Abstract. This paper deals with the Sturm-Liouville operators with distribution potentials of the space \(W_2^{-1}\) on a metric tree. We study an inverse spectral problem that consists in the recovery of the potentials from the characteristic functions related to various boundary conditions. We prove the uniform stability of this inverse problem for potentials in a ball of any fixed radius, as well as the local stability under small perturbations of the spectral data. Our approach is based on a stable algorithm for the unique reconstruction of the potentials relying on the ideas of the method of spectral mappings.

Keywords: inverse Sturm-Liouville problem; quantum tree; stability; method of spectral mappings.

AMS Mathematics Subject Classification (2020): 34A55 34B09 34B45 34L40

1 Introduction↩︎

This paper aims to prove the stability of the inverse Sturm-Liouville problem on a metric tree (i.e. graph without cycles). Inverse problems of spectral analysis consist in the reconstruction of differential operators from their spectral characteristics. The basic results of the inverse spectral theory have been obtained for the Sturm-Liouville operators \(-y'' + q(x)y\) on finite and infinite intervals (see the monographs [1]–[4] and references therein).

Differential operators on graphs, also called quantum graphs, are given by the three components: (i) a metric graph, (ii) differential expressions on the graph edges, and (iii) matching conditions at the graph vertices. Spectral problems for such operators arise in numerous applications including organic chemistry, mesoscopic physics and nanotechnology, the theories of photonic crystals, waveguides, quantum chaos (see [5]–[7] and references therein).

Inverse Sturm-Liouville problems that consist in the reconstruction of the potentials on the edges of quantum trees have been investigated in [8]–[14] and other studies. In particular, Belishev [8] was the first to prove the uniqueness for solution of an inverse problem on an arbitrary tree. He used the eigenvalues and the derivatives of the eigenfunctions at the boundary vertices as the spectral data. Brown and Weikard [9] showed that potentials are uniquely determined by the Dirichlet-to-Neumann map (i.e. the Weyl matrix), which is equivalent to the spectral data in [8]. Yurko [10] has obtained the uniqueness by the minimal amount of data consisting only of the diagonal elements of the Weyl matrix. Moreover, Yurko has developed a constructive approach to solving inverse spectral problems on graphs, based on the method of spectral mappings [15]. Later, this approach was extended to arbitrary compact graphs [16], as well as to a wide class of noncompact graphs [17]. A more detailed overview of inverse spectral theory for quantum graphs can be found in [18], [19]. Among recent advances, we mention the study [20] of the inverse spectral problem for a nonlocal functional-differential operator on a graph and the numerical reconstruction of potentials on quantum trees in [21]. Thus, although constructive procedures for solving inverse problems on graphs are well-understood, there is a lack of stability analysis. In the recent study [22], the author has investigated the stability of the inverse Sturm-Liouville problem on a simple graph with a loop. This paper aims to fill the gap for arbitrary tree graphs, whose geometrical structure is definitely more complicated.

In this paper, we consider the Sturm-Liouville problems with distribution potentials of the class \(W_2^{-1}\) on a metric tree. This class of potentials is wider than the usual \(L_1\). In particular, the space \(W_2^{-1}\) contains perturbations by the Dirac delta-functions and the Coulomb-type singularities \(\frac{1}{x}\), which arise in quantum mechanics (see [23]). Four approaches to defining Sturm-Liouville operators with distribution potentials are described in [24]. The Sturm-Lioville operators with \(W_2^1\)-potentials on metric graphs were considered in [25] and inverse problems for such operators, in [14], [26]–[29]. Basing on the ideas of those studies, we develop a stable algorithm for the unique reconstruction of the potential from spectral data on a metric tree of arbitrary structure. Moreover, we prove the uniform stability of the inverse problem for potentials in a ball of any fixed radius.

An important role in our investigation is played by analytical characteristic functions, whose zeros coincide with eigenvalues of various Sturm-Liouville problems on the whole tree and on its subtrees. As spectral data, we use characteristic functions related to quantum trees with various boundary conditions. Our problem statement generalizes the two-spectra inverse Sturm-Liouville problem by Borg [30] and the inverse problem by Yurko [10] on a quantum tree with regular potentials. Our stability analysis relies on the Lipschitz continuity of the characteristic functions on the tree with respect to the potentials in suitable norms. This auxiliary result is obtained by using transformation operators that were constructed in [31]–[33] for the solutions of the Sturm-Liouville equation with \(W_2^{-1}\)-potential.

Our reconstruction algorithm is based on the following two auxiliary steps: (i) recovery of the potential on a boundary edge, (ii) transition through a vertex, which allows us to pass to internal edges. For step (i), using contour integration, we derive a new reconstruction formula to find distribution potentials on the tree edges and apply it to obtain both uniform and local stability estimates. At step (ii), in contrast to previous studies, we choose a countable set of points in the spectral plane and use sampling to obtain a reconstruction procedure that is valid under any sufficiently small perturbation of the spectral data in a certain space. As a result, we obtain the uniform stability, as well as local stability of the inverse Sturm-Liouville problem on an arbitrary metric tree.

The paper is organized as follows. In Section 2, we state eigenvalue problems on a metric tree and introduce the notation. In Section 3, we describe the construction and some properties of characteristic functions for the Sturm-Liouville problems on the tree. Section 4 contains the formulations of the main results and briefly outlines their proofs. In Section 5, the uniform stability of recovering the potential on a boundary edge is proved. In Section 6, we investigate an auxiliary problem that is related to the transition through a vertex. In Section 7, we develop a stable algorithm for the unique reconstruction of the potentials on the whole tree and prove the corresponding local theorem. Appendix contains the proof of the Lipschitz continuity of the transformation operator kernel with respect to the distributional potential. That auxiliary result has independent significance.

2 Preliminaries↩︎

Denote by \(G\) the metric tree that consists of vertices \(\{ v_j \}_1^{m+1}\) and edges \(\{ e_j \}_1^m\) (\(m \ge 1\)). Assume that each edge \(e_j\) (\(j = \overline{1,m}\)) goes from the vertex \(v_j\) to its parent \(v_{p(j)}\), \(j < p(j) \le m+1\). The vertex \(v_{m+1}\) is the root, it has no outgoing edges. Furthermore, assume that \(v_{m+1}\) has the only incoming edge \(e_m = [v_m, v_{m+1}]\) (see the example in Fig. 1). For each vertex \(v\), denote by \(E_v\) the set of all incoming and outgoing edges incident to \(v\). The size of \(E_v\) is called the degree of \(v\). We call \(v\) a boundary vertex if it has degree \(1\) and an internal vertex otherwise. Denote the sets of the boundary vertices and of the internal vertices of the tree \(G\) by \(\partial G\) and \(int \, G\), respectively. Since the root \(v_{m+1}\) belongs to \(\partial G\), we introduce the set \(\partial G' := \partial G \setminus v_{m+1}\).

Let each edge \(e_j\) have length \(T_j > 0\) and be parameterized by \(x_j \in [0,T_j]\) so that \(x_j = 0\) corresponds to the vertex \(v_j\) and \(x_j = T_j\), to \(v_{p(j)}\). A function on the tree \(G\) is a vector \(y = \{ y_j \}_1^m\) of complex-valued functions \(y_j = y_j(x_j)\), \(x_j \in (0,T_j)\), \(j = \overline{1,m}\).

