Abstract
We study the inverse spectral problems of recovering Dirac-type functional-differential operator with two constant delays \(a_1\) and \(a_2\) not less than one-third of the length the interval. It has been proved that the operator can be recovered uniquely from four spectra when \(2a_1+\frac{a_2}{2}\) is not less than the length of the interval, while it is not possible otherwise.
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1 Introduction
The theory of differential equations with delays is a a very significant area of the theory of ordinary differential equations (see [12, 13]). In last decades, there has been a growing interest in studying inverse spectral problems for different types of operators with one or more delays. It turned out that this type of operators is usually more adequate for modeling different real physical processes, frequently possessing a nonlocal nature. Inverse problems for Sturm–Liouville operators with one delay have been studied in most details (see [1, 5,6,7,8,9, 11, 15,16,17,18]). There is also considerable number of results related to the Sturm–Liouville operators with two constant delays (see [4, 14, 19,20,21,22,23]). In recent years, a significant number of results related to the inverse problems for Sturm–Liouville operators, have been extended to Dirac operators with one delay (see [3, 10, 25, 26]), as well as to Dirac operators with two delays (see [24]).
The key issue in solving inverse problems for operators with delays is the question of inverse problem solution’s uniqueness. Although inverse problem solution’s uniqueness was for long thought to be indisputable, as in the case of inverse problems for classical operators (without delays), it turned out that the solution of inverse problems for operators with delays does not have to be unique. It has been shown in the papers [16, 18] that Sturm–Liouville operator with one delay can be recovered uniquely from two spectra if the delay belongs to \(\left[ \frac{2\pi }{5},\pi \right) ,\) while in the papers [6, 7] has been shown that this is not possible for the delay from \(\left[ \frac{\pi }{3},\frac{2\pi }{5}\right) .\) There are the same results for Dirac operator with one delay (see [3, 10]). For operators with two delays, there are just results related to the inverse problem solution’s uniqueness. So, in the papers [21, 22] it has been proven that Sturm–Liouville operator can be recovered uniquely from four spectra for the delays greater than \(\frac{\pi }{2}\) under Robin and Dirichlet boundary conditions, respectively. In the paper [20] inverse problem solution’s uniqueness has been proven for the Sturm–Liouville operator with two delays from \([\frac{2\pi }{5},\frac{\pi }{2})\) under Robin boundary conditions. Papers [19, 23] deal with Sturm–Liouville operator with two delays such that first delay \(a_1\) belongs to \(\left[ \frac{\pi }{3},\frac{2\pi }{5}\right) \) and the second one to \([2a_1,\pi ) \) under Robin and Dirichlet/Neumann boundary conditions, respectively and uniqueness of inverse problem solution has been proven. In the paper [24] it has been proven that Dirac operator with two delays from \(\left[ \frac{2\pi }{5},\pi \right) \) can be recovered uniquely from four spectra. So far there are no results with non-unique solutions of inverse problems for operators with two delays, even for Sturm–Liouville operators.This paper will be the first result proving that the uniqueness of the inverse problem’s solution does not have to be valid neither for operators with two delays.
In this paper we study Boundary value problems (BVPs) \(D_{j}(P,Q,m), m\in \{0,1\}, j\in \{1,2\},\) for Dirac-type system of the form
where
and \(p_{1}(x),p_{2}(x),q_{1}(x), q_{2}(x)\in L^{2}[0,\pi ]\) are complex-valued functions such that
This paper shall answer the question whether the theorem of uniqueness holds or not in the case when the first delay belongs to \(\left[ \frac{\pi }{3},\frac{2\pi }{5}\right) \) and the second one to \(\left[ \frac{\pi }{3},\pi \right) .\) In this way, the results from the paper [10] dealing with Dirac operator with one delay shall be generalized to Dirac operator with two delays. Besides research in inverse problems for delay(s) less than \(\frac{\pi }{3}\), further research in this area should also answer the question weather the theorem of uniqueness holds or not for Sturm–Liouville operators with two delays greater than one-third of the interval.
Hereinafter we will assume that delays \(a_{1}\) and \(a_{2}\) are known. Also, in the following we will assume that \(j\in \{1,2\} \) and \(m\in \{0,1\}.\)
Let \(\{\lambda _{n,j}^{m}\}\) be the spectra of the BVPs \(D_{j}(P,Q,m).\) The inverse problem of recovering matrix-functions \(P(x), x \in (a_{1},\pi )\) and \(Q(x), x \in (a_{2},\pi )\) from four spectra has been studied.
