Abstract. In this paper, the reconstruction of a linear differential operator of arbitrary order \(n \ge 2\) is studied by using two types of spectral characteristics: (i) eigenvalues and weight numbers, (ii) \((2n-2)\) spectra. We prove the unconditional uniform stability of these inverse problems, generalizing the results of Savchuk and Shkalikov [Funct. Anal. Appl. 44 (2010), no. 4, 270–285] to \(n > 2\). Furthermore, we for the first time obtain sufficient conditions of solvability for the higher-order inverse problem by \((2n-2)\) spectra. By applying our main results, we get new theorems on the necessary and sufficient conditions of solvability and on the uniform stability of the inverse problems for \(n = 3\) and \(n = 4\). Our approach is based on the method of spectral mappings, which provides a constructive solution of the inverse problems.

Keywords: higher-order differential operators; inverse spectral problems; uniform stability; method of spectral mappings.

AMS Mathematics Subject Classification (2020): 34A55 34B05 34B09 34L05

1 Introduction↩︎

Consider the differential equation \[\begin{align} \nonumber \ell_n(y) := & y^{(n)} + \sum_{k = 0}^{\lfloor n/2\rfloor - 1} (\tau_{2k}(x) y^{(k)})^{(k)} \\ \label{eqv} + & \sum_{k = 0}^{\lfloor (n-1)/2\rfloor - 1} \bigl((\tau_{2k+1}(x) y^{(k)})^{(k+1)} + (\tau_{2k+1}(x) y^{(k+1)})^{(k)}\bigr) = \lambda y, \: x \in (0,1), \end{align}\tag{1}\] where \(n \ge 2\), the notation \(\lfloor a \rfloor\) means rounding a real number \(a\) down, \(\tau_{\nu}\) belongs to the Sobolev spaces \(W_2^{\nu}[0,1]\) for \(\nu = \overline{0,n-2}\), and \(\lambda\) is the spectral parameter.

Equation 1 can be equivalently represented in the form \[\label{eqp} \ell_n(y) = y^{(n)} + \sum_{s = 0}^{n-2} p_s(x) y^{(s)} = \lambda y, \quad x \in (0,1),\tag{2}\] where \(p_s \in W_2^s[0,1]\), \(s = \overline{0,n-2}\). However, we use the divergent form 1 , since we focus on the “self-adjoint” case \(i^{n + \nu} \tau_{\nu}(x) \in \mathbb{R}\).

For \(k = \overline{1,n-1}\), denote by \(\{ \lambda_{l,k} \}_{l \ge 1}\) and \(\{ \mu_{l,k} \}_{l \ge 1}\) the eigenvalues (counting with multiplicities) of the boundary value problems \(\mathcal{L}_k\) and \(\mathcal{M}_k\), respectively, for equation 1 with the corresponding boundary conditions \[\begin{align} \tag{3} \mathcal{L}_k \colon & \quad y^{(j-1)}(0) = 0, \quad j = \overline{1,k}, \qquad \qquad \quad \:\:\: y^{(s-1)}(1) = 0, \quad s = \overline{1,n-k}, \\ \tag{4} \mathcal{M}_k \colon & \quad y^{(j-1)}(0) = 0, \quad j = \overline{1,k-1},\,k+1, \quad y^{(s-1)}(1) = 0, \quad s = \overline{1,n-k}. \end{align}\]

We study the inverse spectral problems that consist in the reconstruction of the functions \(\{ \tau_{\nu} \}_{\nu = 0}^{n-2}\) from the two types of spectral data:

(i) the eigenvalues \(\{ \lambda_{l,k} \}_{l \ge 1, k = \overline{1,n-1}}\) and the related weight numbers \(\{ \beta_{l,k} \}_{l \ge 1, \, k = \overline{1,n-1}}\), which are rigorously defined in Section 2;

(ii) the system of \((2n-2)\) spectra \(\{ \lambda_{l,k}, \mu_{l,k} \}_{l \ge 1, \, k = \overline{1,n-1}}\).

The first problem statement goes back to the seminal paper [1] by Gelfand and Levitan for \(n = 2\) and to the studies [2]–[5] by Leibenzon for \(n > 2\). The solution of the second problem, which generalizes the famous Borg inverse problem by two spectra [6], was considered by Yurko [7] for \(n > 2\). In this paper, we prove theorems on the uniform stability of the inverse problems, which have no analogs for higher orders \(n > 2\). Moreover, we for the first time find sufficient conditions for the existence of solution for the inverse problem by the system of spectra.

The most complete results in the inverse spectral theory were obtained for the second-order Sturm-Liouville operators (see monographs [8]–[12] and references therein). Inverse problems for the third-order differential operators have applications to integration of the nonlinear Boussinesq equation (see [13]–[15]). Inverse problems for \(n = 4\) arise in geophysics [16] and vibration theory [17]. However, transformation operators, which played an important role in the development of the inverse spectral theory for \(n = 2\), appeared to be ineffective for higher orders. Only some results were achieved under the analyticity requirements for the coefficients of differential equations (see [18]–[20]).

Leibenzon has developed another approach that allowed him to prove the uniqueness [3] and to create a constructive procedure for the recovery of the coefficients \(\{ p_s \}_{s = 0}^{n-2}\) of equation 2 from the spectral data \(\{ \lambda_{l,k}, \beta_{l,k} \}_{l \ge 1, \, k =\overline{1,n-1}}\). Furthermore, in [5], the necessary and sufficient conditions for solvability of this inverse problem were obtained. However, as was pointed out by Yurko [7], Leibenzon’s spectral data uniquely determine the coefficients only if the separation condition \(\{ \lambda_{l,k} \}_{l \ge 1} \cap \{ \lambda_{l,k+1} \}_{l \ge 1} = \varnothing\) is satisfied. As an alternative, Yurko introduced the Weyl-Yurko matrix, a spectral characteristic that ensures the uniqueness of the solution to the inverse problem for higher-order operators without any additional restrictions on their spectra. This facilitated the development of a general inverse problem theory for equation 2 with \(p_s \in W_2^{s + \nu}\), \(s = \overline{0,n-2}\), \(\nu \ge 0\), on both a finite interval and the half-line (see [7], [21], [22]). Meanwhile, the case of the whole line requires a different approach (see [23], [24]). In recent years, the method of [7] was transferred to arbitrary order differential operators with distribution coefficients (see [25]–[30]). In particular, the most challenging issue regarding the necessary and sufficient conditions for the solvability of inverse spectral problems was resolved. For third- and fourth-order operators, those conditions take their simplest form, involving only the asymptotics of the spectral data and their basic structural properties (see [28], [29]). We also mention recent studies [31]–[33] on inverse scattering and spectral theory for the third-order differential operators. Some specific kinds of inverse problems for \(n = 4\) were considered in [34]–[36].

In recent years, significant progress has been achieved in the investigation of the uniform stability of inverse spectral problems for second-order differential, integro-differential, and functional-differential operators [37]–[43]. Savchuk and Shkalikov [37] were the first to describe the set of spectral data of the Sturm-Liouville equation for which the inverse spectral mapping is uniformly bounded and, as a consequence, the unconditional uniform stability of the inverse problem is satisfied. However, to the best of the author’s knowledge, there were no such kind of results for higher orders. This paper aims to fill this gap.

To reconstruct the parameters \(\{ \tau_{\nu} \}_{\nu = 0}^{n-2}\) of equation 1 from the spectral data \(\{ \lambda_{l,k}, \beta_{l,k} \}\), we develop the approach of the previous studies [7], [22], [26], [27], [29], based on the method of spectral mappings. This method allows us to reduce the nonlinear inverse problem to a linear equation \((I - \tilde{R}(x)) \psi(x) = \tilde{\psi}(x)\) in the Banach space of bounded infinite sequences. In this paper, we apply a new modification of the main equation which makes the operator \(\tilde{R}(x)\) and the right-hand side \(\tilde{\psi}(x)\) continuous with respect to the spectral data \(\lambda_{l,k}\) and \(\beta_{l,k}\). This modification was introduced in [40] and was crucial for proving the uniform stability of the inverse problem in the case \(n = 2\). The subsequent analysis consists of three steps. First, we consider the general non-self-adjoint case and prove that the solution of the inverse problem is uniformly bounded under the constraint \(\| (I - \tilde{R}(x))^{-1} \| \le K\), where \(K\) is a positive constant. Second, basing on the uniform boundedness, we prove the uniform stability under the same assumptions. Third, we consider the self-adjoint case and find the restrictions on the spectral data that guaranty \(\| (I - \tilde{R}(x))^{-1} \| \le K\). As a result, we get the unconditional uniform stability of the inverse spectral problem for equation 1 of arbitrary order \(n\). Furthermore, we reduce the inverse problem by the eigenvalue set \(\{ \lambda_{l,k}, \mu_{l,k} \}_{l \ge 1, \, k = \overline{1,n-1}}\) to the inverse problem by the spectral data \(\{ \lambda_{l,k}, \beta_{l,k} \}_{l \ge 1, \,k = \overline{1,n-1}}\) and transfer our main results to the first problem. Namely, we obtain sufficient conditions for the existence of solution and prove the unconditional uniform stability of the inverse spectral problem by the \((2n-2)\) spectra \(\{ \lambda_{l,k}, \mu_{l,k} \}_{l \ge 1, \, k = \overline{1,n-1}}\).

One of the conditions of our main theorems (Theorems 7, 11, and 13) is the asymptotics of the eigenvalues \(\lambda_{l,k}\), \(\mu_{l,k}\) and of the weight numbers \(\beta_{l,k}\). Derivation of explicit formulas for the coefficients in those asymptotics for higher orders \(n\) is technically complicated. Therefore, we focus on the cases \(n = 3, 4\) and find the corresponding coefficients by using the standard method from [44], [45] and symbolic computations in Python [46]. Finally, we deduce new theorems on the necessary and sufficient conditions of solvability and on the uniform stability of inverse problems for \(n = 3, 4\) from the main results for arbitrary \(n\).

The paper is organized as follows. In Section 2, we formulate the inverse spectral problems and the main theorems. Sections 3–6 are concerned with the inverse problem by eigenvalues and weight numbers. In Section 3, the inverse problem is reduced to a linear equation in the Banach space of bounded infinite sequences. In Section 4, we consider the problem in the general non-self-adjoint case and prove the uniform boundedness of the inverse problem under the bound \(\| (I - \tilde{R}(x))^{-1} \| \le K\) together with other suitable conditions. In Section 5, we prove the uniform stability of the inverse problem under the assumptions of the previous section. In Section 6, the unconditional uniform stability of the inverse problem is obtained in the self-adjoint case. In Section 7, solvability and stability results are transferred to the inverse problem by the \((2n-2)\) spectra. In Sections 8 and 9, we apply our main theorems to the special cases \(n = 3\) and \(n = 4\), respectively. In Appendix 10, asymptotic properties of solutions of equation 1 are described basing on previous studies [7], [25]–[27]. In Appendix 11, we present the derivation of the spectral data asymptotics for \(n = 3, 4\). In Appendix 12, we provide a discrete weighted version of Schur’s test, which is used for proving an auxiliary estimate in Section 7.

Throughout the paper, we use the following notations:

• \(W_2^s[0,1]\) is the Sobolev space with the norm \[\| y \|_{W_2^s[0,1]} = \left( \sum_{j = 0}^s \| y^{(j)} \|^2_{L_2[0,1]} \right)^{1/2}.\]

• \(\delta_{j,k}\) denotes the Kronecker delta.

• Along with \(\tau = \{ \tau_{\nu} \}_{\nu = 0}^{n-2}\), we consider other vector functions \(\tilde{\tau} = \{ \tilde{\tau}_{\nu} \}_{\nu = 0}^{n-2}\), \(\tilde{\tilde{\tau}} = \{ \tilde{\tilde{\tau}}_{\nu} \}_{\nu = 0}^{n-2}\), and \(\tau_{(u)} = \{ \tau_{\nu,(u)} \}_{\nu = 0}^{n-2}\) (\(u \ge 1\)) of the same class. We agree that, if a symbol \(\gamma\) denotes an object related to \(\tau\), then the symbols \(\tilde{\gamma}\), \(\tilde{\tilde{\gamma}}\), and \(\gamma_{(u)}\) will denote the similar objects related to \(\tilde{\tau}\), \(\tilde{\tilde{\tau}}\), and \(\tau_{(u)}\), respectively.