Consider the Sturm-Liouville equations on the tree \(G\): \[\label{eqv} -(y_j^{[1]})' - \sigma_j(x_j) y_j^{[1]} - \sigma_j^2(x_j) y_j = \lambda y_j, \quad x_j \in (0,T_j), \quad j = \overline{1,m},\tag{1}\] where \(\sigma = \{ \sigma_j \}_1^m \in L_2(G)\), \(y_j^{[1]} := y_j' - \sigma_j y_j\) are the quasi-derivatives, and \(\lambda\) is the spectral parameter. We call \(\sigma\) the potential on the graph \(G\) and \(\sigma_j\) the potentials on the corresponding edges \(e_j\) (\(j = \overline{1,m}\)). Equations 1 are considered for functions \(y_j\) such that \(y_j, y_j^{[1]} \in AC[0,T_j]\), \(j = \overline{1,m}\). It is worth mentioning that equations 1 are equivalent to the Schrödinger equations \[\label{schr} -y_j''(x_j) + q_j(x_j) y_j(x_j) = \lambda y_j(x_j), \quad x_j \in (0,T_j),\tag{2}\] with distributional potentials \(q_j \in W_2^{-1}(0,T_j)\) (see [24]).

For a function \(y_j\) on the edge \(e_j = [v_j, v_{p(j)}]\), introduce the notations \[\begin{cases} y_j(v_j) = y_j(0), \quad & y_j(v_{p(j)}) = y_j(T_j), \\ y_j^{[1]}(v_j) = -y_j^{[1]}(0), \quad & y_j^{[1]}(v_{p(j)}) = y_j^{[1]}(T_j). \end{cases}\] Thus, \(y_j(v)\) is the value of a function on the tree in the vertex \(v\) and \(y_j^{[1]}(v)\) is the “normal quasi-derivative”, which is taken in the outgoing direction from the interval \((0, T_j)\). For boundary vertices \(v \in \partial G\), we omit the index \(j\) and write \(y(v)\), \(y^{[1]}(v)\).

This paper focuses on the boundary value problems \(\mathcal{L}\) and \(\mathcal{L}_k\) (\(v_k \in \partial G'\)) for the Sturm-Liouville system 1 with the matching conditions \[\label{mc} \begin{cases} y_j(v) = y_k(v), \quad e_j, e_k \in E_v & \text{(continuity condition)}, \\ \sum\limits_{e_j \in E_v} y_j^{[1]}(v) = 0 & \text{(Kirchhoff's condition)} \end{cases}\tag{3}\] at all the internal vertices \(v \in int\,G\) and the corresponding boundary conditions: \[\begin{align} \tag{4} \mathcal{L} \colon & \quad y(v_j) = 0, \quad v_j \in \partial G, \\ \tag{5} \mathcal{L}_k \colon & \quad y^{[1]}(v_k) = 0, \quad y(v_j) = 0, \quad v_j \in \partial G \setminus v_k. \end{align}\]

Throughout the paper, we use the following notation:

• \(b = |\partial G|\) is the number of the boundary vertices in the tree \(G\).

• \(L_2(G) := \bigl\{ y = \{ y_j \}_1^m \colon y_j \in L_2(0,T_j), \, j = \overline{1,m} \bigr\}\) is the Hilbert space with the scalar product \[(y,z) = \sum_{j = 1}^m \int_0^{T_j} \overline{y_j(x_j)} z_j(x_j) \, dx_j\] and the norm \[\| y \|_{L_2(G)} = \sqrt{\sum_{j = 1}^m \| y_j \|^2_{L_2(0,T_j)}}.\]

• For \(\Omega > 0\), introduce the ball \[\mathcal{B}_{\Omega} := \bigl\{ y = \{ y_j \}_1^m \in L_2(G) \colon \| y \|_{L_2(G)} \le \Omega \bigr\}.\]

• Along with \(\sigma = \{ \sigma_j \}_1^m\), we consider other potentials in \(L_2(G)\): \(\tilde{\sigma} = \{ \tilde{\sigma}_j \}_1^m\), \(\sigma^0 = \{ \sigma_j^0 \}_1^m\), \(\sigma_j^0 \equiv 0\) (\(j = \overline{1,m}\)), and \(\sigma^s = \{ \sigma_j^s \}_{j = 1}^m\) (\(s \ge 1\)). We agree that, if a symbol \(\alpha\) denotes an object related to the potential \(\sigma\), then the notations \(\tilde{\alpha}\), \(\alpha^0\), and \(\alpha^s\) mean the analogous objects related to the potentials \(\tilde{\sigma}\), \(\sigma^0\), and \(\sigma^s\), respectively. Also, \(\hat{\alpha}:= \alpha- \tilde{\alpha}\).

• In estimates, the same symbol \(C\) denotes various positive constants independent of \(\lambda\), \(x\), etc. In proofs, we assume that constants \(C\) depend on the same parameters as in the formulations of the corresponding theorems and lemmas.

• \(\lambda= \rho^2\), \(\mu = \theta^2\).

• \(\mathbf{T} := length(G) = \sum_{j = 1}^m T_j\) is the total length of the graph \(G\).

• For \(T > 0\), we denote by \(PW(T)\) the class of the Paley-Wiener functions of the form \[\label{PW} \mathcal{F}(\rho) = \int_{-T}^T f(t) \exp(i \rho t) \, dt, \quad f \in L_2(-T, T).\tag{6}\]

• The notation \(f(\rho) \asymp g(\rho)\) means that \[\forall \rho \quad c |f(\rho)| \le |g(\rho)| \le C |f(\rho)|,\] where \(c\) and \(C\) are positive constants.

3 Characteristic functions↩︎

In this section, we provide a recursive definition of characteristic functions, whose zeros coincide with the eigenvalues of the corresponding Sturm-Liouville operators on trees. Moreover, we present the asymptotics of the characteristic functions, and study their continuous dependence on the potential \(\sigma\) in suitable norms.

Let \(G\) be a metric tree defined in Section 4, and let \(\sigma = \{ \sigma_j \}_1^m\) be a function on the tree \(G\) of the class \(L_2(G)\). For \(j \in \{ 1, 2, \dots, m \}\), denote by \(\varphi_j(x_j, \lambda)\) and \(S_j(x_j, \lambda)\) the solutions of equation 1 for the fixed \(j\) satisfying the initial conditions: \[\label{ic} \varphi_j(0,\lambda) = S_j^{[1]}(0,\lambda) = 1, \quad \varphi_j^{[1]}(0,\lambda) = S_j(0,\lambda) = 0.\tag{7}\] Note that equation 1 is equivalent to the first-order system \[\frac{d}{dx_j} \begin{bmatrix} y_j \\ y_j^{[1]} \end{bmatrix} = \begin{bmatrix} \sigma_j & 1 \\ -(\sigma_j^2 + \lambda) & -\sigma_j \end{bmatrix} \begin{bmatrix} y_j \\ y_j^{[1]} \end{bmatrix}.\] Therefore, the solutions \(\varphi_j(x_j, \lambda)\) and \(S_j(x_j, \lambda)\) exist and are unique. Moreover, for each fixed \(x_j \in [0,T_j]\), the functions \(\varphi_j(x_j, \lambda)\), \(\varphi^{[1]}_j(x_j, \lambda)\), \(S_j(x_j, \lambda)\), and \(S_j^{[1]}(x_j, \lambda)\) are entire analytic in \(\lambda\).

Suppose that the set of the boundary vertices is divided into two disjoint subsets: \[\partial G = \partial G^D \cup \partial G^N, \quad \partial G^D \cap \partial G^N = \varnothing.\]

Consider the following boundary conditions BC, which are more general than 4 and 5 : \[y(v) = 0, \: v \in \partial G^D, \qquad y^{[1]}(v) = 0, \: v \in \partial G^N.\] Here and below in similar notations, the upper indices \(D\) and \(N\) mean “Dirichlet” and “Neumann”, respectively. The quasi-derivatives on the respective edges are defined by using \(\sigma\).

In the case of the Sturm-Liouville equations with regular potentials, our construction of the characteristic function inductively follows from [34], which implies Proposition 2. For equations 1 , the construction and the proof are analogous. In a number of previous studies (see, e.g., [10], [16], [27]), characteristic functions were constructed as determinants of some linear systems. That definition is equivalent to ours up to a constant multiplier. Anyway, we find the recursive definition of characteristic functions more convenient for investigating their properties.

