Recovering First Order Integro-Differential Operators from Spectral Data

TL;DR

This paper investigates the spectral properties of first order integro-differential operators on finite intervals and proves a uniqueness theorem for their inverse problem, enabling operator reconstruction from spectral data.

Contribution

It establishes spectral properties and proves a uniqueness theorem for recovering first order integro-differential operators from spectral data.

Findings

Spectral characteristics of the operators are characterized.

A uniqueness theorem for the inverse problem is proved.

Operators can be uniquely recovered from spectral data.

Abstract

First order integro-differential operators on a finite interval are studied. Properties of spectral characteristic are established, and the uniqueness theorem is proved for the inverse problem of recovering operators from their spectral data.

Equations

$$\ell y:=i{y}^{\prime }(x)+{\int }_0^{x}Q(x,t)y(t)dt=\lambda y(x),x\in [0,\pi ],$$

$$R(\pi -x)\sim C_{\alpha }{x}^{\alpha },V(x)\sim D_{\beta }{x}^{\beta },x\to +0,C_{\alpha }{D}_{\beta }\ne 0.$$

$$\phi (x,\lambda )=\exp (-i\lambda x)+{\int }_0^{x}K(x,t)\exp (-i\lambda t)dt,$$

$${\Pi }_{+}:=\{\lambda :\text{mbox}Im\lambda \ge 0\},{\Pi }_{-}^{\delta }:=\{\lambda :\arg \lambda \in [\pi +\delta ,2\pi -\delta ]\}.$$

$$\left.\begin{array}[]{c}\varphi^{(\nu)}(x,\lambda)=(-i\lambda)^{\nu}\exp(-i\lambda x)(1+o(1)),\quad\lambda\in\Pi_{+},\quad\nu=0,1,\\[8.53581pt] \varphi^{(\nu)}(x,\lambda)=o(\lambda^{\nu}),\quad\lambda\in\Pi_{-}^{\delta},\quad\nu=0,1.\end{array}\right\}$$

$${\phi }_{\nu }(x,\lambda ):=\frac{1}{\nu !}\frac{\partial \phi (x,\lambda )}{\partial {\lambda }^{\nu }},\nu \ge 0,\Delta (\lambda ):=\phi (\pi ,\lambda ).$$

$$S:=\{n:n-1\in \mathbf{N},{\lambda }_{n-1}\ne {\lambda }_n\}\cup \{1\},s_{n+\nu }(x):={\phi }_{\nu }(x,{\lambda }_n),n\in S,\nu =\overline{m_n-1}.$$

$$i{\eta }^{\prime }(x,\lambda )-R(x){\int }_x^{\pi }V(t)\eta (t,\lambda )dt+R(x)=\lambda \eta (x,\lambda ),\eta (\pi ,\lambda )=0.$$

$$i{\theta }^{\prime }(x,\lambda )+R_0(x){\int }_0^x{V}_0(t)\theta (t,\lambda )dt-R_0(x)=-\lambda \theta (x,\lambda ),\theta (0,\lambda )=0,$$

$$\theta (x,\lambda )={\int }_0^{x}g(x,t,\lambda )R_0(t)dt,$$

$$i{g}_x(x+t,t,\lambda )-\lambda g(x+t,t,\lambda )+R(x+t){\int }_0^{x}V(\tau +t)g(\tau +t,t,\lambda )d\tau =0,g(t,t,\lambda )=-i,$$

$$i{\phi }^{\prime }(x,\lambda ;t)+R(x+t){\int }_0^{x}V(\tau +t)\phi (\tau ,\lambda ;t)d\tau =\lambda \phi (x,\lambda ;t),\phi (0,\lambda ;t)=1.$$

$$\phi (x,\lambda ;t)=\exp (-i\lambda x)+{\int }_0^{x}K(x,\tau ;t)\exp (-i\lambda \tau )d\tau ,$$

$$g(x,t,\lambda )=-i\exp (-i\lambda (x-t))-i{\int }_0^{x-t}K(x-t,\tau ;t)\exp (-i\lambda \tau )d\tau .$$

$$\theta (x,\lambda )={\int }_0^{x}P(x,t)\exp (i\lambda t)dt,$$

$$P(x,t)=-i{R}_0(x-t)-i{\int }_0^{x-t}{R}_0(\tau )K(x-\tau ,t;\tau )d\tau .$$

$$\left.\begin{array}[]{c}\eta^{(\nu)}(x,\lambda)=o(\lambda^{\nu}),\quad\lambda\in\Pi_{+},\quad\nu=0,1,\\[8.53581pt] \eta^{(\nu)}(x,\lambda)=o(\lambda^{\nu}\exp(i\lambda(\pi-x))),\quad\lambda\in\Pi_{-}^{\delta},\quad\nu=0,1.\end{array}\right\}$$

