Abstract
The topic of this paper are non-self-adjoint second-order differential operators with constant delay generated by \(-y''+q(x)y(x-\tau )\) where potential q is complex-valued function, \(q\in L^{2}[0,\pi ]\). We study inverse problems of these operators for \(\tau \in \left[ \frac{2\pi }{5},\pi \right) \). We investigate the inverse spectral problems of recovering operators from their two spectra, firstly under Dirichlet–Dirichlet and second under Dirichlet/Polynomial boundary conditions. We will prove theorem of uniqueness, and we will give procedure for constructing potential. In the first case, for \(\tau \in \left[ \frac{\pi }{2},\pi \right) :\) we will show that Fourier coefficients of a potential are uniquely0 determined by spectra. In the second case for \(\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) ,\) we will construct integral equation under potential and we will prove that this integral equation has a unique solution. Also, we will show that other parameters are uniquely determined by spectra.
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1 Introduction
An intensive development of the spectral theory for various classes of differential and integral operators as well as for operators in abstract spaces took place in the second half of twentieth century and twenty-first century. Within this theory inverse spectral problems take a special place. Inverse spectral problems are problems studying operator from some of its spectral characteristics. The inverse spectral problem can be regarded from three aspects: existence, uniqueness and reconstruction of the operators with specific properties of eigenvalues and eigenfunctions. The Sturm–Liouville-type operators are generated by second-order differential expression and boundary conditions (see [7, 9] and references therein). In this paper, we study Sturm–Liouville operators with one constant delay under Dirichlet–Dirichlet and Dirichlet/Polynomial boundary conditions. In the papers [1,2,3, 6, 8, 13,14,15,16] authors study this differential expression under different types of boundary conditions. Also in many papers authors study boundary conditions with spectral parameter (see [10,11,12]). We will show uniqueness and we will recover the operators from two spectra.
In this paper we generalize the results in the paper [13] and [14].
In this paper, we study two boundary value problems \(L_{k},k = {0,1}\), \(L_{0}\) generated by (1.1),(1.2),(1.3) and \(L_{1}\) generated by (1.1),(1.2),(1.4)
where \(\lambda \) is the spectral parameter, \(\tau \in \left[ \frac{2\pi }{5},\pi \right) \), Function q(x) is a complex-valued function which we call potential, such that \(q \in L^{2}(0,\pi )\), and \(q(x)\equiv 0\) for \(x \in [0,\tau ]\). Function \(P(\lambda )\) is normalized polynomial with degree \(s,s\in N\) and complex coefficients. We will separately study two cases, the first when \(\tau \in \left[ \frac{\pi }{2},\pi \right) \) and the second when \(\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) \).
The spectra of \(L_{0}\) and \(L_{1}\) are countable. We will prove that the potential q and polynomial P are uniquely determined from the spectra of \(L_{0}\) and \(L_{1}\). Let \((\lambda _{n})_{n=1}^{\infty }\) be the eigenvalues of \(L_{0}\) and \((\mu _{n})_{n=1}^{\infty }\) be the eigenvalues of \(L_{1}\).
The inverse problem is to prove that q(x) and \(P(\lambda )\) are uniquely determined from \((\lambda _{n})_{n=1}^{\infty }\) and \((\mu _{n})_{n=1}^{\infty }\), and determine q(x) and \(P(\lambda )\) from \((\lambda _{n})_{n=1}^{\infty }\) and \((\mu _{n})_{n=1}^{\infty }\).
2 Preliminaries
Let the function Y(x) be the solution of the differential equation (1.1) under initial condition \(Y(0)=0, Y'(0)=1\), then the function Y(x) satisfy an integral equation
where \(\lambda =z^{2}\). We will be solving equation (2.1) using \(q(x)\equiv 0\) for \(x \in [0,\tau ]\).
For \(x \in [0,\tau ]\), the solution is
For \(x \in (\tau , 2\tau ]\), the solution is
For \(x\in (2\tau ,3\tau ]\), the solution is
3 Main Results
3.1 Linear Case, \(\tau \in \left[ \frac{\pi }{2},\pi \right) \)
In the case, when \(\tau \in \left[ \frac{\pi }{2},\pi \right) \) we have \(\pi \in (\tau ,2\tau ]\), and using (1.3) and (1.4) from Eq. (2.2), we get
The functions \(\Delta _{0}(\lambda ), \Delta _{1}(\lambda )\) are entire in \(\lambda \) of order 1/2. It is clear that the set of zeros of functions \(\Delta _{0}(\lambda ), \Delta _{1}(\lambda )\) is equivalent to the spectrum of boundary spectral problems \(L_{0}, L_{1}\), respectively. For the spectrum \((\lambda _{n})_{n=1}^{\infty }\) of boundary spectral problems \(L_{0}\), we have the asymptotic formula (see [8]):
For the spectrum \((\mu _{n})_{n=1}^{\infty }\) of boundary spectral problems \(L_{1}\), using well-known method (see [8]), based on Rouche‘s theorem, we have asymptotic formula which depends on the degree of the polynomial \(P(\lambda )\). When degree of the polynomial \(P(\lambda )\) is equal to 1, \(s=1\) we have asymptotic formula
When degree of the polynomial \(P(\lambda )\) is different from 1, \(s>1\), we have asymptotic formula
Using Hadamard’s factorization theorem we conclude that spectra uniquely determine functions \(\Delta _{0}(\lambda ), \Delta _{1}(\lambda )\). We introduce notation
The delay \(\tau \) and the integral \(\int _{\tau }^{\pi } q(t) {\text{ d }}t=I_{1}\) are uniquely determined from the spectrum \((\lambda _{n})_{n=1}^{\infty }\)(see [1]).