Proceed to the asymptotics of characteristic functions.

Using Definition 1, Proposition 3, and Lemma 4, we get the following corollary.

Using transformation operators for the solutions \(S_j(x_j, \lambda)\) and \(\varphi_j(x_j, \lambda)\) (see [31], [32]), we obtain the following lemma on the Lipschitz continuity of these solutions and their quasi-derivatives with respect to the potential \(\sigma_j\).

The proof of Lemma 6 is presented in Appendix. Lemma 6 and Definition 1 imply the following corollary.

According to part (i) of Lemma 4, the zeros of \(\Delta^0(\lambda)\) are real and non-negative. Consequently, taking Corollaries 5 and 7 into account, we get the following properties for the zeros of \(\Delta(\lambda)\).

4 Main results and proof strategy↩︎

Denote by \(\Delta(\lambda)\) and \(\Delta_k(\lambda)\) (\(v_k \in \partial G'\)) the characteristic functions of the boundary value problems problems \(\mathcal{L}\) and \(\mathcal{L}_k\), respectively, constructed according to Definition 1. We study the following inverse spectral problem.

Inverse Problem 9 is equivalent to the recovery of the potentials \(\{ \sigma_j \}_1^m\) from the spectra of the eigenvalue problems \(\mathcal{L}\) and \(\mathcal{L}_k\) (\(k = \overline{1,m}\)), and so generalizes the Borg problem [30]. The solution of Inverse Problem 9 is unique, which is proved similarly to [10], [26], [27].

The first result of this paper is the following theorem, which establishes the uniform stability of Inverse Problem 9 for potentials \(\sigma\) in the ball \(\mathcal{B}_{\Omega}\).

Let us outline the proof of Theorem 10. Fix an index \(k\) of a boundary vertex \(v_k \in \partial G'\) and consider the following auxiliary inverse problem on the edge \(e_k\):

First, we prove the uniform stability of Inverse Problem 11:

In order to prove Theorem 12, we derive a new reconstruction formula in Section 5 within the framework of the method of spectral mappings (see [15]).

Next, let \(v_p\) be the parent of \(v_k\): \(p = p(k)\), \(e_k = [v_k, v_p]\). Denote by \(g_p\) the directed subtree of \(G\) with the root \(v_p\) and by \(G_p\) the directed subtree obtained from \(G\) by excluding \(g_p\) (see Fig. 4). Thus \(v_p\) is a boundary vertex in \(G_p\). Denote by \(\Delta_p^D(\lambda)\) and \(\Delta_p^N(\lambda)\) the characteristic functions constructed by Definition 1 for the Sturm-Liouville problems 1 on the subtree \(G_p\) with the boundary conditions \(y_p(0) = 0\) and \(y^{[1]}_p(0) = 0\), respectively, the Dirichlet conditions \(y(v) = 0\) at the other boundary vertices \(v \in \partial G_p \setminus v_p\), and the matching conditions 3 at the internal vertices \(v \in int \, G_p\). Consider the following auxiliary problem.

Denote by \(B_p\) the number of the boundary vertices in \(G_p\). Then, the uniform stability of Auxiliary Problem 13 holds in the following sense:

Theorems 12 and 14 are proved in Sections 5 and 6, respectively. Note that, using the resulting functions \(\Delta_p^D(\lambda)\) and \(\Delta_p^N(\lambda)\) of Auxiliary Problem 13, one can construct the potential \(\sigma_p\) by solving Inverse Problem 11 for the subtree \(G_p\). In this way, starting from the boundary edges, one can step-by-step recover the potentials on all the edges of the tree \(G\). Consequently, Theorems 12 and 14 directly imply Theorem 10.

In addition, we consider the question about local solvability and stability of Inverse Problem 9:

The answer to the existence question in general is negative, since Inverse Problem 9 is overdetermined. Nevertheless, in this paper, we develop a theoretical algorithm for reconstruction (Algorithm 20), which for every fixed \(\sigma \in L_2(G)\) and any sufficiently small perturbation of the spectral data of \(\sigma\) generates the unique approximation \(\tilde{\sigma} \in L_2(G)\) satisfying ?? . Consequently, we establish the local stability of Inverse Problem 9 under the existence assumption for its solution:

Note that, for each fixed \(\sigma \in L_2(G)\) and every \(\varepsilon> 0\), there exist infinitely many functions \(\tilde{\sigma}\) satisfying the condition \(\delta< \varepsilon\) of Theorem 15. This follows from the continuity of the characteristic functions \(\Delta(\lambda)\) and \(\Delta_k(\lambda)\) with respect to \(\sigma\) (see Corollary 7). Taking a small perturbation of \(\sigma\), one can make \(\delta\) arbitrarily small.

5 Stability on a boundary edge↩︎

In this section, we consider Inverse Problem 11 of the recovery \(\sigma_k\) on a boundary edge \(e_k\). Introduce the Weyl function with respect to the vertex \(v_k\) according to [10]: \[\label{defM} M_k(\lambda) := -\frac{\Delta_k(\lambda)}{\Delta(\lambda)}.\tag{9}\]

The uniqueness of recovering the potential \(\sigma_k\) from the Weyl function \(M_k(\lambda)\) is obtained similarly to [26]. The goal of this section is to prove Theorem 12 on the uniform stability. For this purpose, we derive the new reconstruction formula 19 . The novelty of our case consists in the following two features:

1. Behavior of the spectrum for the Sturm-Liouville operator on a graph is complicated. Eigenvalues can be multiple and/or not asymptotically separated. For simple structure of the spectrum, it is convenient to calculate the contour integrals using the Residue Theorem and then operate with infinite series (see, e.g., [32]). In our case, we only work with contour integrals.

2. The potentials \(q_j\) in the Schrödinger-form equation 2 are distributional, which influence the convergence of the contour integrals. In order to obtain the desired reconstruction formula, we approximate \(q_j\) by integrable potentials.

Consider two problems \(\mathcal{L}\) and \(\tilde{\mathcal{L}}\) on the same three \(G\) with potentials \(\sigma = \{ \sigma_j \}_1^m\) and \(\tilde{\sigma} = \{ \tilde{\sigma}_j \}_1^m\) of \(L_2(G)\), respectively.

For a fixed \(\tau > 0\), introduce the contours \[\label{cont} \gamma:= \{ \rho = s + i \tau \colon -\infty < s < \infty \}, \quad \Gamma := \{ \lambda= \rho^2 \colon \rho \in \gamma \}\tag{10}\] with the circuit corresponding to \(s\) going from \(+\infty\) to \(-\infty\). In view of Corollary 8, one can choose \(\tau > 0\) so large that all the zeros of \(\Delta(\lambda)\) and \(\tilde{\Delta}(\lambda)\) lie inside \(\Gamma\) (see Fig. 5).