• In estimates, the same symbol \(C\) denotes various positive constants independent of \(\lambda\), \(x\), \(n\), etc. The notation \(C(A_1, A_2, \dots)\) means that the constant \(C\) depends on the parameters \(A_1\), \(A_2\), …, e.g., \(C(\Omega,\delta)\).

• \(C_n^k = \dfrac{n!}{k!(n-k)!}\) are binomial coefficients.

• The notations \(\lfloor a \rfloor\) and \(\lceil a \rceil\) are used for rounding a real number \(a\) down and up, respectively.

• \(J = \bigl\{ (l,k) \colon l \ge 1, \, k = \overline{1,n-1} \bigr\}\).

• \(V = \bigl\{ (l,k,\varepsilon) \colon (l,k) \in J, \varepsilon= 0, 1 \bigr\}\).

• \(m\) is the Banach space of bounded infinite sequences \(a = [a_v]_{v \in V}\) with the norm \(\| a \|_m = \sup_{v \in V} |a_v|\).

• \(I\) is the identity operator in \(m\).

• For \(Q > 0\), denote by \(B_Q\) the set of vectors \(\tau = \{ \tau_{\nu} \}_{\nu = 0}^{n-2}\) satisfying \(\| \tau_{\nu} \|_{W_2^{\nu}} \le Q\), \(\nu = \overline{0,n-2}\).

2 Main results↩︎

Let us begin with some preliminaries.

For \(k = \overline{1,n}\), denote by \(\Phi_k(x,\lambda)\) the so-called Weyl solution of equation 1 satisfying the boundary conditions \[\label{bcPhi} \Phi_k^{(j-1)}(0,\lambda) = \delta_{k,j}, \quad j = \overline{1,k}, \qquad \Phi_k^{(s-1)}(1,\lambda) = 0, \quad s = \overline{1,n-k}.\tag{5}\]

Define the Weyl functions as \(M_k(\lambda) := \Phi_k^{(k)}(0,\lambda)\), \(k = \overline{1,n-1}\). The functions \(\Phi_k^{(j-1)}(x,\lambda)\) for each fixed \(x \in [0,1]\), \(j = \overline{1,n}\) and \(M_k(\lambda)\) are meromorphic in \(\lambda\), and their poles coincide with the eigenvalues \(\{ \lambda_{l,k} \}_{l \ge 1}\) (see [7]). Under the assumptions (A-1) and (A-2), these poles are simple and \[\label{defbe} \beta_{l,k} := \mathop{\mathrm{Res}}_{\lambda= \lambda_{l,k}} M_k(\lambda) \ne 0, \quad (l,k) \in J,\tag{6}\] where \(J := \{ (l,k) \colon l \ge 1, \, k = \overline{1,n-1} \}\).

We call \(\{ \beta_{l,k} \}_J\) the weight numbers and the sequence \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) the spectral data of \(\tau \in \mathbf{W}\). Consider the following inverse spectral problem.

The solution of Inverse Problem 2 in the class \(\mathbf{W}_{simp}\) is unique (see [2], [7] and [27]). However, the assumptions (A-1) and (A-2) are crucial for uniqueness. In order to determine \(\tau\) in the general case, one has to specify the Weyl-Yurko matrix (see [7]). Necessary and sufficient conditions for solvability of Inverse Problem 2 in the class \(\tau \in \mathbf{W}_{simp}^+\) have been obtained in [29]. Definitions 3 and 4 below describe the properties that are sufficient for a sequence \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) to be the spectral data of some vector \(\tau \in \mathbf{W}_{simp}^+\).

Note that the spectral data possess the following asymptotics (see [7] and [26]): \[\begin{align} \tag{8} \lambda_{l,k} & = (-1)^{n-k} \bigl(c_{0,k} l^n + c_{1,k} l^{n-1} + c_{2,k} l^{n-2} + \dots + c_{l,k} + \varkappa_{l,k} \bigr), \quad \{ \varkappa_{l,k} \} \in l_2, \\ \tag{9} \beta_{l,k} & = -n \lambda_{l,k} \bigl( 1 + d_{1,k} l^{-1} + \dots + d_{n-1,k} l^{-(n-1)} + l^{-(n-1)} \eta_{l,k} \bigr), \quad \{ \eta_{l,k} \} \in l_2, \end{align}\] where \(c_{j,k}\) and \(d_{j,k}\) are complex constants that depend on \(\tau\), excluding \(c_{1,k}\) and \[\label{c0k} c_{0,k} = \biggl( \frac{\pi}{\sin \frac{\pi k}{n}} \biggr)^n.\tag{10}\] Therefore, the condition 7 in Definition 3 means that \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) satisfy the relations 8 and 9 with the coefficients \(c_{j,k}\) and \(d_{j,k}\) equal to the respective coefficients \(\tilde{c}_{j,k}\) and \(\tilde{d}_{j,k}\) from the spectral data asymptotics of \(\tilde{\tau}\). One can obtain explicit formulas for the coefficients \(c_{j,k}\) and \(d_{j,k}\) by using \(\{ \tau_{\nu} \}_{\nu = 0}^{n-2}\). Examples for \(n = 3\) and \(n = 4\) are provided in Sections 8 and 9, respectively. However, for higher orders \(n\), derivation of explicit formulas is technically complicated. Therefore, we will formulate our main result in terms of the asymptotical proximity of values \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) to the spectral data \(\{ \tilde{\lambda}_{l,k}, \tilde{\beta}_{l,k} \}_J\) of a so-called model vector \(\tilde{\tau}\).

Note that the relations 11 and 12 hold for the spectral data of any \(\tau \in \mathbf{W}_{simp}^+\) (see [29]).

The following proposition provides sufficient conditions for the solvability of Inverse Problem 2.

Strictly speaking, in [29], the case \(\tau_{\nu} \in W_2^{\nu-1}[0,1]\) (\(\nu = \overline{0,n-2}\)) was considered. Higher degree of smoothness \(\tau_{\nu} \in W_2^{\nu}[0,1]\) corresponds to one more term in the asymptotic relations 8 and 9 comparing with [29].

The main result of this paper is the following theorem on the uniform stability of Inverse Problem 2.

Note that the value \(Z\) in ?? is finite, because \(Z^2 \le \Xi_{(1)}^2 + \Xi_{(2)}^2\) due to 7 and ?? .

Proceed to the inverse problem by the system of \((2n-2)\) spectra of the problems \(\mathcal{L}_k\) 1 , 3 and \(\mathcal{M}_k\) 1 , 4 for \(k = \overline{1,n-1}\). We call the sequence of the corresponding eigenvalues \(\{ \lambda_{l,k}, \mu_{l,k} \}_J\) the eigenvalue set of \(\tau = \{ \tau_{\nu} \}_{\nu = 0}^{n-2}\).

As shown in Section 7, Inverse Problem 8 is easily reduced to Inverse Problem 2, which implies the uniqueness of recovering \(\tau \in \mathbf{W}_{simp}\) from the eigenvalue set (see [7]). We show that this reduction is stable and obtain for Inverse Problem 8 results analogous to Proposition 5 and Theorem 7. We begin with the following lemma, which describes some structural properties of the eigenvalue set.

The following definition and theorem show that the conditions of Lemma 9 together with suitable asymptotics, the separation conditions, and 13 in the odd-order case are sufficient for a sequence \(\{ \lambda_{l,k}, \mu_{l,k} \}_J\) to be the eigenvalue set of some \(\tau \in \mathbf{W}_{simp}^+\).

Note that the eigenvalues \(\{ \mu_{l,k} \}_{l \ge 1}\) are not required to be simple.

Next, let us formulate the analog of Definition 6 for eigenvalue sets.

Relying on Theorem 7, we obtain the unconditional uniform stability of Inverse Problem 8:

Theorems 7 and 13 generalize the results of Savchuk and Shkalikov [37], [38] on the uniform stability of the inverse Sturm-Liouville problems to the case of arbitrary order differential operators.

3 Main equation↩︎

In this section, Inverse Problem 2 is reduced to a linear equation in the Banach space of bounded infinite sequences. Our construction of the main equation is analogous to previous studies [7], [26], [27], [29], so we outline it briefly. However, we apply a new modification that appears in [40] for the \(n=2\). This modification makes the components of the operator and the free term in the main equation continuous with respect to the spectral data (see Remark 15 for details). This feature is crucial for investigating the stability of the inverse problem.

Consider two vectors of coefficients \(\tau = \{ \tau_{\nu} \}_{\nu = 0}^{n-2}\) and \(\tilde{\tau} = \{ \tilde{\tau}_{\nu} \}_{\nu = 0}^{n-2}\) of \(\mathbf{W}_{simp}\). The eigenvalues are assumed to be reordered so that, if \(\lambda_{l,k} = \tilde{\lambda}_{s,k}\), then \(l = s\). In addition, we assume that \[\label{assump} \lambda_{l,k} \ne \tilde{\lambda}_{s,k\pm 1} \quad \text{for all l,s \ge 1 and k}.\tag{16}\]

Introduce the notation \[\begin{gather} \nonumber V := \{ (l,k,\varepsilon) \colon l \in \mathbb{N}, \, k = \overline{1,n-1}, \, \varepsilon= 0, 1 \}, \\ \nonumber \lambda_{l,k,0} := \lambda_{l,k}, \quad \lambda_{l,k,1} := \tilde{\lambda}_{l,k}, \quad \beta_{l,k,0} := \beta_{l,k}, \quad \beta_{l,k,1} := \tilde{\beta}_{l,k}, \\ \label{defvv} \varphi_{l,k,\varepsilon}(x) := \Phi_{k+1}(x, \lambda_{l,k,\varepsilon}), \quad \tilde{\varphi}_{l,k,\varepsilon}(x) := \tilde{\Phi}_{k+1}(x, \lambda_{l,k,\varepsilon}), \quad (l,k,\varepsilon) \in V. \end{gather}\tag{17}\]

Recall that the function \(\Phi_{k+1}(x,\lambda)\) is meromorphic in the \(\lambda\)-plane with the poles \(\{ \lambda_{l,k+1} \}_{l \ge 1}\). In view of the assumptions (A-2) and 16 , the points \(\lambda_{l,k}\) and \(\tilde{\lambda}_{l,k}\) are regular for \(\Phi_{k+1}(x,\lambda)\), so the functions \(\varphi_{l,k,\varepsilon}(x)\) are correctly defined. Similar arguments are valid for \(\tilde{\varphi}_{l,k,\varepsilon}(x)\).

For \(k = \overline{1,n}\), denote by \(\Phi_k^{\star}(x, \lambda)\) the solution of the differential equation \[\begin{align} \nonumber \ell_n^{\star}(y) := & (-1)^n y^{(n)} + \sum_{k = 0}^{\lfloor n/2\rfloor - 1} (\tau_{2k}(x) y^{(k)})^{(k)} \\ \label{eqstar} - & \sum_{k = 0}^{\lfloor (n-1)/2\rfloor - 1} \bigl((\tau_{2k+1}(x) y^{(k)})^{(k+1)} + (\tau_{2k+1}(x) y^{(k+1)})^{(k)}\bigr) = \lambda y, \quad x \in (0,1), \end{align}\tag{18}\] satisfying the boundary conditions 5 . Obviously, in the case \(\tau \in \mathbf{W}^+\), we have \(\Phi_k^{\star}(x,\lambda) = \overline{\Phi_k(x, (-1)^n \overline{\lambda})}\).

