# Inverse spectral problems for first order integro-differential operators

## Abstract

Inverse spectral problems are studied for the 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.

## Keywords

integro-differential operators inverse spectral problems uniqueness theorem## MSC

47G20 45J05 44A15## 1 Introduction

*ℓ*on a finite interval. Properties of spectral characteristics are established, and the uniqueness theorem is proved for the inverse problem of recovering the function \(R(x)\) and \(V(x)\) from the given spectral data.

## 2 Preliminary information

*λ*of exponential type, and its zeros \(\Lambda :=\{\lambda _{n}\}_{n\ge 1}\) (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 \(\lambda _{n}\) (\(\lambda _{n}=\lambda _{n+1}= \cdots = \lambda _{n+m_{n}-1}\)). Denote

*L*.

### Example 1

Let \(\lambda _{1}=\lambda _{2}<\lambda _{3}<\lambda _{4}=\lambda _{5}=\lambda _{6}<\lambda _{7}< \lambda _{8}<\cdots \) . Then \(S=\{1,3,4,7,8,\ldots \}\), \(s_{1}(x)=\varphi _{0}(x,\lambda _{1})\), \(s_{2}(x)=\varphi _{1}(x,\lambda _{1})\), \(s_{3}(x)=\varphi _{0}(x,\lambda _{3})\), \(s_{4}(x)=\varphi _{0}(x,\lambda _{4})\), \(s_{5}(x)=\varphi _{1}(x,\lambda _{4})\), \(s_{6}(x)=\varphi _{2}(x,\lambda _{4})\), \(s_{7}(x)=\varphi _{0}(x,\lambda _{7}),\ldots \) .

*λ*of exponential type. Denote \(\Delta_{1}(\lambda ):=\Delta_{0}(\lambda )\exp (-i\lambda \pi )\). Using (8), (9) and (11), by standard arguments (see, e.g., [3]), we obtain that for \(\vert \lambda \vert \to \infty \), the following asymptotical formulae hold:

*λ*of exponential type. By virtue of (3),

*L*, and

*L*. We will consider the following inverse problem.

### Inverse problem 1

*Given the spectral data* \(\{\lambda _{n},\beta _{n}\}_{n\ge 1}\), *construct* *R* *and* *V*.

## 3 The uniqueness theorem

Below we will assume that \(R(x)\ne 0\) 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).

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 different functions \(\tilde{R}(x)\), \(\tilde{V}(t)\). We agree that in what follows if a certain symbol *α* denotes an object related to *L*, then *α̃* will denote the analogous object related to *L̃*.

### 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* \(n\ge 1\), *then* \(R(x)\equiv \tilde{R}(x)\), \(V(x)\equiv \tilde{V}(x)\), \(x\in [0,\pi ]\).

### Proof

*λ*of exponential type. Taking (3), (10) and (13) into account, we obtain for \(\vert \lambda \vert \to \infty \)

*λ*. In particular, (17) yields

*λ*. Using (18) we calculate

### Example 2

Fix \(a\in (0,\pi )\). Let \(R(x)\equiv 0\) for \(x\in [0,a]\) and \(R(x)\ne 0\) for \(x\in (a,\pi )\). Put \(\tilde{R}(x) \equiv R(x)\) for \(x\in [0,\pi ]\), and choose \(V(t)\), \(\tilde{V}(t)\) such that \(V(t)\equiv \tilde{V}(t)\) for \(t\in (a,\pi )\), and \(V(t)\ne \tilde{V}(t)\) 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 \(n\ge 1\).

## Notes

### Acknowledgements

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

## References

- 1.Marchenko, VA: Sturm-Liouville Operators and Their Applications. Naukova Dumka, Kiev (1977) English transl., Birkhäuser, Basel, 1986 MATHGoogle Scholar
- 2.Levitan, BM: Inverse Sturm-Liouville Problems. Nauka, Moscow (1984) English transl., VNU Sci. Press, Utrecht, 1987 MATHGoogle Scholar
- 3.Freiling, G, Yurko, VA: Inverse Sturm-Liouville Problems and Their Applications. Nova Science Publishers, New York (2001) MATHGoogle Scholar
- 4.Beals, R, Deift, P, Tomei, C: Direct and Inverse Scattering on the Line. Math. Surveys and Monographs, vol. 28. Am. Math. Soc., Providence (1988) MATHGoogle Scholar
- 5.Yurko, VA: Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-Posed Problems Series. VSP, Utrecht (2002) CrossRefMATHGoogle Scholar
- 6.Yurko, VA: Inverse Spectral Problems for Differential Operators and Their Applications. Gordon & Breach, New York (2000) MATHGoogle Scholar
- 7.Lakshmikantham, V, Rao, HRM: Theory of Integro-Differential Equations. Stability and Control: Theory and Applications, vol. 1. Gordon & Breach, New York (1995) MATHGoogle Scholar
- 8.Yurko, VA: An inverse problem for integro-differential operators. Mat. Zametki
**50**(5), 134-146 (1991) (Russian); English transl. in Math. Notes**50**(5-6), 1188-1197 (1991) MathSciNetMATHGoogle Scholar - 9.Kuryshova, Yu: An inverse spectral problem for differential operators with integral delay. Tamkang J. Math.
**42**(3), 295-303 (2011) MathSciNetCrossRefMATHGoogle Scholar - 10.Buterin, SA: On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum. Differ. Uravn. (Minsk)
**46**, 146-149 (2010) (Russian); English transl. in Diff. Equ.**46**, 150-154 (2010) MathSciNetMATHGoogle Scholar - 11.Yurko, VA: An inverse problem for integral operators. Mat. Zametki
**37**(5), 690-701 (1985) (Russian); English transl. in Math. Notes**37**(5-6), 378-385 (1985) MathSciNetMATHGoogle Scholar - 12.Buterin, SA, Choque Rivero, AE: On inverse problem for a convolution integro-differential operator with Robin boundary conditions. Appl. Math. Lett.
**48**, 150-155 (2015) MathSciNetCrossRefMATHGoogle Scholar

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