$${\Delta }_0(\lambda ):=1-{\int }_0^{\pi }V(t)\eta (t,\lambda )dt.$$

$$i\eta^{\prime}(x,\lambda)+R(x)\Big{(}\Delta_{0}(\lambda)+\int_{0}^{x}V(t)\eta(t,\lambda)\,dt\Big{)}=\lambda\eta(x,\lambda),\quad\eta(\pi,\lambda)=0.$$

$$\left.\begin{array}[]{c}\Delta_{1}(\lambda)=\exp(-i\lambda\pi)(1+o(1)),\quad\lambda\in\Pi_{+},\\[8.53581pt] \Delta_{1}(\lambda)=B\lambda^{-\gamma-1}(1+o(1)),\quad\lambda\in\Pi_{-}^{\delta},\end{array}\right\}$$

$$\left.\begin{array}[]{c}\Delta(\lambda)=\exp(-i\lambda\pi)(1+o(1)),\quad\lambda\in\Pi_{+},\\[8.53581pt] \Delta(\lambda)=o(1),\quad\lambda\in\Pi_{-}^{\delta}.\end{array}\right\}$$

$$\Delta (\lambda )\equiv {\Delta }_0(\lambda )\exp (-i\lambda \pi ).$$

$${\eta }_{\nu }(x,\lambda ):=\frac{1}{\nu !}\frac{\partial \eta (x,\lambda )}{\partial {\lambda }^{\nu }},\nu \ge 0,{\psi }_{n+\nu }(x):={\eta }_{\nu }(x,{\lambda }_n),n\in S,\nu =\overline{m_n-1}.$$

$${\psi }_{n+\nu }(x)=\sum _{j=0}^{\nu }{\beta }_{n+\nu -j}{s}_{n+j}(x),n\in S,\nu =\overline{m_n-1}.$$

$$A_{j}(x,\lambda)=(\Delta_{0}(\lambda))^{-1}\exp(i\lambda x)\Big{(}\tilde{\varphi}(x,\lambda)\eta^{(j-1)}(x,\lambda)-\tilde{\eta}(x,\lambda)\varphi^{(j-1)}(x,\lambda)\Big{)},\;j=1,2,$$

$$A_1(x,\lambda )=o(1),A_2(x,\lambda )=o(\lambda ),\lambda \in {\Pi }_{+},$$

$$A_1(x,\lambda )=o({\lambda }^{\gamma +1}),A_2(x,\lambda )=o({\lambda }^{\gamma +2}),\lambda \in {\Pi }_{-}^{\delta },$$

$$A_1(x,\lambda )\equiv 0,A_2(x,\lambda )\equiv A(x),$$

$$\tilde{\phi }(x,\lambda )\eta (x,\lambda )\equiv \tilde{\eta }(x,\lambda )\phi (x,\lambda ),$$

$$\tilde{\phi }(x,\lambda ){\eta }^{\prime }(x,\lambda )-\tilde{\eta }(x,\lambda ){\phi }^{\prime }(x,\lambda )\equiv A(x){\Delta }_0(\lambda )\exp (-i\lambda x).$$

Full text

**Recovering First Order Integro-Differential Operators from Spectral Data

** V.A. Yurko

Abstract. First order integro-differential operators on a finite interval are studied. Properties of spectral characteristic are established, and the uniqueness theorem is proved for the inverse problem of recovering operators from their spectral data.

MSC Classification: 47G20 45J05 44A15

Keywords: integro-differential operators, inverse spectral problems, uniqueness theorem

**1. ** Inverse spectral problems consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics and other branches of natural sciences and engineering. The greatest success in the inverse problem theory has been achieved for the Sturm-Liouville operator (see, e.g., [1-3]) and afterwards for higher-order differential operators [4-6] and other classes of differential operators.

For integro-differential and other classes of nonlocal operators inverse problems are more difficult for investigation, and the main classical methods (transformation operator method and the method of spectral mappings [1-6]) either are not applicable to them or require essential modifications, and for such operators the general inverse problem theory does not exist. At the same time, nonlocal and, in particular, integro-differential operators are of great interest, because they have many applications (se, e.g., [7]). We note that some aspects of inverse problems for integro-differential operators were studied in [8-10] and other works. In the present paper we study inverse spectral problem for one class of first order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.