Inverse Problem 1. Given the spectra \(\{\lambda _{n,j}^{m}\}\) of the BVPs \(D_{j}(P,Q,m)\), find the matrix-functions P(x) and Q(x).
The paper is organized as follows: In Sect. 2 we construct characteristic functions and study asymptotic behavior of eigenvalues. Section 3 is devoted to the solving Inverse problem 1. We shall show that Theorem of uniqueness holds in the case when delays meet the condition \(2a_1+\frac{a_2}{2}\ge \pi , \) as well as it does not in the case when \(2a_1+\frac{a_2}{2}< \pi .\)
2 Spectral Properties
Based on the results from the paper [24], we obtain that Eq. (1.1) is equivalent to the integral equation
where
and
is a constant vector. Let
be the the solution of Eq. (1.1) such that
From (2.1) we obtain
where
We solve integral Eq. (2.2) by the method of successive approximation, where representation of the solution depends on the order of delays (see [24]).
Let us for \(k,l \in \{1,2\},\) introduce notations
Eigenvalues of the BVPs \(D_{j}(P,Q,m)\) coincide with zeros of the entire function
which is called characteristic function of BVPs \(D_{j}(P,Q,m)\). It has been shown in [24] that characteristic functions of BVPs \(D_{j}(P,Q,m)\) can be represented in the form
where
and for \(x\in (\frac{a_{1}}{2},\pi -\frac{a_{1}}{2})\)
Now we consider the asymptotic behavior of eigenvalues of BVPs \(D_{j}(P,Q,m).\) Using the standard approach involving Rouché’s theorem or proof from [10], one can show that the next theorem holds.
Theorem 2.1
The boundary value problems \(D_{j}(P,Q,m)\) have infinitely many eigenvalues \(\lambda _{n,j}^{m},\) \(n\in {{\mathbb {Z}}},\) of the form
where for \(\varkappa _{n,j}^{m}\ne {0}\) and for \(|n| \rightarrow \infty \)
3 Recovering of the Matrix-Functions
In order to recover the matrix-functions P(x) and Q(x) from the spectra \(\{\lambda _{n,j}^{m}\}\), at the beginning we construct characteristic functions by Hadamard theorem of factorization (Fig. 1).
Lemma 3.1
The specification of the spectra \((\lambda _{n,j}^{m})\) uniquely determines the characteristic functions \(\Delta _{1}^{m}\) and \(\Delta _{2}^{m}\) of BVPs \(D_{j}(P,Q,m)\) by the formulas
Proof
See Theorem 5 in [2]. \(\square \)
Using approach from the paper [10], we can recover functions \(K^{m}(x)\) and \(G^{m}(x)\) by formulas
where
Now we come to our main result. Using functions \(K^m(x)\) and \(G^m(x)\) from (3.1) and (3.2) respectively, we shall answer the question whether the theorem of uniqueness for Inverse problem 1 holds on the set
It will be shown that it is true on the subset
while on the subset
that is not true, Picture 3.1 (Fig. 1).
Firstly we will prove that the theorem of uniqueness holds on the subset \(R_1.\) We will recover functions \(p_{1}(x),\hspace{1mm} p_{2}(x)\) by formulas
where for \(x\in (\frac{a_1+a_2}{2},\pi -\frac{a_1+a_2}{2})\)
and
Functions \(q_{1}(x),\hspace{1mm} q_{2}(x)\) will be recovered by formulas
where for \(x\in (a_1,a_2)\cup (\pi -a_2,\pi -a_1)\)
for \(x\in (a_2,\pi -a_2)\)
and
Theorem 3.2
Let \(\frac{\pi }{3}\le a_1<\frac{2\pi }{5}<a_2<\pi \) and \(2a_1+\frac{a_2}{2}\ge \pi .\) The spectra \((\lambda _{n,j}^{m})\) of BVPs \(D_{j}(P,Q,m) \) uniquely determine matrix-functions \(P(x), \hspace{1mm} x \in (a_1,\pi )\) and \(Q(x), \hspace{1mm} x \in (a_2,\pi ).\)
Proof
We distinguish cases when the sum of delays is greater or less than \(\pi \).