Let us derive the relation for the difference \(\hat{\sigma}_k(x)\). Begin from the case \[\label{siW21} \sigma_j \in \mathring{W}_2^1[0,T_j], \quad j = \overline{1,m},\tag{11}\] where \[\mathring{W}_2^1[0,T] = \bigl\{ \sigma \in AC[0,T] \colon \sigma' \in L_2(0,T), \, \sigma(0) = \sigma(T) = 0 \bigr\}.\] Then, equations 1 can be represented in the form 2 with \[\label{qL2} q_j = \sigma_j' \in L_2(0,T_j), \quad \int_0^{T_j} q_j(x_j) \, dx_j = 0,\tag{12}\] and \(y_j^{[1]}(x_j) = y_j'(x_j)\) at \(x_j = 0\) and \(x_j = T_j\), \(j = \overline{1,m}\). Thus, the boundary value problem \(\mathcal{L}\) turns into the Sturm-Liouville problem with regular potentials and the standard matching conditions, which has been studied in [10]. In particular, by using the method of spectral mappings (see [10], [15]), the following relation has been obtained: \[\hat{q}_k(x) = \frac{1}{2 \pi i} \int_{\Gamma} \frac{d}{dx}\bigl(S_k(x,\lambda) \tilde{S}_k(x,\lambda)\bigr) \hat{M}_k(\lambda) \, d\lambda.\]

Integration implies \[\label{difsi1} \hat{\sigma}_k(x) = \frac{1}{2 \pi i} \int_{\Gamma} S_k(x,\lambda) \tilde{S}_k(x,\lambda) \hat{M}_k(\lambda) \, d\lambda,\tag{13}\] where the constant of integration is chosen according to the condition \(\sigma_k(0) = \tilde{\sigma}_k(0) = 0\). Note that, for the potentials \(\{ q_j \}_1^m\) and \(\{ \tilde{q}_j \}_1^m\) satisfying the conditions 12 , there hold \[|S_k(x,\rho^2)|, \, |\tilde{S}_k(x,\rho^2)| \le C |\rho|^{-1}, \quad \rho \in \gamma\] and \(\hat{M}_k(\rho^2) \in L_2(\gamma)\) (see [22]). Consequently, the integral \[\label{intga} \int_{\Gamma} S_k(x,\lambda) \tilde{S}_k(x,\lambda) \hat{M}_k(\lambda) \, d\lambda= 2 \int_{\gamma} \rho S_k(x, \rho^2) \tilde{S}_k(x,\rho^2) \hat{M}_k(\rho^2) d \rho\tag{14}\] converges absolutely and uniformly by \(x \in [0,T_k]\).

Proof. Introduce the contours (see Fig. 6): \[\label{defcont} \mathcal{C}_N = \{ \lambda\in \mathbb{C} \colon |\lambda| = R_N \}, \quad c_N = \mathcal{C}_N \cap int \, \Gamma, \quad \Gamma_N := \Gamma \cap int \, \mathcal{C}_N,\tag{15}\] where the radii \(\{ R_N \}\) are chosen so that \(R_N \to +\infty\) as \(N \to \infty\) and \[\label{Dbelow} |\Delta(\lambda)|, \, |\tilde{\Delta}(\lambda)| \ge c |\rho|^{1-b} \exp(|Im \, \rho| \mathbf{T}), \quad c > 0, \quad b = |\partial G|, \quad \lambda= \rho^2 \in \mathcal{C}_N.\tag{16}\]

Taking the condition 12 into account, similarly to Corollary 5, we obtain the asymptotics \[\label{asymptDreg} \begin{cases} \Delta(\rho^2) = \Delta^0(\rho^2) + o\bigl(\rho^{-b} \exp(|Im \, \rho| \mathbf{T})\bigr), \\ \Delta_k(\rho^2) = \Delta_k^0(\rho^2) + o\bigl(\rho^{1-b} \exp(|Im \, \rho| \mathbf{T})\bigr), \end{cases} \quad |\rho| \to \infty.\tag{17}\] Combining 9 , 16 , and 17 implies \[\lim_{N \to \infty} \sup_{\lambda\in \mathcal{C}_N} |\hat{M}_k(\lambda)| = 0.\] Hence \[\label{intCN} \lim_{N \to \infty} \int_{c_N} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda= 0, \quad \lim_{N \to \infty} \oint_{\mathcal{C}_N} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda= 0.\tag{18}\] Consequently \[\int_{\Gamma} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda= \lim_{N \to \infty} \int_{\Gamma_N} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda= \lim_{N \to \infty} \oint_{\Gamma_N \cup c_N} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda.\] In view of the Residue Theorem, we have \[\oint_{\Gamma_N \cup c_N} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda= \oint_{\mathcal{C}_N} \frac{\hat{M}_k(\lambda)}{\lambda} \, d\lambda.\] This together with 18 conclude the proof. ◻

Combining 13 and ?? , we arrive at the relation \[\label{recsi} \hat{\sigma}_k(x) = \frac{1}{2\pi i} \int_{\Gamma} \left( S_k(x, \lambda) \tilde{S}_k(x, \lambda) - \frac{1}{2 \lambda} \right) \hat{M}_k(\lambda) \, d\lambda\tag{19}\] with the absolutely convergent integral in the case of \(\sigma\) and \(\tilde{\sigma}\) satisfying 11 .

6 Uniform stability of the auxiliary problem↩︎

In this section, we study the stability of Auxiliary Problem 13. Our analysis is based on constructing characteristic functions for the subtrees \(G_p\) and \(g_p\) (see Fig. 4) and applying their Lipschitz continuity with respect to the potentials.

Let \(v_k \in \partial G\) be fixed, and let \(v_p\) be the parent of \(v_k\). For the Sturm-Liouville equations 1 on the subtree \(g_p\), introduce the characteristic functions \(\Delta^{DD}(\lambda)\), \(\Delta^{DK}(\lambda)\), \(\Delta^{ND}(\lambda)\), and \(\Delta^{NK}(\lambda)\) according to Definition 1 associated with the following boundary conditions:

• \(\Delta^{DD}(\lambda)\): the Dirichlet conditions \(y_k(v_k) = 0\) and \(y_j(v_p) = 0\) for all the ingoing edges \(e_j\) to \(v_p\);

• \(\Delta^{DK}(\lambda)\): the Dirichlet condition \(y_k(v_k) = 0\) and the matching conditions 3 at \(v_p\);

• \(\Delta^{ND}(\lambda)\): the Neumann condition \(y_k^{[1]}(v_k) = 0\) and the Dirichlet conditions \(y_j(v_p) = 0\) for all the ingoing edges \(e_j\) to \(v_p\);

• \(\Delta^{NK}(\lambda)\): the Neumann condition \(y_k^{[1]}(v_k) = 0\) and the matching conditions 3 at \(v_p\);

wherein \(D\), \(N\), and \(K\) mean “Dirichlet”, “Neumann”, and “Kirchhoff”, respectively. It is assumed that, in the internal vertices \(v \in int\,g_p \setminus v_p\), the matching conditions 3 and, in the boundary vertices \(v \in \partial g_p \setminus \{ v_k, v_p \}\), the Dirichlet conditions \(y(v) = 0\) are imposed. Note that, in the case of the Dirichlet conditions at \(v_p\), the tree \(g_p\) actually splits into several subtrees with the root at \(v_p\).

Using 8 , we obtain the relations \[\label{sys} \begin{cases} \Delta(\lambda) = \Delta^{DD}(\lambda) \Delta_p^N(\lambda) + \Delta^{DK}(\lambda) \Delta_p^D(\lambda), \\ \Delta_k(\lambda) = \Delta^{ND}(\lambda) \Delta_p^N(\lambda) + \Delta^{NK}(\lambda) \Delta_p^D(\lambda). \end{cases}\tag{29}\] Cramer’s rule implies \[\label{cram} \Delta_p^N(\lambda) = \frac{A_1(\lambda)}{A(\lambda)}, \quad \Delta_p^D(\lambda) = \frac{A_2(\lambda)}{A(\lambda)},\tag{30}\] where \[\begin{align} \tag{31} & A(\lambda) := \Delta^{DD}(\lambda) \Delta^{NK}(\lambda) - \Delta^{ND}(\lambda) \Delta^{DK}(\lambda), \\ \tag{32} & A_1(\lambda) := \Delta(\lambda) \Delta^{NK}(\lambda) - \Delta_k(\lambda) \Delta^{DK}(\lambda), \\ \tag{33} & A_2(\lambda) := \Delta^{DD}(\lambda) \Delta_k(\lambda) - \Delta^{ND}(\lambda) \Delta(\lambda). \end{align}\]