2. Consider the integro-differential equation

[TABLE]

where $Q(x,t)=R(x)V(t).$ We assume that $R(x),V(t)$ are continuous complex-valued functions, and

[TABLE]

Let $\varphi(x,\lambda)$ be the solution of Eq. (1) with the condition $\varphi(0,\lambda)=1.$ Then the following representation holds (see [3]):

[TABLE]

where $K(x,t)$ is a continuous function, and $K(x,0)=0.$ Denote

[TABLE]

It follows from (2) that for $|\lambda|\to\infty$ uniformly in $x\in[0,\pi]$:

[TABLE]

Denote

[TABLE]

The function $\Delta(\lambda)$ is entire in $\lambda$ of exponential type, and its zeros $\Lambda:=\{\lambda_{n}\}_{n\geq 1}$ (counting with multiplicities) coincide with the eigenvalues of the boundary value problem $L=L(R,V)$ for Eq. (1) with the condition $y(\pi)=0.$ Let $m_{n}$ be the multiplicity of $\lambda_{n}$ ($\lambda_{n}=\lambda_{n+1}=\ldots=\lambda_{n+m_{n}-1}$). Denote

[TABLE]

The functions $\{s_{n}(x)\}_{n\geq 1}$ are eigen and associated functions for $L.$

**Example 1. ** Let $\lambda_{1}=\lambda_{2}<\lambda_{3}<\lambda_{4}=\lambda_{5}=\lambda_{6}<\lambda_{7}<\lambda_{8}<\ldots$. Then $S=\{1,3,4,7,8,\ldots\},$ $s_{1}(x)=\varphi_{0}(x,\lambda_{1}),$ $s_{2}(x)=\varphi_{1}(x,\lambda_{1}),$ $s_{3}(x)=\varphi_{0}(x,\lambda_{3}),$ $s_{4}(x)=\varphi_{0}(x,\lambda_{4}),$ $s_{5}(x)=\varphi_{1}(x,\lambda_{4}),$ $s_{6}(x)=\varphi_{2}(x,\lambda_{4}),$ $s_{7}(x)=\varphi_{0}(x,\lambda_{7}),\ldots$

Let the function $\eta(x,\lambda)$ be the solution of the problem

[TABLE]

Denote $\theta(x,\lambda):=\eta(\pi-x,\lambda).$ Then

[TABLE]

where $R_{0}(x):=R(\pi-x),\,V_{0}(x):=V(\pi-x).$ It follows from (5) that

[TABLE]

where $g(x,t,\lambda)$ is Green’s function of the Cauchy problem, and

[TABLE]

and consequently, $g(x+t,t,\lambda)=-i\varphi(x,\lambda;t),$ where $\varphi(x,\lambda;t)$ is the solution of the Cauchy problem

[TABLE]

In view of (2) we get

[TABLE]

where $K(x,\tau;t)$ is a continuous function. This yields

[TABLE]

Substituting (7) into (6), we obtain

[TABLE]

where

[TABLE]

Clearly, $P(x,x)=-iR_{0}(0),\;P(x,0)=-iR_{0}(x).$ Using (8)-(9) and (4) we conclude that for $|\lambda|\to\infty$ uniformly in $x\in[0,\pi]$:

[TABLE]

Denote

[TABLE]

Using (4) and (11) we calculate

[TABLE]

In particular, it follows from (12) that zeros of the entire function $\Delta_{0}(\lambda)$ coincide with the zeros of $\Delta(\lambda),$ and multiplicities of zeros of $\Delta_{0}(\lambda)$ are not more than multiplicities of zeros of $\Delta(\lambda).$ Therefore the function $\Delta(\lambda)/\Delta_{0}(\lambda)$ is entire in $\lambda$ of exponential type. Denote $\Delta_{1}(\lambda):=\Delta_{0}(\lambda)\exp(-i\lambda\pi).$ Using (8), (9) and (11), by standard arguments (see, for example, [?]) we obtain that for $|\lambda|\to\infty,$ the following asymptotical formulae hold

[TABLE]

where $B\neq 0,\;\gamma:=\alpha+\beta+1.$ The function $F(\lambda):=\Delta(\lambda)/\Delta_{1}(\lambda)$ is entire in $\lambda$ of exponential type. By virtue of (3),

[TABLE]

Together with (13) this yields that $F(\lambda)\equiv 1,$ i.e. $\Delta(\lambda)\equiv\Delta_{1}(\lambda)$ or

