Inverse spectral problems for first order integro-differential operators
- Vjacheslav Yurko^{1}Email author
Received: 17 May 2017
Accepted: 19 June 2017
Published: 30 June 2017
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
MSC
1 Introduction
2 Preliminary information
Example 1
Let \(\lambda _{1}=\lambda _{2}<\lambda _{3}<\lambda _{4}=\lambda _{5}=\lambda _{6}<\lambda _{7}< \lambda _{8}<\cdots \) . Then \(S=\{1,3,4,7,8,\ldots \}\), \(s_{1}(x)=\varphi _{0}(x,\lambda _{1})\), \(s_{2}(x)=\varphi _{1}(x,\lambda _{1})\), \(s_{3}(x)=\varphi _{0}(x,\lambda _{3})\), \(s_{4}(x)=\varphi _{0}(x,\lambda _{4})\), \(s_{5}(x)=\varphi _{1}(x,\lambda _{4})\), \(s_{6}(x)=\varphi _{2}(x,\lambda _{4})\), \(s_{7}(x)=\varphi _{0}(x,\lambda _{7}),\ldots \) .
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
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 \(n\ge 1\).
Declarations
Acknowledgements
This work was supported by Grant 17-11-01193 of the Russian Science Foundation.
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Authors’ Affiliations
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