In the special case of the graph \(g_p\) consisting of the only edge \(e_k = [v_k, v_p]\), we have \[\begin{align} & \Delta^{DD}(\lambda) = S_k(T_k,\lambda), \quad \Delta^{DK}(\lambda) = S_k^{[1]}(T_k,\lambda), \\ & \Delta^{ND}(\lambda) = \varphi_k(T_k,\lambda), \quad \Delta^{NK}(\lambda) = \varphi_k^{[1]}(T_k,\lambda), \\ & A(\lambda) = S_k(T_k,\lambda) \varphi_k^{[1]}(T_k,\lambda) - S_k^{[1]}(T_k,\lambda) \varphi_k(T_k,\lambda). \end{align}\]

Using equation 1 , one can easily show that the generalized Wronskian \(W_k(x,\lambda) = (S_k \varphi_k^{[1]} - S_k^{[1]} \varphi_k)(x,\lambda)\) does not depend on \(x\). The initial conditions 7 imply \(W_k(0,\lambda) \equiv -1\), so \(A(\lambda) = W_k(T_k,\lambda) \equiv -1\). Thus, the case of a single edge in \(g_p\) is quite simple. In the following, we focus on the case of \(g_p\) containing more than one edge.

Exclude the edge \(e_k\) from the graph \(g_p\) and denote the resulting graph by \(g_p^*\). Let \(\Delta^D(\lambda)\) and \(\Delta^K(\lambda)\) be the characteristic function of the Sturm-Liouville system 1 on the graph \(g_p^*\) with the Dirichlet conditions \(y_j(v_p) = 0\) for all \(e_j = [v_j, v_p]\) and the matching conditions 3 at the vertex \(v_p\), respectively. We assume that the Dirichlet boundary conditions \(y(v) = 0\) are imposed at the other boundary vertices \(v \in \partial g_p^* \setminus v_p\) and the matching conditions 3 are satisfied at all \(v \in int \, g_p^*\).

According to Corollary 5, we have the asymptotics \[\label{asymptDp} \Delta_p^N(\rho^2) = \Delta_p^{N,0}(\rho^2) + \rho^{2-B_p} \kappa_p^N(\rho), \quad \Delta_p^D(\rho^2) = \Delta_p^{D,0}(\rho^2) + \rho^{1-B_p} \kappa_p^D(\rho),\tag{34}\] where \(\kappa_p^N(\rho)\) and \(\kappa_p^D(\rho)\) are Paley-Wiener functions of the class \(PW(\mathcal{T})\), \(\mathcal{T} := length(G_p)\).

Now, we are ready to prove the uniform stability of Auxiliary Problem 13.

7 Stable algorithm↩︎

In this section, we develop an algorithm for unique and stable reconstruction of the potential \(\sigma\) on the tree \(G\). Moreover, we prove Theorem 15 on the local stability of Inverse Problem 9.

7.1 Reconstruction on a boundary edge↩︎

For reconstruction on the boundary edges, we derive the main equation of the method of spectral mappings. Our technique is quite similar to [26], so we outline it briefly.

Consider a potential \(\sigma \in L_2(G)\) and the zero potential \(\sigma^0 \equiv 0\). Let an index \(k\) of a boundary vertex \(v_k \in \partial G'\) be fixed. Choose the contour \(\Gamma\) according to 10 so that the zeros of \(\Delta(\lambda)\) and \(\Delta^0(\lambda)\) lie inside \(\Gamma\). Furthermore, choose radii \(\{ R_N \}\) so that the estimates 16 hold for \(\Delta(\lambda)\) and \(\Delta^0(\lambda)\) and define the contours 15 . The contour integration implies the relation \[S_k^0(x,\lambda) = S_k(x, \lambda) + \frac{1}{2\pi i}\lim_{N \to \infty} \oint_{\Gamma_N \cup c_N} D_k^0(x, \lambda, \mu) (M_k - M_k^0)(\mu) S_k(x,\mu) \, d\mu, \quad x \in [0,T_k],\] where \[D_k^0(x,\lambda,\mu) := \int_0^x S_k^0(x,\lambda) S_k^0(x,\mu) \, dx.\]

Recall that \(S_k^0(x,\lambda) = \dfrac{\sin \rho x }{\rho}\), \(\lambda= \rho^2\), \(\mu = \theta^2\). For fixed \(x \in [0,\pi]\), \(\lambda\), and \(\mu \in c_N\), the following estimates hold: \[|S_k(x, \mu)| \le \frac{C}{|\theta|}, \quad |D_k^0(x, \lambda, \mu)| \le \frac{C}{|\rho||\theta|}, \quad |(M_k - M_k^0)(\mu)| \le C|\theta|.\] Here and below, constants \(C\) depend only on \(\sigma\). Also, we have \(length(c_N) \sim \sqrt{R_N}\). Consequently \[\lim_{N \to \infty} \int_{c_N} D_k^0(x, \lambda, \mu) (M_k - M_k^0)(\mu) S_k(x,\mu) \, d\mu = 0.\] Hence \[S_k^0(x,\lambda) = S_k(x, \lambda) + \frac{1}{2\pi i} \int_{\Gamma} D_k^0(x, \lambda, \mu) (M_k - M_k^0)(\mu) S_k(x,\mu) \, d\mu\] for \(x \in [0,T_k]\) and \(\lambda\in \mathbb{C}\). Representing \(S_k(x, \lambda)\) in the form ?? and passing to the contour \(\gamma\) in the \(\rho\)-plane, we obtain \[\label{maineq} \psi_k(x, \rho) + \frac{1}{\pi i} \int_{\gamma} r_k^0(x, \rho, \theta) \psi_k(x, \theta) \, d\theta = F_k(x, \rho),\tag{40}\] where \[\begin{gather} \tag{41} r_k^0(x, \rho, \theta) := \mathscr M_k(\theta) \int_0^x \sin \rho t \sin \theta t \, dt, \quad \mathscr M_k(\theta) := \theta^{-1} (M_k - M_k^0)(\theta^2), \\ \tag{42} F_k(x, \rho) := \int_0^x f_k(x, t) \sin \rho t \, dt, \quad f_k(x, t) := -\frac{1}{\pi i} \int_{\gamma} \mathscr M_k(\theta) \sin \theta x \sin \theta t \, d \theta. \end{gather}\]

For each fixed \(x \in [0,T_k]\), the relation 40 can be considered as an equation in \(L_2(\gamma)\) with respect to \(\psi_k(x,.)\). Analogously to 23 , we get \(\mathscr M_k \in L_2(\gamma)\). Therefore, we obtain \[\int_{\gamma} \int_{\gamma} |r_k^0(x, \rho, \theta)|^2 |d\theta| |d\rho| \le C,\] similarly to [26]. Furthermore, we have \(f_k(x, .) \in L_2(0,x)\) and so \(F_k(x, .) \in L_2(\gamma)\).

Thus, equation 40 can be rewritten in the form \[\label{mainop} (I + H_k^0(x)) \psi_k(x) = F_k(x), \quad x \in [0,T_k],\tag{43}\] where \(I\) is the identity operator in \(L_2(\gamma)\) and \[\label{defHk0} H_k^0(x) \kappa(\rho) := \frac{1}{\pi i} \int_{\gamma} r_k^0(x, \rho, \theta) \kappa(\theta) \, d \theta\tag{44}\] is the Hilbert-Schmidt operator in \(L_2(\gamma)\) for each fixed \(x \in [0,T_k]\).