Define the functions \[\label{defD} \tilde{D}_{k,k_0}(x, \mu, \lambda) = \begin{cases} \int_0^x \tilde{\Phi}_k^{\star}(t, \mu) \tilde{\Phi}_{k_0}(t, \lambda) \, dt, & k + k_0 > n + 1, \\ \frac{(-1)^{k+1}}{\lambda- \mu} + \int_0^x \tilde{\Phi}_k^{\star}(t, \mu) \tilde{\Phi}_{k_0}(t, \lambda) \, dt, & k + k_0 = n + 1, \\ -\int_x^1 \tilde{\Phi}_k^{\star}(t, \mu) \tilde{\Phi}_{k_0}(t, \lambda) \, dt, & k + k_0 < n + 1. \end{cases}\tag{19}\]

In [26] and [29], the following relation has been obtained: \[\label{relPhik} \Phi_{k_0}(x, \lambda) = \tilde{\Phi}_{k_0}(x, \lambda) + \sum_{(l,k,\varepsilon) \in V} (-1)^{\varepsilon+ n - k} \beta_{l,k,\varepsilon} \varphi_{l,k,\varepsilon}(x) \tilde{D}_{n-k+1,k_0}(x, \lambda_{l,k,\varepsilon}, \lambda), \quad k_0 = \overline{1,n}.\tag{20}\]

Let us discuss the correctness of the terms \(\tilde{D}_{n-k+1,k_0}(x, \lambda_{l,k,\varepsilon}, \lambda)\), which contain the functions \(\tilde{\Phi}^{\star}_{n-k+1}(x,\lambda_{l,k,\varepsilon})\). The eigenvalues of the problems \(\mathcal{L}_k^{\star}\) for equation 18 with the boundary conditions 3 satisfy the relation \(\lambda_{l,k}^{\star} = \lambda_{l,n-k}\), \(l \ge 1\), \(k = \overline{1,n-1}\) (see [29]). Therefore, the Weyl solution \(\tilde{\Phi}^{\star}_{n-k+1}(x,\lambda)\) has the poles \(\{ \tilde{\lambda}_{l,k-1} \}_{l \ge 1}\) if \(k > 1\). The assumptions (A-2) and 16 imply that \(\lambda_{l,k}\) and \(\tilde{\lambda}_{l,k}\) are regular points of \(\tilde{\Phi}^{\star}_{n-k+1}(x,\lambda)\), so the functions \(\tilde{D}_{n-k+1,k_0}(x, \lambda_{l,k,\varepsilon}, \lambda)\) are correctly defined.

Basing on the relation 20 , we provide a new construction of the main equation for Inverse Problem 2, employing the approach of [40]. Put \(\chi_{l,k} := l^{1-n} (\lambda_{l,k} - \tilde{\lambda}_{l,k})\) and introduce the matrices \[P_{l,k} := \begin{bmatrix} \chi_{l,k} & 1 \\ 0 & 1 \end{bmatrix}, \quad P_{l,k}^{-1} = \begin{bmatrix} \chi_{l,k}^{-1} & -\chi_{l,k}^{-1} \\ 0 & 1 \end{bmatrix} \:\: \text{if \lambda_{l,k} \ne \tilde{\lambda}_{l,k}}.\] Define the functions \[\label{defw} w_{l,k}(x) := l^{-k} \exp(-xl \cot(k\pi/n)),\tag{21}\] which are related to the growth of \(\varphi_{l,k,\varepsilon}(x)\): \(|\varphi_{l,k,\varepsilon}(x)| \le C w_{l,k}(x)\) (see Lemma 43).

For \((l,k,\varepsilon), (l_0,k_0,\varepsilon_0) \in V\), denote \[\begin{align} \nonumber \tilde{G}_{(l,k,\varepsilon), (l_0, k_0, \varepsilon_0)}(x) & := (-1)^{n-k} \beta_{l,k,\varepsilon} \tilde{D}_{n-k+1, k_0 + 1}(x, \lambda_{l,k,\varepsilon}, \lambda_{l_0,k_0,\varepsilon_0}), \\ \label{defdvv} \dot{\varphi}_{l,k,\varepsilon}(x) & := l^{n-1}\frac{d}{d\lambda} \Phi_{k+1}(x,\lambda)\Big|_{\lambda= \lambda_{l,k,\varepsilon}}, \\ \nonumber \dot{\tilde{G}}_{(l,k,\varepsilon), (l_0,k_0,\varepsilon_0)}(x) & := (-1)^{n-k} l^{l-1} \beta_{l,k,\varepsilon} \frac{d}{d \lambda} \tilde{D}_{n-k+1, k_0 + 1}(x, \lambda_{l,k,\varepsilon}, \lambda)\Big|_{\lambda= \lambda_{l_0,k_0,\varepsilon_0}}. \end{align}\tag{22}\]

Introduce the functions \[\begin{align} \begin{bmatrix} \tag{23} \psi_{l,k,0}(x) \\ \psi_{l,k,1}(x) \end{bmatrix} & := w_{l,k}^{-1}(x) P_{l,k}^{-1} \begin{bmatrix} \varphi_{l,k,0}(x) \\ \varphi_{l,k,1}(x) \end{bmatrix}, \: \lambda_{l,k} \ne \tilde{\lambda}_{l,k}, \\ \tag{24} \begin{bmatrix} \psi_{l,k,0}(x) \\ \psi_{l,k,1}(x) \end{bmatrix} & := w_{l,k}^{-1}(x) \begin{bmatrix} \dot{\varphi}_{l,k,1}(x) \\ \varphi_{l,k,1}(x) \end{bmatrix}, \: \lambda_{l,k} = \tilde{\lambda}_{l,k}, \end{align}\] \[\begin{gather} \label{defR1} \begin{bmatrix} \tilde{R}_{(l_0,k_0,0),(l,k,0)}(x) & \tilde{R}_{(l_0,k_0,0),(l,k,1)}(x) \\ \tilde{R}_{(l_0,k_0,1),(l,k,0)}(x) & \tilde{R}_{(l_0,k_0,1),(l,k,1)}(x) \end{bmatrix} \\ := \frac{w_{l,k}(x)}{w_{l_0,k_0}(x)} P_{l_0,k_0}^{-1} \begin{bmatrix} \tilde{G}_{(l,k,0),(l_0,k_0,0)}(x) & -\tilde{G}_{(l,k,1),(l_0,k_0,0)}(x) \\ \tilde{G}_{(l,k,0),(l_0,k_0,1)}(x) & -\tilde{G}_{(l,k,1),(l_0,k_0,1)}(x) \end{bmatrix} P_{l,k}, \quad \lambda_{l_0,k_0} \ne \tilde{\lambda}_{l_0,k_0}. \end{gather}\tag{25}\] \[\begin{gather} \label{defR2} \begin{bmatrix} \tilde{R}_{(l_0,k_0,0),(l,k,0)}(x) & \tilde{R}_{(l_0,k_0,0),(l,k,1)}(x) \\ \tilde{R}_{(l_0,k_0,1),(l,k,0)}(x) & \tilde{R}_{(l_0,k_0,1),(l,k,1)}(x) \end{bmatrix} \\ := \frac{w_{l,k}(x)}{w_{l_0,k_0}(x)} \begin{bmatrix} \dot{\tilde{G}}_{(l,k,0),(l_0,k_0,1)}(x) & -\dot{\tilde{G}}_{(l,k,1),(l_0,k_0,1)}(x) \\ \tilde{G}_{(l,k,0),(l_0,k_0,1)}(x) & -\tilde{G}_{(l,k,1),(l_0,k_0,1)}(x) \end{bmatrix} P_{l,k}, \quad \lambda_{l_0,k_0} = \tilde{\lambda}_{l_0,k_0}. \end{gather}\tag{26}\] Analogously to \(\psi_{l,k,\varepsilon}(x)\), define \(\tilde{\psi}_{l,k,\varepsilon}(x)\) by using \(\tilde{\varphi}_{l,k,\varepsilon}(x)\) instead of \(\varphi_{l,k,\varepsilon}(x)\).

It follows from the relation 20 that \[\label{relpsi} \psi_{l_0,k_0,\varepsilon_0}(x) = \tilde{\psi}_{l_0,k_0,\varepsilon_0}(x) + \sum_{(l,k,\varepsilon) \in V} \tilde{R}_{(l_0,k_0,\varepsilon_0), (l,k,\varepsilon)}(x) \psi_{l,k,\varepsilon}(x), \quad (l_0,k_0,\varepsilon_0) \in V, \: x \in [0,1].\tag{27}\]

The relations 27 can be treated as a system of main equations for solving Inverse Problem 2. Indeed, suppose that the spectral data \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) and the model vector \(\tilde{\tau}\) are given. Then, one can construct the functions \(\tilde{\psi}_{l,k,\varepsilon}(x)\) and \(\tilde{R}_{(l_0,k_0,\varepsilon_0), (l,k,\varepsilon)}(x)\) by using 23 , 24 and 25 , 26 , respectively. By solving the system 27 , the solution \(\{ \psi_{l,k,\varepsilon}(x) \}_V\) is obtained, and then one can find \[\label{findvv} \begin{bmatrix} \varphi_{l,k,0}(x) \\ \varphi_{l,k,1}(x) \end{bmatrix} = w_{l,k}(x) P_{l,k} \begin{bmatrix} \psi_{l,k,0}(x) \\ \psi_{l,k,1}(x) \end{bmatrix}\tag{28}\] by inverting 23 and 24 . Using \(\varphi_{l,k,\varepsilon}(x)\), one obtains \(\Phi_{k_0}(x,\lambda)\) (\(k_0 = \overline{1,n}\)) by the formula 20 and recovers \(\{ \tau_{\nu} \}_{\nu = 0}^{n-2}\).

Let us represent the system 27 in the operator form. Put \(v = (l,k,\varepsilon)\) and \(v_0 = (l_0,k_0,\varepsilon_0)\). Denote \[\label{defxi} \xi_l := \sum_{k = 1}^{n-1} \left( l^{-(n-1)} |\lambda_{l,k} - \tilde{\lambda}_{l,k}| + l^{-n} |\beta_{l,k} - \tilde{\beta}_{l,k}| \right), \quad l \ge 1.\tag{29}\]

There hold the estimates \[\label{estpsiR} |\psi_v(x)|, \, |\tilde{\psi}_v(x)| \le C, \quad |\tilde{R}_{v_0,v}(x)| \le \frac{C \xi_l}{|l - l_0| + 1}, \quad v, v_0 \in V, \quad x \in [0,1].\tag{30}\]

Recall that \(m\) is the Banach space of bounded infinite sequences \(a = [a_v]_{v \in V}\) with the norm \(\| a \|_m = \sup_{v \in V} |a_v|\). In view of 30 , the vectors \(\psi(x) := [\psi_v(x)]_V\) and \(\tilde{\psi}(x) := [\tilde{\psi}_v(x)]_V\) belong to \(m\) for each fixed \(x \in [0,1]\). Moreover, the linear operator \(\tilde{R}(x)\) given by the rule \[\label{opR} (\tilde{R}(x) a)_{v_0} = \sum_{v \in V} \tilde{R}_{v_0, v}(x) a_v, \quad a \in m,\tag{31}\] is bounded from \(m\) to \(m\): \[\label{estop} \| \tilde{R}(x) \|_{m \to m} = \sup_{v_0 \in V} \sum_{v \in V} |\tilde{R}_{v_0,v}(x)| \le C \left( \sum_{l= 1}^{\infty} \xi_l^2 \right)^{1/2} < \infty, \quad x \in [0,1].\tag{32}\]

Thus, the system 27 can be represented in the operator form: \[\label{maineq} (I - \tilde{R}(x)) \psi(x) = \tilde{\psi}(x), \quad x \in [0,1],\tag{33}\] where \(I\) is the identity operator.

Due to [26], the operator \((I - \tilde{R}(x))\) has a bounded inverse on \(m\) for each fixed \(x \in [0,1]\), so equation 33 has the unique solution \(\psi(x)\). The operator \((I - \tilde{R}(x))\) plays an important role in the proof of uniform stability theorems in the next sections.

4 Uniform bounds↩︎

In this section, we consider Inverse Problem 2 in the non-self-adjoint case and prove that its solution is uniformly bounded under some a priori conditions on the spectral data.

Assume that \(\tilde{\tau} \in \mathbf{W}_{simp}\) is fixed. Then, for any sequence \(\{ \lambda_{l,k}, \beta_{l,k} \}_J \in \mathcal{S}(\tilde{\tau})\) satisfying the conditions 16 , one can construct the linear bounded operator \(\tilde{R}(x)\) in the Banach space \(m\) following the arguments of Section 3.

Note that the conditions 35 imply 16 , which is essential for the operator \(\tilde{R}(x)\) to be correctly defined. Furthermore, the bounds 32 , 34 , and 35 imply the uniform bound for the operator: \[\label{uniR} \| \tilde{R}(x) \|_{m \to m} \le C(\Omega,\delta), \quad x \in [0,1].\tag{37}\]

The goal of this section is to prove the following theorem.

Theorem 17 is proved analogously to Theorem 2 in [27]. The most principal differences are as follows:

• The coefficients \(\tau_{\nu}\) in this paper have one more derivative than in [27], and so do auxiliary functions such as \(\psi_{l,k,\varepsilon}(x)\), \(\varphi_{l,k,\varepsilon}(x)\), etc.