1. Let \(a_1+a_2\ge \pi .\) We differ two cases
and
We will prove the theorem for the first case, i.e. assuming that \(a_2<2a_1,\) while the proof for the second case differs only in the order of the intervals on which functions will be determined. From (2.5) and (2.6), for \(x\in (\frac{a_{1}}{2},\frac{a_{2}}{2})\cup (\pi -\frac{a_{2}}{2},\pi -\frac{a_{1}}{2}),\) we have
i.e. we determine functions
If \( x\in (\frac{a_{2}}{2},a_1)\cup (\pi -a_1,\pi -\frac{a_{2}}{2}),\) functions \(K^m(x)\) and \(G^m(x)\) have the form
and
Then we recover functions
by formulas (3.3) for \(A_1(x)=A_2(x)=0\), as well as functions
by formulas (3.4) for \(B_1(x)=B_2(x)=0.\) Finally, from (2.5) and (2.6) we obtain that for \(x\in ({a_{1}},\pi -{a_{1}})\) functions \(K^m(x)\) and \(G^m(x)\) have the form
Then we recover functions
by formulas (3.3) for \(A_1(x)=A_2(x)=0.\) In this way functions \(p_{1}(x)\) and \(p_{2}(x)\) are recovered on \((a_{1},\pi ).\) Then integrals \(\alpha _{p_{k}p_{l}}^{1}(x), k,l\in \{1,2\}\) are known, too. Now, using formulas (3.4) for \(B_1(x)=\alpha _{p_{1}p_{2}}^{1}(x)- \alpha _{p_{2}p_{1}}^{1}(x)\) and \(B_2(x)=\alpha _{p_{1}p_{1}}^{1}(x)+ \alpha _{p_{2}p_{2}}^{1}(x),\) we determine functions
so they are also completely recovered on \((a_2,\pi ).\)
2. Let us now consider the case \(a_{1}+a_{2}<\pi .\) Taking the definition of the subset \(R_1\) into account, we haveFootnote 1Footnote 2
At the beginning, in the same way as in the proof of previous case, for \(x\in (\frac{a_{1}}{2},\frac{a_{2}}{2})\cup (\pi -\frac{a_{2}}{2},\pi -\frac{a_{1}}{2})\) and \( x\in (\frac{a_{2}}{2},a_1)\cup (\pi -a_1,\pi -\frac{a_{2}}{2}),\) using formulas (3.3) and (3.4), we determine functions
and functions
For \(x\in (a_1,\frac{a_{1}+a_2}{2})\cup (\pi -\frac{a_{1}+a_2}{2},\pi -a_1)\) functions \(K^m(x)\) and \(G^m(x)\) have the form (3.5) and (3.6) and we determine functions
by formulas (3.3) for \(A_1(x)=A_2(x)=0.\) In order to determine functions \(q_1(x), q_2(x)\), it is needed to show that integrals \(\alpha _{p_{k}p_{l}}^{1}(x), k,l\in \{1,2\},\) are known. For arguments of subintegral functions \(p_1(x), p_2(x)\) is valid
due to the assumption
Therefore, arguments of subintegral functions \(p_{1}(x), p_{2}(x)\) belong to the interval \( (a_{1},a_1+\frac{a_{2}}{2})\cup (\pi -\frac{a_{2}}{2},\pi ),\) hence integrals \(\alpha _{p_{k}p_{l}}^{1}(x) \) are known.Then, from (3.4) for \(B_1(x)=\alpha _{p_{1}p_{2}}^{1}(x)- \alpha _{p_{2}p_{1}}^{1}(x)\) and \(B_2(x)=\alpha _{p_{1}p_{1}}^{1}(x)+ \alpha _{p_{2}p_{2}}^{1}(x),\) we can determine functions
In the following we have in mind that the functions \(p_1(x)\) and \(p_2(x)\) are recovered on the interval \((a_1, a_1+\frac{a_2}{2})\cup (\pi -\frac{a_2}{2}, \pi )\) and functions \(q_1(x)\) and \(q_2(x)\) on the interval \((a_2, a_2+\frac{a_1}{2})\cup (\pi -\frac{a_1}{2}, \pi ). \) We differ two cases, depending on the condition whether the second delay is grater or less than \(\frac{\pi }{2}.\)
2.1. Let \(\frac{\pi }{2}\le a_2<\frac{2\pi }{3}.\) Then it remains to recover matrix-functions P(x) and Q(x) on the interval \( (\frac{a_1+a_{2}}{2},\pi -\frac{a_1+a_{2}}{2})\). We have
and
One can easily obtain that integrals \(\alpha _{p_{k}p_{l}}^{1}(x)\) are known. We have