Analogously to [26], one can show that \((I + H_k^0(x))\) has a bounded inverse operator on \(L_2(\gamma)\): \[\label{invHk0} (I + H_k^0(x))^{-1} = I - H_k(x),\tag{45}\] where \[\begin{gather} H_k(x) \kappa(\rho) := \frac{1}{\pi i} \int_{\gamma} r_k(x, \rho, \theta) \kappa(\theta) \, d \theta, \\ r_k(x, \rho, \theta) := \rho \theta \mathscr M_k(\theta) \int_0^x S_k(t,\rho^2) S_k(t,\theta^2) \, dt. \end{gather}\] Hence, the main equation 40 is uniquely solvable. Using its solution \(\psi_k(x, \rho)\), one can find \(S_k(x,\lambda)\) by ?? and then recover the potential by the formula \[\label{recsi0} \sigma_k(x) = \frac{1}{2 \pi i} \int_{\Gamma} \left( S_k(x, \lambda) S_k^0(x, \lambda) - \frac{1}{2\lambda}\right)(M_k - M_k^0)(\lambda) \, d\lambda,\tag{46}\] which is a special case of 19 .

7.2 Transition through a vertex↩︎

The solution of Auxiliary Problem 13 is based on the linear algebraic system 29 . However, not every small perturbation of \(\Delta(\lambda)\) and \(\Delta_k(\lambda)\) preserves a solution \(\Delta_p^N(\lambda)\) and \(\Delta_p^D(\lambda)\) of 29 in the class of entire analytic functions having the asymptotics 34 . Therefore, in order to develop a stable algorithm for the unique reconstruction, we use the sampling approach.

Introduce the points \[\label{defnu} \nu_n := \frac{\pi n}{\mathcal{T}} + i \tau, \quad \mu_n = \nu_n^2, \quad n \in \mathbb{Z},\tag{47}\] where \(\tau > 0\) is sufficiently large. Then, the sequence \(e_n(t) := \exp(i \nu_n t)\) (\(n \in \mathbb{Z}\)) is a Riesz basis in \(L_2(-\mathcal{T}, \mathcal{T})\). We readily obtain the following analog of Whittaker-Kotel’nikov-Shannon Theorem, which is a special case of Kramer’s Lemma (see [35] and [36]).

Using the relations 30 and 34 , we get \[\label{findkap} \kappa_p^N(\nu_n) = \nu_n^{B_p-2} \left( \frac{A_1(\mu_n)}{A(\mu_n)} - \Delta_p^{N,0}(\mu_n)\right), \quad \kappa_p^D(\nu_n) = \nu_n^{B_p-1} \left( \frac{A_2(\mu_n)}{A(\mu_n)} - \Delta_p^{D,0}(\mu_n)\right).\tag{49}\]

Thus, one can construct the characteristic functions \(\Delta_p^{N,0}(\lambda)\) and \(\Delta^{D,0}_p(\lambda)\) for \(\sigma \equiv 0\), find \(\kappa_p^N(\rho)\) and \(\kappa_p^D(\rho)\) using 49 and ?? , and then obtain \(\Delta_p^N(\lambda)\) and \(\Delta_p^D(\lambda)\) by 34 .

7.3 Reconstruction on the whole tree↩︎

Summarizing the above results, we obtain the following algorithm for solving Inverse Problem 9 on the whole tree.

In the case of \(\Delta(\lambda)\) and \(\Delta_k(\lambda)\) (\(v_k \in \partial G'\)) being the spectral data of some potential \(\sigma \in L_2(G)\), Algorithm 20 allows one to uniquely reconstruct this potential according to the arguments of Sections 5–7. Let us show that, moreover, Algorithm 20 is stable under small perturbations of the spectral data.

Note that Theorem 21 does not assert that \(\tilde{\Delta}(\lambda)\) and \(\tilde{\Delta}_k(\lambda)\) are the characteristic functions corresponding to the constructed potential \(\tilde{\sigma}\). Moreover, choosing different \(v_k\) at step 4 of Algorithm 20, we may get different results \(\tilde{\sigma}\). Anyway, under the assumption of existence for the inverse problem solution, Theorem 21 immediately yields Theorem 15.

Appendix↩︎

Here we provide the proof of Lemma 6, based on the technique of [32] for constructing transformation operators.

Consider the Sturm-Liouville equation \[\label{eqv1} -(y^{[1]})' - \sigma(x) y^{[1]} - \sigma^2(x) y = \lambda y, \quad x \in (0,T),\tag{53}\] where \(\sigma\) is a complex-valued function of \(L_2(0,T)\), \(y^{[1]} := y' - \sigma y\) is the quasi-derivative, and \(\lambda= \rho^2\) is the spectral parameter.

Denote by \(\varphi(x, \lambda)\) the solution of equation 53 satisfying the initial conditions \(\varphi(0,\lambda) = 1\), \(\varphi^{[1]}(0,\lambda) = 0\). In [32], the following representations have been obtained: \[\begin{align} \tag{54} & \varphi(x, \lambda) = \cos \rho x + \int_0^x \mathscr K(x, t) \cos \rho t \, dt, \\ \tag{55} & \varphi^{[1]}(x, \lambda) = -\rho \sin \rho x + \rho \int_0^x \mathscr N(x, t) \sin \rho t \, dt + \mathscr C(x), \end{align}\] where the functions \(\mathscr K(x,t)\), \(\mathscr N(x,t)\), and \(\mathscr C(x)\) are constructed as the series: \[\label{ser} \mathscr K = \sum_{n = 0}^{\infty} \mathscr K_n, \quad \mathscr N = \sum_{n = 0}^{\infty} \mathscr N_n, \quad \mathscr C = \sum_{n = 0}^{\infty} \mathscr C_n,\tag{56}\] whose terms are defined recursively (see [32], [33]): \[\begin{align} \tag{57} \mathscr K_0(x, t) & = \tfrac{1}{2} \sigma\left( \tfrac{x + t}{2}\right) + \tfrac{1}{2} \sigma\left( \tfrac{x - t}{2}\right) - \tfrac{1}{2} \int_0^x \sigma^2(s) \, ds - \tfrac{1}{4} \int_t^x \left( \sigma^2(\tfrac{x - s}{2}) - \sigma^2(\tfrac{x + s}{2})\right) \, ds, \\ \tag{58} \mathscr N_0(x, t) & = \tfrac{1}{2} \sigma\left( \tfrac{x + t}{2}\right) - \tfrac{1}{2} \sigma\left( \tfrac{x - t}{2}\right) + \tfrac{1}{2} \int_0^x \sigma^2(s) \, ds + \tfrac{1}{4} \int_t^x \left( \sigma^2(\tfrac{x - s}{2}) + \sigma^2(\tfrac{x + s}{2})\right) \, ds, \\ \tag{59} \mathscr C_0(x) & = -\tfrac{1}{2} \int_0^x \sigma^2(t)\,dt - \tfrac{1}{4} \int_0^x (\sigma^2(\tfrac{x + t}{2}) + \sigma^2(\tfrac{x - t}{2})) \, dt, \\ \nonumber \mathscr K_{n+1}(x,t) & = \frac{1}{2} \int_{x - t}^x \bigl( \mathscr K_n(s, t - x + s) + \mathscr N_n(s, t-x+s) \bigr) \sigma(s) \, ds \\ \nonumber & + \frac{1}{2}\int_{\frac{x - t}{2}}^{x - t} \bigl( \mathscr K_n(s, x - s - t) - \mathscr N_n(s, x - s - t) \bigr) \sigma(s) \, ds \\ \nonumber & + \frac{1}{2}\int_{\frac{x + t}{2}}^x \bigl( \mathscr K_n(s, x - s + t) - \mathscr N_n(s, x - s + t) \bigr) \sigma(s) \, ds \\ \nonumber & - \frac{1}{2} \biggl( \int_0^x \sigma^2(s) \, ds \int_0^{\min\{s, x-t\}} \mathscr K_n(s, s-\xi) \, d\xi \\ \nonumber & + \int_0^{x-t} \sigma^2(s) \, ds \int_s^{\min\{ 2s, x-t\}} \mathscr K_n(s, \xi - s) \, d\xi \\ \tag{60} & - \int_{\frac{x+t}{2}}^x \sigma^2(s) \, ds \int_t^{2s-x} \mathscr K_n(s, x + \xi - s) \, d\xi\biggr) - \int_0^{x - t} \mathscr C_n(s) \sigma(s) \, ds, \\ \nonumber \mathscr N_{n+1}(x,t) & =-\frac{1}{2} \int_{x - t}^x \bigl( \mathscr K_n(s, t - x + s) + \mathscr N_n(s, t - x + s) \bigr) \sigma(s) \, ds \\ \nonumber & - \frac{1}{2}\int_{\frac{x - t}{2}}^{x - t} \bigl( \mathscr K_n(s, x - s - t) - \mathscr N_n(s, x - s - t) \bigr) \sigma(s) \, ds \\ \nonumber & + \frac{1}{2}\int_{\frac{x + t}{2}}^x \bigl( \mathscr K_n(s, x - s + t) - \mathscr N_n(s, x - s + t) \bigr) \sigma(s) \, ds \\ \nonumber & +\frac{1}{2} \biggl( \int_0^x \sigma^2(s) \, ds \int_0^{\min\{s, x-t\}} \mathscr K_n(s, s-\xi) \, d\xi \\ \nonumber & + \int_0^{x-t} \sigma^2(s) \, ds \int_s^{\min\{ 2s, x-t\}} \mathscr K_n(s, \xi - s) \, d\xi \\ \nonumber & + \int_{\frac{x+t}{2}}^x \sigma^2(s) \, ds \int_t^{2s-x} \mathscr K_n(s, x + \xi - s) \, d\xi\biggr) + \int_0^{x - t} \mathscr C_n(s) \sigma(s) \, ds, \\ \nonumber \mathscr C_{n+1}(x) & = -\frac{1}{2} \int_0^x \sigma^2(t) \, dt \biggl( \int_{x-t}^x \mathscr K_n(t, \xi - x + t) \, d\xi + \int_{x - 2t}^{x - t} \mathscr K_n(t, x-\xi-t) \, d\xi\biggr) \\ \nonumber & - \int_0^x \mathscr C_n(s) \sigma(s) \, ds. \end{align}\]