• We need estimates to be uniform on \(\mathcal{S}_{\Omega,K,\delta}\), which is achieved using the conditions 7 , 35 , and 36 in Definition 16.

Thus, we outline the proof of Theorem 17 briefly, not elaborating into technical details.

5 Uniform stability↩︎

In this section, we continue the discussion of Inverse Problem 2 in the general non-self-adjoint case, started in the previous section. We consider two sets of spectral data and estimate the differences of the corresponding differential expression parameters. Specifically, we prove the following theorem on the uniform stability of Inverse Problem 2 in the general non-self-adjoint case.

Thus, Theorem 20 establishes the uniform stability of Inverse Problem 2 on the set \(\mathcal{S}_{\Omega,K,\delta}\) of spectral data. Let us discuss the constraints 35 and 36 forming this set. The condition 36 assumes the existence of the inverse operator \((I - \tilde{R}(x))^{-1}\) and its uniform bound. In the non-self-adjoint case, conditions of this kind are essential. Even for the second-order Sturm-Liouville operator with complex-valued potential, there are unknown necessary and sufficient conditions on spectral data that guarantee the unique solvability of the main equation 33 . Therefore, the known characterization theorems for the non-self-adjoint case (see, e.g., [11]) require the existence of the bounded inverse operator \((I - \tilde{R}(x))^{-1}\). Moreover, the uniform stability of the inverse Sturm-Liouville problem in the non-self-adjoint case has been obtained in [40] under the uniform bound 36 for the inverse operator.

Note that the condition 36 is related to a fixed model vector \(\tilde{\tau}\). Furthermore, for \(n > 2\), we have to impose the requirement 35 of separation from the eigenvalues of the model vector. In fact, for every \(\tau \in \mathbf{W}_{simp}\) there exists a model vector satisfying this requirement. However, in Definition 16, the model vector \(\tilde{\tau}\) is fixed, since it is used in construction of the operator \(\tilde{R}(x)\) in the constraint 36 . For the self-adjoint case, we will show in the next sections that the requirement 36 can be omitted, and so 35 can also be removed. But in the non-self-adjoint case the both constraints 35 and 36 are needed.

6 Self-adjoint case↩︎

This section aims to prove Theorem 7 basing on Theorem 20.

Suppose that \(\Omega, \delta> 0\) and \(\tilde{\tau} \in \mathbf{W}_{simp}^+\). Let \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) be any sequence of the set \(\mathcal{S}_{\Omega,\delta}^+(\tilde{\tau})\) introduced by Definition 6. Under the condition 35 , the operator \(\tilde{R}(x)\) is correctly defined. Moreover, each sequence \(\{ \lambda_{l,k}, \beta_{l,k} \}_J \in \mathcal{S}_{\Omega,\delta}^+ \subset \mathcal{S}^+\) is the spectral data of some \(\tau \in \mathbf{W}_{simp}^+\) by Proposition 5. Therefore, there exists a bounded inverse operator \((I - \tilde{R}(x))^{-1}\) on \(m\) for each fixed \(x \in [0,1]\). The following lemma shows that these operators are uniformly bounded on \(\mathcal{S}_{\Omega,\delta}^+\).

7 Reconstruction by \((2n-2)\) spectra↩︎

In this section, we study Inverse Problem 8 by the eigenvalue set \(\{ \lambda_{l,k}, \mu_{l,k} \}_J\). Following the approach of [7], we construct the weight numbers \(\{ \beta_{l,k} \}_J\) and so reduce Inverse Problem 8 to Inverse Problem 2 by the spectral data. Next, we investigate the properties of the eigenvalues \(\{ \mu_{l,k} \}_J\) and prove Theorems 11 and 13 basing on the corresponding results for Inverse Problem 2.

For \(k = \overline{1,n}\), let \(\mathcal{C}_k(x,\lambda)\) be the solution of equation 1 under the initial conditions \[\mathcal{C}_k^{(j-1)}(0,\lambda) = \delta_{k,j}, \quad j = \overline{1,n}.\] Clearly, the functions \(\mathcal{C}_k^{(j-1)}(x,\lambda)\) are entire in \(\lambda\) for each fixed \(x \in [0,1]\), \(k,j = \overline{1,n}\).

The functions \(\{ \mathcal{C}_k(x, \lambda) \}_{k = 1}^n\) and \(\{ \Phi_k(x, \lambda) \}_{k = 1}^n\) form fundamental systems of solutions of equation 2 . Therefore, the following relations hold: \[\Phi_k(x,\lambda) = \sum_{j = 1}^n m_{j,k}(\lambda) \mathcal{C}_j(x, \lambda), \quad k = \overline{1,n},\] where the matrix of the coefficients \(M(\lambda) = [m_{j,k}(\lambda)]_{j,k = 1}^n\) is called the Weyl-Yurko matrix.

The elements of the Weyl-Yurko matrix satisfy the following relations (see [7]): \[\label{relmjk} m_{j,k}(\lambda) = \delta_{j,k}, \quad j \le k, \qquad m_{j,k}(\lambda) = -\frac{\Delta_{j,k}(\lambda)}{\Delta_{k,k}(\lambda)}, \quad j > k,\tag{60}\] where \[\label{defDelta} \Delta_{k,k}(\lambda) := \det([\mathcal{C}_j^{(n-s)}(1,\lambda)]_{s,j = k+1}^n), \quad k = \overline{1,n-1},\tag{61}\] and \(\Delta_{j,k}(\lambda)\) is obtained from \(\Delta_{k,k}(\lambda)\) by replacing \(\mathcal{C}_j\) by \(\mathcal{C}_k\). Obviously, the functions \(\Delta_{j,k}(\lambda)\) (\(1 \le k \le j \le n\)) are entire in \(\lambda\).

For each \(k \in \{ 1, 2, \dots, n-1 \}\), the eigenvalues \(\{ \lambda_{l,k} \}_{l \ge 1}\) and \(\{ \mu_{l,k} \}_{l \ge 1}\) of the corresponding boundary value problems \(\mathcal{L}_k\) 1 , 3 and \(\mathcal{M}_k\) 1 , 4 coincide with the zeros of the functions \(\Delta_{k,k}(\lambda)\) and \(\Delta_{k+1,k}(\lambda)\), respectively (counting with multiplicities). In view of 6 and 60 , the functions \(M_k(\lambda) = m_{k,k}(\lambda)\) are meromorphic in \(\lambda\), their poles belong to the set \(\{ \lambda_{l,k} \}_{l \ge 1}\), and \[\beta_{l,k} = -\mathop{\mathrm{Res}}_{\lambda= \lambda_{l,k}} \frac{\Delta_{k+1,k}(\lambda)}{\Delta_{k,k}(\lambda)}, \quad (l,k) \in J.\]

Under the assumptions (A-1) and (A-2), we have \[\label{beDe} \beta_{l,k} = -\frac{\Delta_{k+1,k}(\lambda_{l,k})}{\dot{\Delta}_{k,k}(\lambda_{l,k})} \ne 0, \quad (l,k) \in J,\tag{62}\] where \(\dot{\Delta}(\lambda) = \tfrac{d}{d\lambda}\Delta(\lambda)\).

The characteristic functions can be constructed as infinite products by using their zeros (see [11]): \[\begin{align} \tag{63} \Delta_{k,k}(\lambda) & = \Delta_{k,k}^0(0) \prod_{s = 1}^{\infty} \frac{\lambda_{s,k} - \lambda}{\lambda_{s,k}^0}, \\ \tag{64} \Delta_{k+1,k}(\lambda) & = \Delta_{k+1,k}^0(0) \prod_{s = 1}^{\infty} \frac{\mu_{s,k} - \lambda}{\mu_{s,k}^0}, \end{align}\] where the values \(\Delta_{k,k}^0(0) \ne 0\), \(\Delta_{k+1,k}^0(0) \ne 0\), \(\lambda_{s,k}^0\), and \(\mu_{s,k}^0\) correspond to the zero vector \(\tau\).

Using 63 , 64 , and 62 , Inverse Problem 8 by the eigenvalue set \(\{ \lambda_{l,k}, \mu_{l,k}\}_J\) is reduced to Inverse Problem 2 by the spectral data \(\{ \lambda_{l,k}, \beta_{l,k}\}_J\).

Proceed to proving the main results for Inverse Problem 8.

8 Case \(n = 3\)↩︎

For \(n = 3\), equation 1 takes the form \[\label{eqv3} y''' + (\tau_1(x) y)' + \tau_1(x) y' + \tau_0(x) y = \lambda y, \quad x \in (0,1),\tag{71}\] where \(\tau = \{ \tau_0, \tau_1 \} \in \mathbf{W}\) means \(\tau_0 \in L_2[0,1]\), \(\tau_1 \in W_2^1[0,1]\). The spectral data \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) correspond the the eigenvalue problems \(\mathcal{L}_1\) and \(\mathcal{L}_2\) with the boundary conditions \[\begin{align} \label{bc31} & \mathcal{L}_1 \colon \quad y(0) = 0, \quad y(1) = y'(1) = 0, \\ \nonumber & \mathcal{L}_2 \colon \quad y(0) = y'(0) = 0, \quad y(1) = 0, \end{align}\tag{72}\] and \(\{ \mu_{l,k}\}_J\) are the eigenvalues of the corresponding problems \(\mathcal{M}_1\) and \(\mathcal{M}_2\) with the boundary conditions \[\begin{align} & \mathcal{M}_1 \colon \quad y'(0) = 0, \quad y(1) = y'(1) = 0, \\ & \mathcal{M}_2 \colon \quad y(0) = y''(0) = 0, \quad y(1) = 0, \end{align}\]

The proof of Theorem 25 is provided in Appendix 11.2.

Suppose that \(\tau \in \mathbf{W}^+\), that is, \(\tau_1 \in W_2^1[0,1]\) is real-valued and \(\tau_0 \in L_2[0,1]\) is purely imaginary. In view of 11 , the spectral data \(\{ \lambda_l, \beta_l \}_{l \ge 1} := \{ \lambda_{l,1}, \beta_{l,1}\}_{l \ge 1}\) are sufficient for the unique reconstruction of \(\tau \in \mathbf{W}_{simp}^+\). Proposition 5 implies the following sufficient conditions for the existence of the inverse problem solution.

Theorem 26 is consistent with [28] for the class \(\tau_0 \in W_2^{-1}[0,1]\), \(\tau_1 \in L_2[0,1]\).

Analogously, Theorem 11 implies the following sufficient conditions for the unique reconstruction of \(\tau = \{ \tau_0, \tau_1\} \in \mathbf{W}^+_{simp}\) from the two spectra \(\{ \lambda_l, \mu_l \}_{l \ge 1} := \{ \lambda_{l,1}, \mu_{l,1}\}_{l \ge 1}\).

Similarly, Theorems 25, 7, and 13 imply the following theorems on the uniform stability of Inverse Problem 2 in the case \(n = 3\).

In a special case, we obtain the following lemma on additional properties of the spectral data \(\{ \lambda_l, \beta_l \}_{l \ge 1}\).

Analogously, we get the sufficient conditions on the two spectra:

The uniform stability of the inverse problems in the special case \(\tau_0 \equiv 0\) holds due to Theorems 28 and 29.

9 Case \(n = 4\)↩︎

For \(n = 4\), we have the equation \[\label{eqv4} y^{(4)} + (\tau_2(x)y')' + (\tau_1(x)y)' + \tau_1(x)y' + \tau_0(x)y = \lambda y, \quad x \in (0,1),\tag{74}\] where \(\tau = \{ \tau_0, \tau_1, \tau_2 \} \in \mathbf{W}^+\) means that the functions \(\tau_0 \in L_2[0,1]\) and \(\tau_2 \in W_2^2[0,1]\) are real-valued and \(\tau_1 \in W_2^1[0,1]\) is purely imaginary. For the recovery of \(\tau_0\), \(\tau_1\), and \(\tau_2\), we use the eigenvalues of the boundary value problems \(\mathcal{L}_k\) (\(k = 1, 2, 3\)) for equation 74 with the boundary conditions \[\begin{align} \tag{75} \mathcal{L}_1 \colon & \quad y(0) = 0, \quad y(1) = y'(1) = y''(1) = 0, \\ \tag{76} \mathcal{L}_2 \colon & \quad y(0) = y'(0) = 0, \quad y(1) = y'(1) = 0, \\ \nonumber \mathcal{L}_3 \colon & \quad y(0) = y'(0) = y''(0) = 0, \quad y(1) = 0. \end{align}\]

Note that \(\mathcal{L}_2 = \mathcal{L}_2^*\) and \(\mathcal{L}_1 = \mathcal{L}_3^*\).