since
Then, from (3.4) we can determine functions
In that way functions \(q_{1}(x),\hspace{1mm} q_{2}(x)\) are completely recovered on \((a_2,\pi ).\) Now it is not difficult to show that "mixed" integrals \(\alpha _{p_{k}q_{l}}^{12}(x)\) and \(\alpha _{q_{k}p_{l}}^{12}(x) \) are also known. Indeed, for arguments of subintegral functions \(p_1(x), p_2(x)\) in "mixed" integrals is valid
since
Therefore, arguments of subintegral functions \(p_{1}(x), p_{2}(x)\) belong to the interval \( (a_{1},a_1+\frac{a_{2}}{2})\cup (\pi -\frac{a_{2}}{2},\pi )\) and "mixed" integrals are known. Then, using formulas (3.3), we can determine functions
so they are also completely recovered on \((a_1,\pi ).\)
2.2. Let \(\frac{2\pi }{5}\le a_2<\frac{\pi }{2}.\) It remains to show that theorem of uniqueness is valid on the intervals \( (\frac{a_1+a_{2}}{2},a_2)\cup (\pi -a_2, \pi -\frac{a_1+a_{2}}{2})\) and \((a_2,\pi -a_2). \) On the interval \( (\frac{a_1+a_{2}}{2},a_2)\cup (\pi -a_2, \pi -\frac{a_1+a_{2}}{2})\) functions \(K^{m}(x)\) and \(G^{m}(x)\) have the form (3.7) and (3.8) respectively. In the same way as in the proof for the case 2.1, one can show that integrals \(\alpha _{p_{k}p_{l}}^{1}(x)\) are known, so we can determine functions
by formulas (3.4). Now we will show that “mixed” integrals \(\alpha _{p_{k}q_{l}}^{12}(x)\) and \(\alpha _{q_{k}p_{l}}^{12}(x)\) are also known. In the same way as in previous case we show that arguments of subintegral functions \(p_{1}(x), p_{2}(x)\) belong to the interval \( (a_{1},a_1+\frac{a_{2}}{2})\cup (\pi -\frac{a_{2}}{2},\pi )\). For arguments of subintegral functions \(q_{1}(x), q_{2}(x)\) in "mixed" integrals we have
since
Then, using formulas (3.3), we can determine functions
Let us finally consider the interval \( (a_2,\pi -a_2)\). Then functions \(K^{m}(x), G^{m}(x)\) have the form
and
Let us show that integrals \(\alpha _{p_{k}p_{l}}^{1}(x),\) as well as integrals \(\alpha _{q_{k}q_{l}}^{2}(x),\) are known. For arguments of subintegral functions \(p_1(x), p_2(x)\) we have
since
For arguments of subintegral functions \(q_1(x), q_2(x)\) the following is valid
since
In that way, using formulas (3.4), we can recover functions
so they are completely recovered on \((a_2,\pi )\). It remains to show that "mixed" integrals \(\alpha _{p_{k}q_{l}}^{1}(x)\) and \(\alpha _{q_{k}p_{l}}^{1}(x)\) are also known. Due to the condition \(\frac{5a_2}{2}>\pi ,\) for arguments of subintegral functions \(p_1(x), p_2(x)\) is valid
Then we can determine functions
by formulas (3.3), and they are also completely recovered on \((a_1,\pi ).\) Theorem is proved. \(\square \)
Now we will show that theorem of uniqueness does not hold on the subset \(R_2.\) For that purpose let us for fixed \(a_{1},a_{2}\) and define the integral operator \(M:L^{2}\left( a_{1}+\frac{a_{2}}{2},\pi -a_{1}\right) \rightarrow L^{2}\left( a_{1}+\frac{a_{2}}{2},\pi -a_{1}\right) \)
for some non-zero real function \(h(x) \in L^{2}(2a_1+\frac{a_2}{2},\pi ).\) Operator M(f(x)) is self-adjoint since
We can choose function h(x) such that the operator M(f(x)) has eigenvalue \(\eta _{1}=1\) with corresponding eigenfunction \(e_1(x)\) (see [6]), i.e.
