0,\lambda)\neq 0$.

If $w,v\in V$ are two end points of an edge $e\in E$, then we denote

Note that, by the assumption (Q-3), we have $\phi_{e0}(z,\lambda)=\phi_{e1}(1-z,\lambda)$, hence $\psi_{wv}(1,\lambda)=\psi_{vw}(1,\lambda)$. In particular, if $q_{e}=0$, then $\psi_{wv}(z,\lambda)=\frac{\sin\sqrt{\lambda}z}{\sqrt{\lambda}}$.

We define the reduced vertex Laplacian $\Delta_{V,\lambda}$ on $V$ by

for $u\in L^{2}_{loc}(V)$. We also define the scalar multiplication operator:

Consider the vertex set $\Omega$ defined by (1) for $\mathring{\Omega}=D_{N}$ and the interior boundary value problem

By virtue of Lemma 3.1, if a function $u=\{u_{e}\}_{e\in E_{\Omega}}$ solves the edge boundary value problem (2), then its values $\{u(v)\}_{v\in\Omega}$ in the vertices satisfy (10).

Define the vertex degree in the subgraph $(\Omega,E_{\Omega})$ as follows:

Then, the vertex D-N map is defined by

Below we use a shorter notation $\Lambda_{V}:=\Lambda_{V,\Omega}$. Note that, for the region $\Omega=D_{N}$, the degrees $\mbox{deg}_{\Omega}(v)$ of the boundary vertices $v\in\partial\Omega$ equal to $1$. Hence

The edge and the vertex D-N maps are closely related to each other, which is shown in the following lemma.

Therefore, Inverse Problem 1.1 of recovering the potential from the edge D-N map can be easily reduced to the following inverse problem.

We are now in a position to construct the solution of Inverse Problem 3.1. For this purpose, we first obtain some special solutions of the vertex Schrödinger equation

The boundary $\partial\Omega$ of the domain $\mathring{\Omega}=D_{N}$ consists of the four parts, see Figure 3:

Recall that $\Lambda_{V}=\Lambda_{V,\Omega}$ is the vertex D-N map defined by (11). The key to the inverse procedure is the following partial data problem.

Now, for $N+1\leqslant k\leqslant 2N$, let us consider the diagonal line

The vertices on $A_{k}\cap\Omega$ are written as

where $\alpha_{k,0}=t_{k-(N+1)}=k-(N+1)+\mathrm{i}(N+1)$.

An important feature of the solution $u$ in Lemma 4.2 is that $u$ vanishes below the line $A_{k}$. For such solutions, we obtain the following property.

Let $u$ be a solution of the equation

which vanishes in the region $x_{1}+x_{2}<k$. Let $a,b,b^{\prime},c,d\in V$ and $e,e^{\prime}\in E$ be as in Figure 4.

Then, evaluating the equation (29) at $v=a$ and using (6), (7), we obtain

The special solutions of Lemma 4.2 and their property (30) play a crucial role in the proof of the main theorem and in the reconstruction procedure in the next section.

The goal of this section is to prove Theorem 2.1 on the uniqueness of the inverse spectral problem solution. The proof consists in a reconstruction procedure, which is based on the reduction to the discrete inverse problem by the vertex D-N map (i.e. Inverse Problem 3.1), on specific properties of the square lattice, and on applying the classical results for the recovery of the Schrödinger potential on each edge.

Before proceeding to the reconstruction, we provide two well-known results for inverse spectral problems on a finite interval. Let $e=(w,v)$ be a fixed edge. Clearly, the eigenvalues of the Dirichlet problem

coincide with the zeros of the characteristic function $\psi_{wv}(1,\lambda)$.

The both inverse problems of Lemmas 5.1 and 5.2 can be solved constructively by the well-known methods, e.g., the Gelfand-Levitan method and the method of spectral mappings (see [15]).

Now, let us prove Theorem 2.1 by providing a reconstruction procedure for solving Inverse Problem 1.1.

Reconstruction procedure. Suppose that the potential $q$ on the square lattice fulfills the conditions (Q-1), (Q-2), and (Q-3). Let $\Omega$ be the square region defined in Section 2 and let the corresponding edge D-N map $\Lambda_{E}$ be given.

Step 1. We obtain the vertex D-N map $\Lambda_{V}$ by the formula (12).

Step 2. For $k=2N,2N-1,\dots,N+1$, implement the following steps 3–5.

Step 3. For the value of $k$ fixed at step 2, draw the line $A_{k}$ defined by (24) and take the boundary data $f$ having the properties in Lemma 4.2. Under the assumption (Q-2), we have $q_{e}(z)=0$ on all the boundary edges $\partial E_{\Omega}$. So, we know the functions $\psi_{wv}(1,\lambda)=\frac{\sin\sqrt{\lambda}}{\sqrt{\lambda}}$ for $(w,v)\in\partial E_{\Omega}$. By Lemma 4.1 (2), we can find $u$ on $(\partial\Omega)_{R}$ with the vertex D-N map $\Lambda_{V}$. Thus, we know $u$ on $\partial\Omega$.

Step 4. If $k=2N$, using (11) and the value of $u(\alpha_{k,1}+\mathrm{i})$, we find

By (30), we have