The following theorem provides the spectral data characterization for \(\tau \in \mathbf{W}_{simp}^+\) in the case \(n = 4\).

In the sufficiency part of Theorem 33, the constants \(\theta\), \(\theta_0\), \(\theta_1\), \(\theta_0'\), \(\theta_1'\), and \(\phi\) can be arbitrary real numbers and the constants \(\sigma\), \(\sigma_0\), and \(\sigma_1\), arbitrary purely imaginary complex numbers. Then the relations ?? are a part of the conclusion of the theorem.

Theorems 7 and 33 together imply the uniform stability of Inverse Problem 2. For clarity, we formulate the corresponding theorem by using the spectral data \(\{ \lambda_{l,k}, \beta_{l,k} \}\) only for \(k = 1, 2\), since \(\lambda_{l,3}\) and \(\beta_{l,3}\) are uniquely specified by 11 .

Consider the special case \(\tau_1 \equiv 0\) arising in applications (see [16], [17]). Similarly to Lemma 30 and Theorem 31, we obtain the following result.

Note that, under the conditions of Theorem 35, there holds \(\{ \lambda_{l,1}, \beta_{l,1} \}_{l \ge 1} = \{ \lambda_{l,3}, \beta_{l,3} \}_{l \ge 1}\) in view of 11 .

Proceed to Inverse Problem 8 by the eigenvalue set \(\{ \lambda_{l,k}, \mu_{l,k} \}_J\). For \(n = 4\), \(\{ \mu_{l,k}\}_{l \ge 1}\) are the eigenvalues of the problems \(\mathcal{M}_k\) (\(k = 1, 2, 3\)) for equation 74 with the boundary conditions: \[\begin{align} \label{bcmu1} \mathcal{M}_1 \colon & \quad y'(0) = 0, \quad y(1) = y'(1) = y''(1) = 0, \\ \nonumber \mathcal{M}_2 \colon & \quad y(0) = y''(0) = 0, \quad y(1) = y'(1) = 0, \\ \nonumber \mathcal{M}_3 \colon & \quad y(0) = y'(0) = y'''(0) = 0, \quad y(1) = 0. \end{align}\tag{77}\]

The following theorem is proved in Appendix 11.3.

Theorems 11, 33, and 36 imply the following sufficient conditions for the existence of solution of Inverse Problem 8 in the case \(n = 4\).

Theorems 13 and 37 together imply the uniform stability of Inverse Problem 8. The corresponding theorem is formulated by using the eigenvalues \(\{ \lambda_{l,k}, \mu_{l,k} \}\) only for \(k = 1, 2\), since \(\lambda_{l,3}\) and \(\mu_{l,3}\) are uniquely specified by ?? .

Analogously to Theorem 35, we obtain sufficient conditions on the eigenvalue set for the special case \(\tau_1 \equiv 0\).

The conditions of Theorem 39, in particular, imply that \(\{ \lambda_{l,1}, \mu_{l,1}\}_{l \ge 1} = \{ \lambda_{l,3}, \mu_{l,3}\}_{l \ge 1}\) and these values are real for sufficiently large \(l\).

Inverse Problems 2 and 8 in the case \(n = 4\), \(\tau_1 \equiv 0\) are uniformly stable according to Theorems 34 and 38.

10 Estimates for Weyl solutions↩︎

In this appendix, we recall asymptotic properties of the Weyl solutions \(\Phi_k(x,\lambda)\) from [7], [25]. Moreover, we present the estimates for \(\varphi_{l,k,\varepsilon}(x)\) and other functions of Sections 3 and 4 from previous studies [26], [27] and discuss the conditions under which these estimates are uniform.

Put \(\lambda= \rho^n\) and divide the \(\rho\)-plane into the sectors \[\label{defGa} \Gamma_s := \left\{ \rho \in \mathbb{C} \colon \frac{\pi(s-1)}{n} < \arg \rho < \frac{\pi s}{n}\right\}, \quad s = \overline{1,2n}.\tag{78}\]

In each fixed sector \(\Gamma_s\), denote by \(\{ \omega_k \}_{k = 1}^n\) the roots of the equation \(\omega^n = 1\) numbered so that \[\label{order} Re \, (\rho \omega_1) < Re \, (\rho \omega_2) < \dots < Re \, (\rho \omega_n), \quad \rho \in \Gamma_s.\tag{79}\]

For \(s \in \{ 1, 2, \dots, 2n \}\), \(h > 0\), and \(\rho_* > 0\), define the region \[\Gamma_{s,h,\rho_*} := \left\{ \rho \in \mathbb{C} \colon \rho + h \exp\bigl( \tfrac{i \pi (s - 1/2)}{n}\bigr) \in \Gamma_s, \, |\rho| > \rho_*\right\}.\]

In order to estimate the Weyl solutions \(\Phi_k(x,\rho)\), we use the Birkhoff solutions described by the following proposition.

Assume that \(\mathbf{p} = \{ p_s \}_{s = 0}^{n-2}\) in \(B_Q\) for some \(Q > 0\), that is, \(\| p_s \|_{W_2^s[0,1]} \le Q\) for \(s = \overline{0,n-2}\). Then the value \(\rho_*\) in Proposition 41 and the constant \(C\) in ?? depend only on \(h\) and \(Q\).

For \(\rho \in \overline{\Gamma}_{s,h,\rho_*}\), the Weyl solutions can be expanded as \[\label{expPhik} \Phi_k(x,\lambda) = \sum_{j = 1}^n b_{j,k}(\rho) y_j(x,\rho),\tag{80}\] where the coefficients \(b_{j,k}(\rho)\) are analytic for \(\rho^n \ne \lambda_{l,k}\).

Denote by \(\{ \lambda_{l,k}^0 \}_{l \ge 1}\) the eigenvalues of the boundary value problems \(\mathcal{L}_k^0\) for the equation \(y^{(n)} = \lambda y\) with the boundary conditions 3 . For sufficiently large \(l\), the eigenvalues \(\{ \lambda_{l,k}^0 \}\) are real (see [29]). Let \(\rho_*\) be sufficiently large and \(l_*\) be such that \(|\lambda_{l,k}^0| \ge \rho_*^n\) for all \(l \ge l_*\). Then, for \(l \ge l_*\), one can choose the roots \(\rho_{l,k}^0 := \sqrt[n]{\lambda_{l,k}^0}\) lying in \(\Gamma_{s,h,\rho_*}\). So, define the regions \[\Gamma_{s,h,\rho_*}^{k,\delta} := \bigl\{ \rho \in \Gamma_{s,h,\rho_*} \colon \forall l \ge l_* \: |\rho - \rho_{l,k}^0| \ge \delta\}.\]

In view of the asymptotics 8 , there holds \(\rho_{l,k} - \rho_{l,k}^0 = o(1)\) as \(l \to \infty\), where \(\rho_{l,k} = \sqrt[n]{\lambda_{l,k}} \in \Gamma_{s,h,\rho_*}\). Hence, for sufficiently large \(\rho_*\) and sufficiently small \(h\) and \(\delta\), the function \(\Phi_k(x, \rho^n)\) is analytic in \(\Gamma_{s,h,\rho_*}^{k,\delta}\).

By using Proposition 41 and 80 , the following lemma is deduced.

Using Lemma 42, we obtain the estimates for the functions \(\varphi_{l,k,\varepsilon}(x)\) and \(\dot{\varphi}_{l,k,\varepsilon}(x)\) defined by 17 and 22 , respectively, summarized in the following lemma.

Clearly, Lemma 42 is applicable to the Weyl solutions \(\Phi_k^{\star}(x,\lambda)\) of equation 18 . Under the assumptions of Lemma 43, we get the following estimates (see [27]): \[\label{estPhi42} |\Phi_{n-k+1}^{\star(j)}(x, \lambda_{l,k,\varepsilon})| \le C l^{j-n} w_{l,k}^{-1}(x), \quad |\Phi_{n-k+1}^{\star(j)}(x, \lambda_{l,k,0}) - \Phi_{n-k+1}^{\star(j)}(x, \lambda_{l,k,1})| \le C l^{\nu - n} w_{l,k}^{-1}(x) \xi_l,\tag{82}\] where \(l \ge 1\), \(k = \overline{1,n-1}\), \(x \in [0,1]\), and \(C = C(Q,\delta)\). Using 82 together with 9 and 29 , we obtain Lemma 19.

Applying 9 , 82 , and the estimates of Lemma 42 to the definitions of the functions \(\psi_v(x)\), \(\tilde{\psi}_v(x)\), and \(\tilde{R}_{v_0, v}(x)\) from Section 3, we arrive at 30 . Suppose that the model vector \(\tilde{\tau} \in W_{simp}\) is fixed and the data \(\{ \lambda_{l,k}, \beta_{l,k} \}_J\) satisfy the assumptions \(\Xi \le \Omega\) and 35 . Then the constant \(C\) in the estimates 30 for \(\tilde{\psi}_v(x)\) and \(\tilde{R}_{v_0,v}(x)\) depends only on \(\Omega\) and \(\delta\). Under these conditions, the relation 32 implies 37 .

11 Spectral data asymptotics↩︎

In this appendix, we derive the asymptotic relations for the spectral characteristics \(\lambda_{l,k}\), \(\mu_{l,k}\), and \(\beta_{l,k}\) in the cases \(n = 3\) and \(n = 4\) by using the standard method [44].

11.1 Basic strategy↩︎

Let \(n \ge 2\) and \(k \in \{1, 2, \dots, n-1 \}\) be fixed. Recall that the eigenvalues \(\{ \lambda_{l,k} \}_{l \ge 1}\) of the problem \(\mathcal{L}_k\) are the zeros of the determinant \(\Delta(\lambda) := \Delta_{k,k}(\lambda)\) given by 61 .

Consider the sector \(\Gamma_1\) given by 78 and the Birkhoff solutions \(y_k(x,\rho)\) (\(k = \overline{1,n}\)) defined in the extended sector \(\rho \in \Gamma_{1,h,\rho_*}\) according to Proposition 41. Suppose that \(\lambda= \rho^n\), \(\rho \in \Gamma_{1,h,\rho_*}\). Expand the solutions \(\mathcal{C}_k(x,\lambda)\) (\(k = \overline{1,n}\)) over the Birkhoff solutions: \[\label{expandC} [\mathcal{C}_k^{(j-1)}(x,\lambda)]_{k,j = 1}^n = [y_k^{(j-1)}(x,\rho)]_{k,j=1}^n A(\rho),\tag{83}\] where \(A(\rho)\) is an \((n \times n)\) matrix function. Then, we get the relation \[\label{DelD} \Delta(\lambda) = D(\rho) \det A(\rho),\tag{84}\] where \(D(\rho)\) is some determinant composed of the functions \(y_k^{(\nu)}(x,\rho)\) at \(x \in \{ 0, 1\}\) and \(\det A(\rho) \ne 0\) for sufficiently large \(|\rho|\), \(\rho \in \Gamma_{1,h,\rho_*}\). Consequently, for large \(l\), there holds \(\lambda_{l,k} = \rho_{l,k}^n\), where \(\rho_{l,k}\) are the zeros of \(D(\rho)\). Thus, it remains to find the asymptotics of \(D(\rho)\) and its zeros.

Analogously, the eigenvalues \(\{ \mu_{l,k} \}_{l \ge 1}\) of the problem \(\mathcal{M}_k\) 1 , 4 coincide with the zeros of \(\Delta^+(\lambda) := \Delta_{k+1,k}(\lambda)\). Using 83 , we get the representation \[\label{DelDp} \Delta^+(\lambda) = -D^+(\rho) \det A(\rho),\tag{85}\] so it remains to find the asymptotics of the zeros of \(D^+(\rho)\).