We construct the family of functions
where
Using this family of functions, we will prove that the solution of Inverse problem 1 is not unique if \(2a_1+\frac{a_2}{2}<\pi .\) Let us for this purpose denote
Theorem 3.3
Let \(\frac{\pi }{3}\le a_1<\frac{2\pi }{5},\; \frac{\pi }{3}\le a_{2}<\frac{2\pi }{3},\; a_1<a_2\) and \(2a_1+\frac{a_2}{2}<\pi .\) The spectra \((\lambda _{n,j}^{m})\) of BVPs \(D_{j}(P_{\beta },Q_{\beta },m) \) is independent of \(\beta \).
Proof
Notice that in this case obviously \(a_1+a_2<\pi \) and functions \(K^m(x)\) and \(G^m(x)\) have the form (3.7) and (3.8) or (3.9) and (3.10) respectively, depending on the condition whether the second delay is less or not than \(\frac{\pi }{2}\). Taking into account that \(p_1(x)=q_1(x)=0\), as well as that function \(q_2^\beta (x)\) is vanishing for \(x\in (\pi -a_1,\pi ),\) we obtain that
while in functions \(G^m(x),\) besides functions \(p_2(x)\) and \(q_2(x),\) only integrals \(\alpha _{p_{2}p_{2}}^{1}(x)\) and \(\alpha _{p_{2}q_{2}}^{12}(x)\) remain. Since
and
we obtain
and
Indeed, for \(p_2(x)\) subintegral function’s argument in integral \(\alpha _{p_{2}p_{2}}^{1}(x)\) on the interval \((\pi -a_1-\frac{a_2}{2},\pi -a_1)\) the following is valid
while \(q_2(x)\) function’s argument in \(\alpha _{p_{2}q_{2}}^{12}(x)\) on the interval \((\pi -\frac{3a_1}{2},\pi -\frac{a_1+a_2}{2})\) satisfies condition
Then we have
Then from (2.3), (3.15) and (3.16), we obtain that characteristic functions \(\Delta _{1}^{m}(\lambda )\) for family of functions \(D_\beta \) have the form
Let us show that
Using the definition and properties of the integral operator M(f) from (3.11) and (3.12), as well as the form of functions \(p_1^\beta (x)\) and \(q_2^\beta (x)\) from (3.13) and (3.14), we obtain
Since
we obtain
In the same way one can show that
Indeed, we have
Then we obtain
In the same way from (2.4) we obtain
i.e. characteristic functions are independent of \(\beta .\) Theorem is proved. \(\square \)
Remark 3.4
If we consider the case when the first delay is greater than the second one, then both delays are less then \(\frac{2\pi }{5}\) and Theorem of uniqueness does not hold on the triangle which completes the set R up to the rectangle (Picture 3.1). But if we do not limit the first delay to the interval \([\frac{\pi }{3},\frac{2\pi }{5})\) and consider the set
it is clear that the theorem of uniqueness holds on the subset
On the subset
the theorem of uniqueness does not hold which follows from the results for Dirac operator with one delay. Indeed, if we take \(P(x)=0\) in (1.1), then we obtain two BVPs with one delay \(a_2.\) It is known that inverse problem’s solution is not unique in the case \(a_2<\frac{2\pi }{5}\) (see [10]) and then we conclude that the theorem of uniqueness does not hold on the subset \(S_2.\) Taking into account this result, as well as the result from the Theorem 3.2, we conclude that position of \((-1)^m\) in the BVPs setting is very important and significantly affects the size of the set on which the theorem of uniqueness is valid.
Notes
The order of delays \(2a_{1}<a_{2}<a_{1}+a_{2}<\pi \) is not possible since \(2a_{1}<a_{2}\implies \pi<3a_{1}<a_{1}+a_{2} \)
The order of delays \(2a_{1}<a_{2}<a_{1}+a_{2}<\pi \) is not possible since \(2a_{1}<a_{2}\implies \pi<3a_{1}<a_{1}+a_{2} \)
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Vojvodić, B., Djurić, N. & Vladičić, V. On Recovering Dirac Operators with Two Delays. Complex Anal. Oper. Theory 18, 105 (2024). https://doi.org/10.1007/s11785-024-01543-z
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DOI: https://doi.org/10.1007/s11785-024-01543-z