[TABLE]

Denote

[TABLE]

The functions $\{\psi_{n}(x)\}_{n\geq 1}$ are eigen and associated functions for the boundary value problem $L,$ and

[TABLE]

The coefficients $\{\beta_{n}\}_{n\geq 1}$ are called Levinson’s weight numbers, and the data $\{\lambda_{n},\beta_{n}\}_{n\geq 1}$ are called the spectral data for the boundary value problem $L.$ We will consider the following inverse problem:

**Inverse problem 1. **Given the spectral data $\{\lambda_{n},\beta_{n}\}_{n\geq 1}$, construct $R$ and $V.$

3. Below we will assume that $R(x)\neq 0$ a.e. on $(0,\pi).$ If this condition does not hold, then the specification of the spectral data does not uniquely determine $L$ (see Example 2 below).

Let us formulate the uniqueness theorem for this inverse problem. For this purpose, together with $L$ we consider the boundary value problem $\tilde{L}:=L(\tilde{R},\tilde{V})$ of the same form but with a different functions $\tilde{R}(x),\tilde{V}(t).$ We agree that everywhere below if a certain symbol $\alpha$ denotes an object related to $L,$ then $\tilde{\alpha}$ will denote the analogous object related to $\tilde{L}.$

**Theorem 1. **Let $\{\tilde{\lambda}_{n},\tilde{\beta}_{n}\}$ be the spectral data for the problem $\tilde{L}=L(\tilde{R},\tilde{V}).$ If $\lambda_{n}=\tilde{\lambda}_{n}$, $\beta_{n}=\tilde{\beta}_{n}$ for all $n\geq 1,$ then $R(x)\equiv\tilde{R}(x),$ $V(x)\equiv\tilde{V}(x),$ $x\in[0,\pi].$

*Proof. * Using (14)-(15) and Hadamard’s factorization theorem we get $\Delta_{0}(\lambda)\equiv\tilde{\Delta}_{0}(\lambda).$ Taking (16) into account, we deduce that the functions

[TABLE]

are entire in $\lambda$ of exponential type. Taking (3), (10) and (13) into account we obtain for $|\lambda|\to\infty$:

[TABLE]

[TABLE]

and consequently,

[TABLE]

where the function $A(x)$ does not depend on $\lambda.$ In particular, (17) yields

[TABLE]

[TABLE]

Similarly, we obtain

[TABLE]

where $A^{*}(x)$ does not depend on $\lambda.$ Using (18) we calculate

[TABLE]

Together with (19)-(20) this yields

[TABLE]

Taking (3) into account, we conclude that $A(x)\equiv A^{*}(x),$ and

[TABLE]

Furthermore, using (18), (19) and equations (1) and (4), we infer

[TABLE]

[TABLE]

Hence, for $|\lambda|\to\infty,\;\lambda\in\Pi_{+}$, we get $A(x)\equiv iR(x).$ In view of (21), one has

[TABLE]

Since $R(x)\neq 0$ a.e. on $(0,\pi),$ it follows from (22) that $\tilde{\varphi}(x,\lambda)\equiv\varphi(x,\lambda).$ By virtue of (18), $\tilde{\eta}(x,\lambda)\equiv\eta(x,\lambda).$ Then, according to (4),

[TABLE]

For $|\lambda|\to\infty$ this yields $R(x)\equiv\tilde{R}(x),$ $x\in[0,\pi],$ and consequently $V(x)\equiv\tilde{V}(x),$ $x\in[0,\pi].$ Theorem 1 is proved.

**Example 2. ** Fix $a\in(0,\pi).$ Let $R(x)\equiv 0$ for $x\in[0,a],$ and $R(x)\neq 0$ for $x\in(a,\pi).$ Put $\tilde{R}(x)\equiv R(x)$ for $x\in[0,\pi],$ and chose $V(t),\tilde{V}(t)$ such that $V(t)\equiv\tilde{V}(t)$ for $t\in(a,\pi),$ and $V(t)\neq\tilde{V}(t)$ for $t\in[0,a].$ Then $\tilde{\varphi}(x,\lambda)\equiv\varphi(x,\lambda)$ and $\tilde{\eta}(x,\lambda)\equiv\eta(x,\lambda)$; hence $\tilde{\lambda}_{n}=\lambda_{n}$, $\tilde{\beta}_{n}=\beta_{n}$ for all $n\geq 1.$

Acknowledgment. This work was supported by Grant 17-11-01193 of the Russian Science Foundation.

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