Proceed to the weight numbers. For large \(l\), the eigenvalues \(\lambda_{l,k}\) are simple zeros of \(\Delta(\lambda)\), so the relation 62 holds. Taking 84 and 85 into account, we obtain \[\label{calcbe} \beta_{l,k} = n \rho_{l,k}^{n-1} \frac{D^+(\rho_{l,k})}{\frac{d}{d\rho} D(\rho_{l,k})}.\tag{86}\]

In this appendix, we use the notation \(\Upsilon(\rho)\) for various functions satisfying \(\Upsilon(\rho) \to 0\) as \(|\rho| \to \infty\) in \(\Gamma_{1,h,\rho_*}\) and \(\{ \Upsilon(\rho_l) \} \in l_2\) for any non-condensing sequence \(\{ \rho_l \} \subset \Gamma_{1,h,\rho_*}\). A sequence \(\{ \rho_l \}_{l = 1}^{\infty}\) is called non-condensing if \[\sup_{r > 0}(N(r+1) - N(r)) < \infty, \quad N(r) := \# \{ l \in \mathbb{N} \colon |\rho_l| \le r \}.\]

The notation \(\Upsilon(x,\rho)\) denotes any function that possesses the same properties as \(\Upsilon(\rho)\) for each fixed \(x \in [0,1]\).

In order to derive asymptotic formulas for \(D(\rho)\) and \(D^+(\rho)\), we use the well-known asymptotics for the Birkhoff solutions and their derivatives (see, e.g., [44]): \[\label{asympty} \def\arraystretch{2.2} \left. \begin{array}{l} y_k(x,\rho) = \exp(\rho \omega_k x) \left( 1 + \displaystyle\sum\limits_{s = 1}^{n-1} \dfrac{q_s(x)}{(\rho \omega_k)^s} + \varepsilon(x,\rho)\right), \\ y_k^{(\nu)}(x,\rho) = (\rho \omega_k)^{\nu} \exp(\rho \omega_k x) \left( 1 + \displaystyle\sum\limits_{s = 1}^{n-1} \dfrac{q_{s \nu}(x)}{(\rho \omega_k)^s} + \varepsilon(x,\rho) \right), \quad \nu = \overline{1,n-1}, \end{array} \quad \right\}\tag{87}\] for \(k = \overline{1,n}\), \(|\rho| \to \infty\), \(\rho \in \Gamma_{1,h,\rho_*}\). Here the notation \(\varepsilon(x,\rho)\) means various functions of form \(\dfrac{\Upsilon(x,\rho)}{\rho^{n-1}}\). This form of the remainder terms in 87 follows from the results of [48]. We use the formulas for the functions \(q_s(x)\) and \(q_{s\nu}(x)\) presented in [45]: \[\label{defq} \def\arraystretch{1.5} \left. \begin{array}{c} q_1'(x) = -\dfrac{1}{n} p_{n-2}(x), \quad q_2'(x) = -\dfrac{1}{n} \bigl( C_n^2 q_1''(x) + p_{n-2}(x) q_1(x) + p_{n-3}(x)\bigr), \\ q_3'(x) = -\dfrac{1}{n} \Bigl( C_n^2 q_2''(x) + C_n^3 q_1'''(x) + p_{n-2}(x)\bigl(q_2(x) + C_{n-2}^1 q_1'(x)\bigr) + p_{n-3}(x) q_1(x) + p_{n-4}(x)\Bigr), \\ q_{s\nu}(x) = \displaystyle\sum\limits_{r = 0}^{\nu} C_{\nu}^r q_{s-r}^{(r)}(x), \quad q_0(x) = 1, \quad q_s(x) = 0 \: \text{for} \: s < 0. \end{array} \right\}\tag{88}\]

Integration constants for the differential equations in 88 can be chosen arbitrarily.

For finding the coefficients in the asymptotic expansions of the characteristic functions and their zeros, we use symbolic computations in Python [46].

By virtue of [26], the Weyl functions associated with equations 1 and 18 satisfy the relation \(M_k(\lambda) = M_{n-k}^{\star}(\lambda)\) for \(k = \overline{1,n-1}\). Consequently, in view of 6 and 60 , there holds \[\label{relstar} \lambda_{l,k} = \lambda_{l,n-k}^{\star}, \quad \mu_{l,k} = \mu_{l,n-k}^{\star}, \quad \beta_{l,k} = \beta_{l,n-k}^{\star}, \quad k = \overline{1,n-1},\tag{89}\] which allows us to reduce the amount of computations.

11.2 Case \(n = 3\)↩︎

This subsection contains the proof of Theorem 25. Here, we use the notations:

• \(\omega_1 = \exp(2 \pi i/3)\), \(\omega_2 = \exp(-2 \pi i/3)\), and \(\omega_3 = 1\) are the roots \(\sqrt[3]{1}\) satisfying 79 for \(\rho \in \Gamma_1\).

• \(\varepsilon(\rho) = \dfrac{\Upsilon(\rho)}{\rho^2}\).

• \(\{ \varepsilon_l \}\) denotes various sequences such that \(\{ l^2 \varepsilon_l \} \in l_2\).

• \(t := \frac{1}{3} \int_0^1 \tau_1(x) \, dx\), \(t_0 := \frac{1}{3} \tau_1(0)\), \(t_1 := \frac{1}{3} \tau_1(1)\), \(s := \frac{1}{3} \int_0^1 \tau_0(x)\,dx\).

Using 88 and the relations \(p_1 = 2 \tau_1\), \(p_0 = \tau_1' + \tau_0\), we find \[\begin{gather} q_1(x) = -\frac{2}{3} \int_0^x \tau_1(\xi) \, d\xi, \quad q_2(x) = \frac{2}{9} \left( \int_0^x \tau_1(\xi) \, d\xi\right)^2 + \frac{1}{3} \tau_1(x) - \frac{1}{3} \int_0^x \tau_0(\xi) d\xi, \\ q_{11}(x) = q_1(x), \quad q_{21}(x) = q_2(x) - \frac{2}{3} \tau_1(x). \end{gather}\]

According to our notations, we have \[\label{valq} \def\arraystretch{1.3} \left.\begin{array}{c} q_1(0) = q_{11}(0) = 0, \quad q_2(0) = t_0, \quad q_{21}(0) = -t_0, \\ q_1(1) = q_{11}(1) = -2t, \quad q_2(1) = 2 t^2 + t_1 - s, \quad q_{21}(1) = 2 t^2 - t_1 - s. \end{array}\quad\right\}\tag{90}\]

The eigenvalues \(\{ \lambda_{l,1} \}_{l \ge 1}\) of the problem \(\mathcal{L}_1\) 71 –72 coincide with the zeros of the characteristic determinant \[\Delta(\lambda) := \Delta_{1,1}(\lambda) = \begin{vmatrix} \mathcal{C}_1(0,\lambda) & \mathcal{C}_2(0,\lambda) & \mathcal{C}_3(0,\lambda) \\ \mathcal{C}_1'(1,\lambda) & \mathcal{C}_2'(1,\lambda) & \mathcal{C}_3'(1,\lambda) \\ \mathcal{C}_1(1,\lambda) & \mathcal{C}_2(1,\lambda) & \mathcal{C}_3(1,\lambda) \end{vmatrix}.\]

The relation 84 holds with \[\label{defD1} D(\rho) := \begin{vmatrix} y_1(0,\rho) & y_2(0,\rho) & y_3(0,\rho) \\ y_1'(1,\rho) & y_2'(1,\rho) & y_3'(1,\rho) \\ y_1(1,\rho) & y_2(1,\rho) & y_3(1,\rho) \end{vmatrix}.\tag{91}\]

For brevity, denote \(\omega:= \omega_1\). Substituting 87 for \(n = 3\) together with 90 into 91 , we get \[\begin{align} & D(\rho) = \rho \exp(\rho (\omega_2 + \omega_3)) d(\rho), \\ & d(\rho) = r_1(\rho) - r_2(\rho) \exp(\rho(\omega_1-\omega_2)) + \varepsilon(\rho), \\ & r_1(\rho) = \omega^2 - 1 + \frac{2t(\omega- \omega^2)}{\rho} + \frac{(1-\omega)(s + 2t^2 + t_0 + t_1)}{\rho^2}, \\ & r_2(\rho) = \omega-1 + \frac{2 t(\omega^2 - \omega)}{\rho} + \frac{(1-\omega^2)(s + 2t^2 + t_0 + t_1)}{\rho^2}. \end{align}\]

The zeros of \(d(\rho)\) for large \(|\rho|\) satisfy the relation \[\label{rho21} \rho = \frac{1}{\omega_2 - \omega_1} \ln \frac{r_2(\rho)}{r_1(\rho)} + \varepsilon(\rho).\tag{92}\] Consequently, we get \[\rho_{l,1} = \frac{1}{2 i \sin \frac{\pi}{3}} \left( 2 \pi i l + \frac{\pi i}{3} + \frac{2t(\omega^2 - \omega)}{\rho_{l,1}} + \frac{(s + t_0 + t_1)(\omega- \omega^2)}{\rho_{l,1}^2} + \varepsilon_l \right),\] which implies \[\rho_{l,1} = \frac{\pi}{\sin\frac{\pi}{3}} \left( l + \frac{1}{6} - \frac{\theta}{2 \pi^2 \bigl(l + \frac{1}{6}\bigr)} + \frac{\sqrt 3 (\sigma + \theta_0 + \theta_1)}{8 \pi^3 l^2} + \varepsilon_l \right),\] according to the notations ?? . Finding \(\rho_{l,1}^3\), we arrive at ?? .

Proceed to obtaining the asymptotics for \(\mu_{l,1}\). We have \[\Delta^+(\lambda) := \Delta_{2,1}(\lambda) = -\begin{vmatrix} \mathcal{C}_1'(0,\lambda) & \mathcal{C}_2'(0,\lambda) & \mathcal{C}_3'(0,\lambda) \\ \mathcal{C}_1'(1,\lambda) & \mathcal{C}_2'(1,\lambda) & \mathcal{C}_3'(1,\lambda) \\ \mathcal{C}_1(1,\lambda) & \mathcal{C}_2(1,\lambda) & \mathcal{C}_3(1,\lambda) \end{vmatrix}, \quad D^+(\rho) = \begin{vmatrix} y_1'(0,\rho) & y_2'(0,\rho) & y_3'(0,\rho) \\ y_1'(1,\rho) & y_2'(1,\rho) & y_3'(1,\rho) \\ y_1(1,\rho) & y_2(1,\rho) & y_3(1,\rho) \end{vmatrix}.\]

Substituting 87 and 90 into the formula for \(D^+(\rho)\), we derive \[\begin{align} & D^+(\rho) = \rho^2 \exp(\rho (\omega_2 + \omega_3)) d^+(\rho), \\ & d^+(\rho) = r_1^+(\rho) - r_2^+(\rho) \exp(\rho(\omega_1 - \omega_2)) + \varepsilon(\rho), \\ & r_1^+(\rho) = 1 - \omega+ \frac{2t(\omega^2 - 1)}{\rho} + \frac{(\omega-\omega^2)(s + 2t^2 - t_0 + t_1)}{\rho^2}, \\ & r_2^+(\rho) = 1 - \omega^2 + \frac{2t(\omega-1)}{\rho} + \frac{(\omega^2 - \omega)(s + 2t^2 - t_0 + t_1)}{\rho^2}. \end{align}\]

The zeros of \(d^+(\rho)\) for large \(|\rho|\) satisfy the relation \[\label{rho21p} \rho = \frac{1}{\omega_2 - \omega_1} \ln \frac{r_2^+(\rho)}{r_1^+(\rho)} + \varepsilon(\rho).\tag{93}\] Consequently, they have the form \[\begin{align} \varrho_{l,1} & = \frac{1}{2 i \sin \frac{\pi}{3}} \left( 2 \pi i l - \frac{\pi i}{3} + \frac{2t(\omega^2 - \omega)}{\varrho_{l,1}} + \frac{(s - t_0 + t_1)(\omega- \omega^2)}{\varrho_{l,1}^2} + \varepsilon_l \right) \\ & = \frac{\pi}{\sin\frac{\pi}{3}} \left( l - \frac{1}{6} - \frac{\theta}{2 \pi^2 \bigl(l - \frac{1}{6}\bigr)} + \frac{\sqrt 3 (\sigma - \theta_0 + \theta_1)}{8 \pi^3 l^2} + \varepsilon_l \right) \end{align}\] according to the notations ?? . Computing \(\mu_{l,1} = \varrho_{l,1}^3\) implies ?? .