The functions \(\mathscr K_n(x,t)\) and \(\mathscr N_n(x,t)\) (\(n \ge 1\)) are continuous on \(\mathbf{D} := \bigl\{ (x,t) \colon 0 \le t \le x \le T \bigr\}\) and the functions \(\mathscr C_n(x)\) (\(n \ge 0\)), on \([0,T]\). Moreover, the following estimates hold: \[\label{estKn} |\mathscr K_n(x, t)|, \, |\mathscr N_n(x, t)|, \, |\mathscr C_n(x)| \le a^n Q^n(x) \sqrt{\tfrac{x^{n-1}}{(n-1)!}}, \quad n \ge 1, \quad (x,t) \in \mathbf{D},\tag{61}\] where \(Q(x) = \| \sigma \|_{L_2(0,x)}\) and \(a\) is some constant depending on \(\| \sigma \|_{L_2(0,T)}\) (see [33]). Consequently, the series of \(\mathscr K_n\) and \(\mathscr N_n\) starting from \(n \ge 1\) and the series of \(\mathscr C_n\) in 56 converge absolutely and uniformly to continuous functions. Adding the corresponding terms \(\mathscr K_0\) and \(\mathscr N_0\), we conclude that, for each fixed \(x \in (0,T]\), the functions \(\mathscr K(x,.)\) and \(\mathscr N(x,.)\) belong to \(L_2(0,x)\).

Now, along with 1 , consider the similar equation with another complex-valued potential \(\tilde{\sigma} \in L_2(0,T)\). Let us show that, for each fixed \(x \in (0,T]\), there hold \[\label{difest} \| \hat{\mathscr K}(x, .) \|_{L_2(0,x)}, \, \| \hat{\mathscr N}(x, .) \|_{L_2(0,x)}, \, |\hat{\mathscr C}(x)| \le C \| \hat{\sigma} \|_{L_2(0,x)},\tag{62}\] where the constant \(C\) depends only on \(\max \bigl\{ \| \sigma \|_{L_2(0,T)}, \| \tilde{\sigma} \|_{L_2(0,T)} \bigr\}\).

Using 57 –59 , we get \[\label{est0} \| \hat{\mathscr K}_0(x,.) \|_{L_2(0,x)}, \, \| \hat{\mathscr N}_0(x,.) \|_{L_2(0,x)}, \, |\hat{\mathscr C}_0(x)| \le a \| \hat{\sigma} \|_{L_2(0,x)}.\tag{63}\]

Next, by induction, we obtain the estimates \[\label{estdifK} |\hat{\mathscr K}_n(x, t)|, \, |\hat{\mathscr N}_n(x, t)|, \, |\hat{\mathscr C}_n(x)| \le a^n \| \hat{\sigma} \|_{L_2(0,x)} Q^{n-1}(x) \sqrt{\tfrac{x^{n-1}}{(n-1)!}}, \quad n \ge 1, \quad (x,t) \in \mathbf{D},\tag{64}\] where \[\label{defQ} Q(x) = \| \sigma_{max} \|_{L_2(0,x)}, \quad \sigma_{max}(x) = \max\{ |\sigma(x)|, |\tilde{\sigma}(x)| \}\tag{65}\] and \(a\) is a constant that depends only on \(\max \bigl\{ \| \sigma \|_{L_2(0,T)}, \| \tilde{\sigma} \|_{L_2(0,T)} \bigr\}\).

For instance, let us prove 64 for \(\hat{\mathscr K}_{n+1}(x,t)\) (\(n \ge 0\)) assuming that the claimed estimates for \(\hat{\mathscr K}_n\), \(\hat{\mathscr N}_n\), \(\hat{\mathscr C}_n\) are already established. Consider the first term in 60 : \[\label{defIn} I_n(x,t) := \frac{1}{2} \int_{x - t}^x \bigl( \mathscr K_n(s, t - x + s) + \mathscr N_n(s, t-x+s) \bigr) \sigma(s) \, ds.\tag{66}\]

Combining 57 and 58 , we get \[\mathscr K_0(x,t) + \mathscr N_0(x,t) = \sigma\left( \tfrac{x+t}{2} \right) + \tfrac{1}{2} \int_t^x \sigma^2\left( \tfrac{x+s}{2}\right) \,ds.\] Hence \[\begin{align} I_0(x, t) & = \frac{1}{2} \int_{x-t}^x \sigma\left( \tfrac{t-x+2s}{2}\right) \sigma(s) \, ds + \frac{1}{4} \int_{x-t}^x \sigma(s) \, ds \int_{t-x+s}^s \sigma^2 \left( \tfrac{s+\xi}{2}\right) d\xi, \\ \hat{I}_0(x,t) & = \frac{1}{2} \int_{x-t}^x \hat{\sigma}\left( \tfrac{t-x+2s}{2}\right) \sigma(s) \, ds + \frac{1}{2} \int_{x-t}^x \tilde{\sigma}\left( \tfrac{t-x+2s}{2}\right) \hat{\sigma}(s) \, ds \\ & + \frac{1}{4} \int_{x-t}^x \hat{\sigma}(s) \, ds \int_{t-x+s}^s \sigma^2 \left( \tfrac{s+\xi}{2}\right) d\xi + \frac{1}{4} \int_{x-t}^x \tilde{\sigma}(s) \, ds \int_{t-x+s}^s \hat{\sigma}\left( \tfrac{s+\xi}{2}\right) \Bigl( \sigma\left( \tfrac{s +\xi}{2}\right) + \tilde{\sigma}\left( \tfrac{s+\xi}{2}\right)\Bigr) d\xi \end{align}\] Applying the Cauchy-Bunyakovsky-Schwarz inequality, we estimate \[\label{estI1} |\hat{I}_0(x,t)| \le a \| \hat{\sigma} \|_{L_2(0,x)}, \quad (x,t) \in \mathbf{D}.\tag{67}\]

Analogously, consider the fourth term from 60 : \[\begin{align} \nonumber J_{n}(x, t) & := \frac{1}{2} \int_0^x \sigma^2(s) \, ds \int_0^{\min\{s, x-t\}} \mathscr K_n(s, s-\xi) \, d\xi, \\ \nonumber \hat{J}_n(x,t) & = \frac{1}{2} \int_0^x \hat{\sigma}(s) (\sigma(s) + \tilde{\sigma}(s))\, ds \int_0^{\min\{s, x-t\}} \mathscr K_n(s, s-\xi) \, d\xi \\ \label{difJn} & + \frac{1}{2} \int_0^x \tilde{\sigma}^2(s) \, ds \int_0^{\min\{s, x-t\}} \hat{\mathscr K}_n(s, s-\xi) \, d\xi. \end{align}\tag{68}\]

In view of 57 , the function \(\mathscr K_0(x,t)\) is square-integrable for each fixed \(x \in (0,T]\) and \(\| \mathscr K_0(x, .) \|_{L_2(0,x)} \le a\). Using the latter estimate and 63 , we obtain \[\label{estintK0} \left| \int_0^{\min\{s, x-t\}} \mathscr K_n(s, s-\xi) \, d\xi \right| \le a, \quad \left| \int_0^{\min\{s, x-t\}} \hat{\mathscr K}_n(s, s-\xi) \, d\xi \right| \le a \| \hat{\sigma} \|_{L_2(0,x)}.\tag{69}\]

Using 68 for \(n = 0\) and 69 , we deduce \[\label{estJ1} |\hat{J}_0(x,t)| \le a \| \hat{\sigma} \|_{L_2(0,x)}, \quad (x,t) \in \mathbf{D}.\tag{70}\]

Estimates similar to 67 and 70 can be obtained for the other terms in 60 , so we arrive at 64 for \(\hat{\mathscr K}_1(x,t)\).

Proceed to proving 64 for \(\mathscr K_{n+1}(x,t)\) with \(n \ge 1\). As above, we confine ourselves by analyzing the terms \(I_n\) and \(J_n\), since the other terms in 60 can be treated analogously. It follows from 66 that \[\begin{align} \nonumber \hat{I}_n(x,t) & = \frac{1}{2} \int_{x-t}^x \bigl(\hat{\mathscr K}_n(s,t-x+s) + \hat{\mathscr N}_n(s,t-x+s)\bigr) \sigma(s) \, ds \\ \label{difIn} & + \frac{1}{2} \int_{x-t}^x \bigl(\tilde{\mathscr K}_n(s,t-x+s) + \tilde{\mathscr N}_n(s,t-x+s)\bigr) \hat{\sigma}(s) \, ds. \end{align}\tag{71}\]

Note that, for \(\tilde{\mathscr K}_n(x,t)\) and \(\tilde{\mathscr N}_n(x,t)\), the estimates 61 with \(Q(x)\) given by 65 are valid. So, using 61 , 64 , and 71 , we obtain \[\begin{align} \label{smIn} |\hat{I}_n(x,t)| & \le a^n \| \hat{\sigma} \|_{L_2(0,x)} \int_{x-t}^x Q^{n-1}(s) \sqrt{\frac{s^{n-1}}{(n-1)!}} |\sigma(s)| \, ds + a^n \int_{x-t}^x Q^n(s) \sqrt{\frac{s^{n-1}}{(n-1)!}} |\hat{\sigma}(s)| \, ds. \end{align}\tag{72}\] Taking 65 into account and applying the Cauchy-Bunyakovsky-Schwarz inequality, we get \[\begin{align} \nonumber & \int_0^x Q^{n-1}(s) \sqrt{\frac{s^{n-1}}{(n-1)!}} |\sigma(s)| \, ds \\ \tag{73} & \le \sqrt{\int_0^x \sigma_{max}^2(s) \, ds \int_0^s \sigma_{\max}^{2(n-1)}(\xi) \, d\xi} \cdot \sqrt{\int_0^x \frac{s^{n-1}}{(n-1)!}\, ds} = \frac{Q^n(x)}{\sqrt{n}} \cdot \sqrt{\frac{x^n}{n!}}, \\ \nonumber & \int_0^x Q^n(s) \sqrt{\frac{s^{n-1}}{(n-1)!}} |\hat{\sigma}(s)| \, ds \\ \tag{74} & \le Q^n(s) \sqrt{\int_0^x |\hat{\sigma}(s)|^2 \, ds} \cdot \sqrt{\int_0^x \frac{s^{n-1}}{(n-1)!}\, ds} = \| \hat{\sigma}\|_{L_2(0,x)} Q^n(x) \sqrt{\frac{x^n}{n!}}. \end{align}\] Combining 72 , 73 , and 74 , we obtain \[\label{estIn} |\hat{I}_n(x,t)| \le a^{n+1} \| \hat{\sigma} \|_{L_2(0,x)} Q^n(x) \sqrt{\frac{x^n}{n!}}, \quad (x,t) \in \mathbf{D}.\tag{75}\]

Proceed to estimating \(\hat{J}_n\). It follows from 61 , 64 , and 68 that \[\begin{align} |\hat{J}_n(x,t)| & \le a^n \int_0^x |\hat{\sigma}(s)| \sigma_{max}(s) Q^n(s) \sqrt{\frac{s^{n-1}}{(n-1)!}} s \, ds \\ & + \frac{a^n}{2} \int_0^x |\tilde{\sigma}(s)|^2 \| \hat{\sigma} \|_{L_2(0,s)} Q^{n-1}(s) \sqrt{\frac{s^{n-1}}{(n-1)!}} s \, ds \\ & \le a^n \sqrt{\frac{x^{n+1}}{(n-1)!}} \Biggl( \int_0^x |\hat{\sigma}(s)| \cdot \sigma_{max}(s) Q^n(s) \, ds \\ & + \frac{1}{2} \| \hat{\sigma} \|_{L_2(0,x)} \int_0^x |\tilde{\sigma}(s)| \cdot \sigma_{max}(s) Q^{n-1}(s) \, ds\Biggr). \end{align}\] Applying the Cauchy-Bunyakovsky-Schwarz inequality to the latter integrals similarly to 73 , we arrive at the estimate for \(\hat{J}_n\) analogous to 75 . Consequently, we obtain 64 for \(\hat{\mathscr K}_{n+1}(x,t)\).

Summing up the series \[\hat{\mathscr K} = \sum_{n = 0}^{\infty} \hat{\mathscr K}_n, \quad \hat{\mathscr N} = \sum_{n = 0}^{\infty} \hat{\mathscr N}_n, \quad \hat{\mathscr C} = \sum_{n = 0}^{\infty} \hat{\mathscr C}_n\] and using 63 –64 , we prove 62 . Clearly, the estimates 62 imply ?? in Lemma 6. The estimates ?? are obtained analogously.

Funding. This work was supported by Grant 24-71-10003 of the Russian Science Foundation, https://rscf.ru/en/project/24-71-10003/.

Acknowledgement. The author is grateful to Professor Maria A. Kuznetsova for her careful reading and valuable comments.

Natalia Pavlovna Bondarenko
. Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia,
. Department of Applied Mathematics, Samara National Research University,
Moskovskoye Shosse 34, Samara 443086, Russia,
. S.M. Nikolskii Mathematical Institute, RUDN University, 6 Miklukho-Maklaya St, Moscow, 117198, Russia,
. Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, Moscow 119991, Russia.
e-mail: bondarenkonp@sgu.ru

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