Note that the numbering in ?? and ?? starts from \(l = 1\), since this does not depend on \(\tau\) and can be directly verified for \(\tau = \{ 0, 0 \}\). Moreover, this fact for \(\lambda_{l,1}\) is known from previous studies (see, e.g., [28]).

Proceed to obtaining the asymptotics for the weight numbers \(\beta_{l,1}\). Taking 86 and the equality \(D(\rho_{l,k}) = 0\) into account, we derive \[\label{calcbe1} \beta_{l,k} = n \lambda_{l,k} \frac{d^+(\rho_{l,k})}{\frac{d}{d\rho}d(\rho_{l,k})}.\tag{94}\]

Symbolic computations show that \[\begin{align} \label{smcalcbe} \frac{d^+(\rho)}{\frac{d}{d\rho} d(\rho)} & = \frac{r_1^+(\rho) r_2(\rho) - r_2^+(\rho) r_1(\rho)}{\frac{d}{d\rho}r_1(\rho) r_2(\rho) - \frac{d}{d\rho} r_2(\rho) r_1(\rho) + (\omega_2 - \omega_1) r_1(\rho) r_2(\rho)} \\ \nonumber & = -\left( 1 + \frac{2(t+t_0)}{\rho^2} + \varepsilon(\rho)\right). \end{align}\tag{95}\]

This together with ?? , ?? , and 94 imply the asymptotic formula ?? for \(\beta_{l,1}\).

The asymptotics for \(\lambda_{l,2}\), \(\mu_{l,2}\), and \(\beta_{l,2}\) follow from the relations 89 : \(\lambda_{l,2} = \lambda_{l,1}^{\star}\), \(\mu_{l,2} = \mu_{l,1}^{\star}\), and \(\beta_{l,2} = \beta_{l,1}^{\star}\). The spectral characteristics \(\lambda_{l,1}^{\star}\), \(\mu_{l,1}^{\star}\), and \(\beta_{l,1}^{\star}\) of equation 18 have the asymptotics similar to \(-\lambda_{l,1}\), \(-\mu_{l,1}\), and \(-\beta_{l,1}\), respectively, for the vector \(\{ -\tau_0, \tau_1 \}\). Thus, one immediately gets ?? , ?? , and ?? for \(k = 2\) from ?? , ?? , and ?? for \(k = 1\), respectively. Also, we have obtained the asymptotics for \(\lambda_{l,2}\), \(\mu_{l,2}\), and \(\beta_{l,2}\) in the straightforward way by symbolic computations in [46] for checking.

11.3 Case \(n = 4\)↩︎

In this subsection, we derive the asymptotic formulas for the data \(\lambda_{l,k}\), \(\mu_{l,k}\), and \(\beta_{l,k}\) associated with any vector \(\tau = \{ \tau_0, \tau_1, \tau_2 \} \in \mathbf{W}\). Here, we use the notations:

• \(\omega_1 = -1\), \(\omega_2 = i\), \(\omega_3 = -i\), and \(\omega_4 = 1\) are the roots \(\sqrt[4]{1}\) satisfying 79 for \(\rho \in \Gamma_1\).

• \(\varepsilon(\rho) = \dfrac{\Upsilon(\rho)}{\rho^3}\).

• \(\{ \varepsilon_l \}\) denotes various sequences such that \(\{ l^3 \varepsilon_l \} \in l_2\).

• \(t := \dfrac{1}{4} \displaystyle\int_0^1 \tau_2(x) \, dx\), \(t_0 := \dfrac{1}{8} \tau_2(0)\), \(t_1 := \dfrac{1}{8} \tau_2(1)\), \(t_0' := \dfrac{1}{16}\tau_2'(0)\), \(t_1' := \dfrac{1}{16} \tau_2'(1)\),
  \(s := \dfrac{1}{2} \displaystyle\int_0^1 \tau_1(x) \, dx\), \(s_0 := \dfrac{1}{2} \tau_1(0)\), \(s_1 := \dfrac{1}{2} \tau_1(1)\), \(u := \dfrac{1}{32} \displaystyle\int_0^1 \tau_2^2(x) \, dx - \dfrac{1}{4} \displaystyle\int_0^1 \tau_0(x) \, dx\).

• \(\tau_{\nu}^{(-1)}(x) := \displaystyle\int_0^x \tau_{\nu}(\xi) \, d\xi\), \(\nu = 0, 1, 2\).

Integrating the equations in 88 and taking into account the relations \(p_2 = \tau_2\), \(p_1 = \tau_2' + 2 \tau_1\), \(p_0 = \tau_1' + \tau_0\), we obtain \[q_1(x) = -\frac{1}{4} \tau_2^{(-1)}(x), \quad q_2(x) = \frac{1}{8} \tau_2(x) + \frac{1}{32} \bigl( \tau_2^{(-1)}(x) \bigr)^2 - \frac{1}{2} \tau_1^{(-1)}(x),\] \[\begin{gather} q_3(x) = \frac{1}{16} \tau_2'(x) - \frac{1}{32} \tau_2(x) \tau_2^{(-1)}(x) + \frac{1}{2} \tau_1(x) + \frac{1}{32} \int_0^x \tau_2^2(\xi) \,d\xi \\ + \frac{1}{8} \tau_1^{(-1)}(x) \tau_2^{(-1)}(x) - \frac{1}{3 \cdot 2^7} \bigl( \tau_2^{(-1)}(x)\bigr)^3 - \frac{1}{4} \tau_0^{(-1)}(x), \end{gather}\] \[\begin{gather} q_{1\nu}(x) = q_1(x), \quad q_{2\nu}(x) = q_2(x) + \nu q_1'(x), \quad \nu = 1, 2, \\ q_{31}(x) = q_3(x) + q_2'(x), \quad q_{32}(x) = q_3(x) + 2 q_2'(x) + q_1''(x). \end{gather}\] Consequently, we have \[\label{valq4} \def\arraystretch{1.3} \left. \begin{array}{c} q_1(0) = q_{1\nu}(0) = 0, \quad q_1(1) = q_{1\nu}(1) = -t, \quad \nu = 1, 2, \\ q_1'(0) = -2 t_0, \quad q_1'(1) = -2 t_1, \quad q_1''(0) = -4t_0', \quad q_1''(1) = -4t_1', \\ q_2(0) = t_0, \quad q_2(1) = t_1 + \tfrac{1}{2} t^2 - s, \quad q_2'(0) = 2 t_0' - s_0, \quad q_2'(1) = 2 t_1' - s_1 + 2 t_1 t, \\ q_{21}(0) = -t_0, \quad q_{21}(1) = \tfrac{1}{2} t^2- s - t_1, \quad q_{22}(0) = -3t_0, \quad q_{22}(1) = -3t_1 + \tfrac{1}{2} t^2 - s, \\ q_3(0) = t_0' + s_0, \quad q_3(1) = t_1' + s_1 - t_1 t + u + ts - \tfrac{1}{6} t^3, \\ q_{31}(0) = 3 t_0', \quad q_{31}(1) = 3 t_1' + t_1 t + u + ts - \tfrac{1}{6} t^3, \\ q_{32}(0) = t_0' - s_0, \quad q_{32}(1) = t_1' - s_1 + 3 t_1 t + u + ts - \tfrac{1}{6} t^3. \end{array} \right\}\tag{96}\]

The eigenvalues \(\lambda_{l,1}\) of the problem \(\mathcal{L}_1\) 75 for equation 74 coincide with the zeros of the characteristic determinant \[\Delta(\lambda) := \Delta_{1,1}(\lambda) = \begin{vmatrix} \mathcal{C}_1(0,\lambda) & \mathcal{C}_2(0,\lambda) & \mathcal{C}_3(0,\lambda) & \mathcal{C}_4(0,\lambda) \\ \mathcal{C}_1''(1,\lambda) & \mathcal{C}_2''(1,\lambda) & \mathcal{C}_3''(1,\lambda) & \mathcal{C}_4''(1,\lambda) \\ \mathcal{C}_1'(1,\lambda) & \mathcal{C}_2'(1,\lambda) & \mathcal{C}_3'(1,\lambda) & \mathcal{C}_4'(1,\lambda) \\ \mathcal{C}_1(1,\lambda) & \mathcal{C}_2(1,\lambda) & \mathcal{C}_3(1,\lambda) & \mathcal{C}_4(1,\lambda) \end{vmatrix}\]

Therefore, the function \(D(\rho)\) in 84 has the form \[\label{D41} D(\rho) = \begin{vmatrix} y_1(0,\rho) & y_2(0,\rho) & y_3(0,\rho) & y_4(0,\rho) \\ y_1''(1, \rho) & y_2''(1,\rho) & y_3''(1,\rho) & y_4''(1,\rho) \\ y_1'(1, \rho) & y_2'(1,\rho) & y_3'(1,\rho) & y_4'(1,\rho) \\ y_1(1, \rho) & y_2(1,\rho) & y_3(1,\rho) & y_4(1,\rho) \end{vmatrix}\tag{97}\]

Substituting 87 for \(n = 4\) together with 96 into 97 , we obtain \[\begin{align} & D(\rho) = \rho^2 \exp(\rho (\omega_2 + \omega_3 + \omega_4)) d(\rho), \\ & d(\rho) = r_1(\rho) - r_2(\rho) \exp(\rho(\omega_1 - \omega_2)) + \varepsilon(\rho), \\ & r_1(\rho) = 4i - \frac{4 i t}{\rho} + \frac{4 i\bigl(s + \tfrac{1}{2} t^2 + t_0 + t_1 \bigr)}{\rho^2} - \frac{4i (t_0' - t_1' + s_0 + s_1 + t(t_0 + t_1) + \tfrac{1}{6} t^3 + st - u)}{\rho^3},\\ & r_2(\rho) = -4 + \frac{4 i t}{\rho} + \frac{4 \bigl(s + \tfrac{1}{2} t^2 + t_0 + t_1 \bigr)}{\rho^2} - \frac{4i (t_0' - t_1' + s_0 + s_1 + t(t_0 + t_1) + \tfrac{1}{6} t^3 + st - u)}{\rho^3}. \end{align}\]

The zeros of \(d(\rho)\) for large \(|\rho|\) satisfy the relation 92 , which implies \[\begin{gather} \rho_{l,1} = \frac{e^{-\frac{\pi i}{4}}}{\sqrt 2} \left( 2 \pi i l + \frac{\pi i}{2} + \frac{t(1-i)}{\rho_{l,1}} - \frac{2(s + t_0 + t_1)}{\rho_{l,1}^2} + \frac{(1 + i)(t_0' - t_1' + s_0 + s_1 - u)}{\rho_{l,1}^3} + \varepsilon_l\right). \end{gather}\] Hence \[\label{rho41} \rho_{l,1} = e^{\frac{\pi i}{4}} \Biggl( \sqrt 2 \pi l + \frac{\pi}{2\sqrt 2} - \frac{t}{\sqrt 2 \pi l + \frac{\pi}{2\sqrt 2}} + \frac{\sqrt 2(s + t_0 + t_1)}{\bigl( \sqrt 2 \pi l + \frac{\pi}{2 \sqrt 2}\bigr)^2} - \frac{t_0' - t_1' + s_0 + s_1 - u + t^2}{(\sqrt 2 \pi l)^3} + \varepsilon_l\Biggr).\tag{98}\]

Computing \(\rho_{l,1}^4\) and taking ?? into account, we arrive at the asymptotics ?? for \(\lambda_{l,1}\).

In order to obtain the asymptotics for the eigenvalues \(\mu_{l,1}\) of the boundary value problem \(\mathcal{M}_1\) 74 , 77 , we consider the functions \[\begin{align} \nonumber \Delta^+(\lambda) := \Delta_{2,1}(\lambda) & = -\begin{vmatrix} \mathcal{C}_1'(0,\lambda) & \mathcal{C}_2'(0,\lambda) & \mathcal{C}_3'(0,\lambda) & \mathcal{C}_4'(0,\lambda) \\ \mathcal{C}_1''(1,\lambda) & \mathcal{C}_2''(1,\lambda) & \mathcal{C}_3''(1,\lambda) & \mathcal{C}_4''(1,\lambda) \\ \mathcal{C}_1'(1,\lambda) & \mathcal{C}_2'(1,\lambda) & \mathcal{C}_3'(1,\lambda) & \mathcal{C}_4'(1,\lambda) \\ \mathcal{C}_1(1,\lambda) & \mathcal{C}_2(1,\lambda) & \mathcal{C}_3(1,\lambda) & \mathcal{C}_4(1,\lambda) \end{vmatrix}, \\ \label{D41p} D^+(\rho) & = \begin{vmatrix} y_1'(0,\rho) & y_2'(0,\rho) & y_3'(0,\rho) & y_4'(0,\rho) \\ y_1''(1, \rho) & y_2''(1,\rho) & y_3''(1,\rho) & y_4''(1,\rho) \\ y_1'(1, \rho) & y_2'(1,\rho) & y_3'(1,\rho) & y_4'(1,\rho) \\ y_1(1, \rho) & y_2(1,\rho) & y_3(1,\rho) & y_4(1,\rho) \end{vmatrix} \end{align}\tag{99}\]

Substituting 87 for \(n = 4\) and 96 into 99 , we find \[\begin{align} & D^+(\rho) = \rho^4 \exp(\rho(\omega_2 + \omega_3 + \omega_4)) d^+(\rho), \\ & d^+(\rho) = r_1^+(\rho) - r_2^+(\rho) \exp(\rho(\omega_1 - \omega_2)) + \varepsilon(\rho), \\ & r_1^+(\rho) = -4i + \frac{4it}{\rho} - \frac{4i\bigl( s + \frac{1}{2} t^2 + t_1 - t_0\bigr)}{\rho^2} + \frac{4i \bigl( 3 t_0' - t_1' + s_1 + t(t_1 - t_0) + \frac{1}{6} t^3 + st - u\bigr)}{\rho^3},\\ & r_2^+(\rho) = -4i - \frac{4t}{\rho} + \frac{4i \bigl(s + \frac{1}{2} t^2 + t_1 - t_0 \bigr)}{\rho^2} + \frac{4 \bigl(3 t_0' - t_1' + s_1 + t(t_1 - t_0) + \frac{1}{6} t^3 + st - u\bigr)}{\rho^3}. \end{align}\]

The zeros \(\{ \varrho_{l,1}\}\) of \(d^+(\rho)\) for large \(l\) have the asymptotics 93 , which implies \[\begin{align} \varrho_{l,1} & = \frac{e^{-\frac{\pi i}{4}}}{\sqrt 2} \left( 2 \pi i l + \frac{t(1-i)}{\varrho_{l,1}} - \frac{2(s - t_0 + t_1)}{\varrho_{l,1}^2} + \frac{(1 + i)(s_1 + 3 t_0' - t_1' - u)}{\varrho_{l,1}^3} + \varepsilon_l\right) \\ & = e^{\frac{\pi i}{4}} \Biggl( \sqrt 2 \pi l - \frac{t}{\sqrt 2 \pi l} + \frac{\sqrt 2(s - t_0 + t_1)}{(\sqrt 2 \pi l)^2} - \frac{s_1 + 3t_0' - t_1' - u + t^2}{(\sqrt 2 \pi l)^3} + \varepsilon_l\Biggr). \end{align}\] Computing \(\mu_{l,1} = \varrho_{l,1}^4\) and taking ?? into account, we arrive at ?? for \(\mu_{l,1}\).

Using the relations 94 and 95 , which have the same form as in the case \(n = 3\), we obtain \[\label{smbe1} \beta_{l,1} = -4 \lambda_{l,1} \left( 1 + \frac{i (t + 2t_0)}{\rho_{l,1}^2} + \frac{(1-i)(2s + s_0 + 2t_0 + 2t_1 - 2t_0')}{\rho_{l,1}^3} + \varepsilon_l\right).\tag{100}\]

Substituting 98 and ?? into 100 , we arrive at the asymptotics ?? for \(\beta_{l,1}\). The formulas ?? , ?? , and ?? for \(k = 3\) follow from the relations 89 : \(\lambda_{l,3} = \lambda_{l,1}^{\star}\), \(\mu_{l,3} = \mu_{l,1}^{\star}\), and \(\beta_{l,3} = \beta_{l,1}^{\star}\), respectively. Indeed, for \(n = 4\), equation 18 has the form 74 with the coefficients \(\{ \tau_0, -\tau_1, \tau_2 \}\). Therefore, in formulas ?? , ?? , and ?? , only the sign of \(\sigma\), \(\sigma_0\), and \(\sigma_1\) changes. For verification, the asymptotics for the case \(k = 3\) have been explicitly derived in [46].

Similarly, we get the asymptotics of the eigenvalues \(\lambda_{l,2}\) of the boundary value problem \(\mathcal{L}_2\) 76 . They coincide with the zeros of the characteristic function \[\Delta(\lambda) := \Delta_{2,2}(\lambda) = \begin{vmatrix} \mathcal{C}_1(0,\lambda) & \mathcal{C}_2(0,\lambda) & \mathcal{C}_3(0,\lambda) & \mathcal{C}_4(0,\lambda) \\ \mathcal{C}_1'(0,\lambda) & \mathcal{C}_2'(0,\lambda) & \mathcal{C}_3'(0,\lambda) & \mathcal{C}_4'(0,\lambda) \\ \mathcal{C}_1'(1,\lambda) & \mathcal{C}_2'(1,\lambda) & \mathcal{C}_3'(1,\lambda) & \mathcal{C}_4'(1,\lambda) \\ \mathcal{C}_1(1,\lambda) & \mathcal{C}_2(1,\lambda) & \mathcal{C}_3(1,\lambda) & \mathcal{C}_4(1,\lambda) \end{vmatrix}.\] Therefore, the function \(D(\rho)\) in 84 has the form \[\label{D42} D(\rho) = \begin{vmatrix} y_1(0,\rho) & y_2(0,\rho) & y_3(0,\rho) & y_4(0,\rho) \\ y_1'(0,\rho) & y_2'(0,\rho) & y_3'(0,\rho) & y_4'(0,\rho) \\ y_1'(1,\rho) & y_2'(1,\rho) & y_3'(1,\rho) & y_4'(1,\rho) \\ y_1(1,\rho) & y_2(1,\rho) & y_3(1,\rho) & y_4(1,\rho) \end{vmatrix}.\tag{101}\]

Substituting 87 for \(n = 4\) together with 96 into 101 , we obtain \[\begin{align} & D(\rho) = \rho^2 \exp(\rho(\omega_3 + \omega_4)) d(\rho), \\ & d(\rho) = r_1(\rho) - r_2(\rho) \exp(\rho(\omega_2 - \omega_3)) + \varepsilon(\rho), \\ & r_1(\rho) = -2i - \frac{2t(1-i)}{\rho} + \frac{4\bigl(t_0 + t_1 + \frac{1}{2}t^2\bigr)}{\rho^2} - \frac{(1 + i) \bigl(6(t_1' - t_0') + 4t(t_0 + t_1) + \frac{2}{3} t^3 + 2u\bigr)}{\rho^3}\\ & r_2(\rho) = 2i - \frac{2t(1+i)}{\rho} + \frac{4\bigl(t_0 + t_1 + \frac{1}{2} t^2\bigr)}{\rho^2} - \frac{(1-i)\bigl(6(t_1' - t_0') + 4t(t_0 + t_1) + \frac{2}{3} t^3 + 2u\bigr)}{\rho^3}. \end{align}\]

The zeros of \(d(\rho)\) for large \(|\rho|\) satisfy the relation \[\rho = \frac{1}{\omega_3 - \omega_2} \ln \frac{r_2(\rho)}{r_1(\rho)} + \varepsilon(\rho).\]

Symbolic computations show that \[\rho_{l,2} = \pi l + \frac{\pi}{2} - \frac{t}{\rho_{l,2}} + \frac{2(t_1 + t_0)}{\rho_{l,2}^2} - \frac{3(t_1' - t_0') + u}{\rho_{l,2}^3} + \varepsilon_l.\] Hence \[\label{rho42} \rho_{l,2} = \pi l + \frac{\pi}{2} - \frac{t}{\pi l + \frac{\pi}{2}} + \frac{2(t_0 + t_1)}{\bigl(\pi l + \frac{\pi}{2} \bigr)^2} - \frac{3(t_1' - t_0') + u + t^2}{(\pi l)^3} + \varepsilon_l.\tag{102}\]

Computing \(\rho_{l,2}^4\) and using ?? , we arrive at the asymptotics ?? .

For obtaining the asymptotics for the eigenvalues \(\mu_{l,2}\), we have \[\label{D42p} D^+(\rho) = \begin{vmatrix} y_1(0,\rho) & y_2(0,\rho) & y_3(0,\rho) & y_4(0,\rho) \\ y_1''(0,\rho) & y_2''(0,\rho) & y_3''(0,\rho) & y_4''(0,\rho) \\ y_1'(1,\rho) & y_2'(1,\rho) & y_3'(1,\rho) & y_4'(1,\rho) \\ y_1(1,\rho) & y_2(1,\rho) & y_3(1,\rho) & y_4(1,\rho) \end{vmatrix}.\tag{103}\]

Substituting 87 for \(n = 4\) together with 96 into 103 , we obtain \[\begin{align} & D^+(\rho) = \rho^3 \exp(\rho(\omega_3 + \omega_4)) d^+(\rho), \\ & d^+(\rho) = r_1^+(\rho) - r_2^+(\rho) \exp(\rho(\omega_2 - \omega_3)) + \varepsilon(\rho), \\ & r_1^+(\rho) = 2 + 2i - \frac{4it}{\rho} - \frac{4(1-i)\bigl(t_1 + \frac{1}{2}t^2 \bigr)}{\rho^2} + \frac{4\bigl( 3 t_1' - t_0' + 2 t_1 t + \frac{1}{3} t^3 + u\bigr)}{\rho^3},\\ & r_2^+(\rho) = 2-2i + \frac{4it}{\rho} - \frac{4(1+i)\bigl( t_1 + \frac{1}{2} t^2 \bigr)}{\rho^2} + \frac{4\bigl(3 t_1' - t_0' + 2 t_1 t+ \frac{1}{3} t^3 + u \bigr)}{\rho^3}. \end{align}\]

The zeros of \(d^+(\rho)\) for large \(|\rho|\) satisfy the relation \[\rho = \frac{1}{\omega_3 - \omega_2} \ln \frac{r_2^+(\rho)}{r_1^+(\rho)} + \varepsilon(\rho).\] Consequently, we obtain \[\begin{align} \varrho_{l,2} & = \pi l + \frac{\pi}{4} - \frac{t}{\varrho_{l,2}} + \frac{2 t_1}{\varrho_{l,2}^2} - \frac{3 t_1' - t_0' + u}{\varrho_{l,2}^3} + \varepsilon_l \\ & = \pi l + \frac{\pi}{4} - \frac{t}{\pi l + \frac{\pi}{4}} + \frac{2 t_1}{\bigl(\pi l + \frac{\pi}{4} \bigr)^2} - \frac{3 t_1' - t_0' + u + t^2}{(\pi l)^3} + \varepsilon_l. \end{align}\] Computing \(\mu_{l,2} = \varrho_{l,2}^4\) and using ?? , we arrive at ?? .

It is known from the previous studies [49]–[51] that the numbering in ?? and ?? starts from \(l = 1\). The same conclusion for ?? and ?? follows from the explicit expressions of characteristic functions for \(\tau = \{ 0, 0, 0\}\).

Proceed to estimating \(\beta_{l,2}\). Symbolic computations show that \[\begin{align} \frac{d^+(\rho)}{\frac{d}{d\rho} d(\rho)} & = \frac{r_1^+(\rho) r_2(\rho) - r_2^+(\rho) r_1(\rho)}{\frac{d}{d\rho}r_1(\rho) r_2(\rho) - \frac{d}{d\rho} r_2(\rho) r_1(\rho) + (\omega_3 - \omega_2) r_1(\rho) r_2(\rho)} \\ \nonumber & = -\left( 1 + \frac{t + 2 t_0}{\rho^2} + \frac{4(t_0' - t_0 - t_1)}{\rho^3} + \varepsilon(\rho)\right). \end{align}\]

Substituting 102 and using 94 , ?? , we arrive at ?? .

12 Schur’s test↩︎

In this appendix, we present a discrete weighted version of Schur’s test, which was originally proposed for integral operators in [52]. Applying Schur’s test, we obtain an auxiliary estimate, which is used in the proof of 69 in Section 7.

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

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,
e-mail: bondarenkonp@sgu.ru

References↩︎