Lemma 3.1
The spectra \((\mu _{n})_{n=1}^{\infty }\) and \((\lambda _{n})_{n=1}^{\infty }\) of boundary value problem \(L_{1}\) and \(L_{0}\) uniquely determine polynomial \(P(\lambda )\).
Proof
Function \(F_{1}(z)\) is uniquely determined by spectrum \((\mu _{n})_{n=1}^{\infty }\) and function \(F_{0}(z)\) is uniquely determined by spectrum \((\lambda _{n})_{n=1}^{\infty }\).
Let \(P(\lambda )=\lambda ^{s}+p_{s-1}\lambda ^{s-1}+\cdots +p_{0},p_{i}\in C\,.\) We have
Now, we put \(z=2m+\frac{1}{2},m\in N\) and we have
Since
we conclude \( \lim \limits _{m\rightarrow \infty } \frac{ \left( p_{s-2}\left( 2m+\frac{1}{2}\right) ^{2s-4}+\cdots +p_{0}\right) F_{0}\left( 2m+\frac{1}{2}\right) }{\left( 2m+\frac{1}{2}\right) ^{2s-2}} =0 \, ,\) and from this equation, we have
Now, when we showed that coefficient \(p_{s-1}\) is known from spectra \((\mu _{n})_{n=1}^{\infty }\) and \((\lambda _{n})_{n=1}^{\infty }\), we have
We repeat this procedure and we have all coefficients \(p_{s-1},...,p_{1}\) are determined by spectra \((\mu _{n})_{n=1}^{\infty }\) and \((\lambda _{n})_{n=1}^{\infty }\).
Now, we will show that coefficient \(p_{0}\) is determined by spectra \((\mu _{n})_{n=1}^{\infty }\) and \((\lambda _{n})_{n=1}^{\infty }\).
First we transform product trigonometric functions to sum and we have
if we put \(z=2m+\frac{1}{2},m\in N\), from Riemann–Lebesgue lemma we have
Finally, we have
and polynomial function \(P(\lambda )\) is ordered by spectra \((\mu _{n})_{n=1}^{\infty }\) and \((\lambda _{n})_{n=1}^{\infty }\). \(\square \)
Since \(q(x) = 0\) for \(x\in [0, \tau )\), we have \(a_{0}=\int \limits _{0}^{\pi } q(t) {\text{ d }}t=\int \limits _{\tau }^{\pi } q(t) {\text{ d }}t = I_{1}\) and coefficient \(a_{0}\) is ordered by spectrum \((\lambda _{n})_{n=1}^{\infty }\). Now we will prove that the other coefficients \(a_{m}=\int \limits _{0}^{\pi }q(t) \cos 2mt {\text{ d }}t \) and \(b_{m}=\int \limits _{0}^{\pi } q(t) \sin 2mt {\text{ d }}t\) of the potential q are also uniquely determined by spectra.
Theorem 3.2
Let \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\) be the spectra of boundary spectral problems \(L_{k}, k=0,1\), respectively, then potential q is uniquely determined by \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\) if \(\frac{\pi }{2} \leqslant \tau < \pi \).
Proof
Using transformations product of trigonometric functions to sum and addition formulas, we have
and
Now we put \(z = m, m \in N\) and using \(q(x)=0\) for \(x\in [0, \tau )\), we have
This is linear system of two variables \(a_{m}\) and \(b_{m}\) with determinant \(D=\frac{1}{4m}\ne 0\), and which has unique solution
Since \(\tau \), \(\int _{\tau }^{\pi }q(t){\text{ d }}t\), polynomial \(P(\lambda )\) and functions \(F_{0}, F_{1}\) are determined from spectra \((\lambda _{n})_{n=1}^{\infty }\) and \((\mu _{n})_{n=1}^{\infty }\), coefficients \(a_{m}\) and \(b_{m}\) are also determined from this spectra. Since \(q\in L^{2}[0, \pi ]\), we have
where is \(c_{m}= \frac{1}{\pi }a_{m} - \frac{i}{\pi }b_{m} \), and we finally conclude that potential q is uniquely determine from spectra \((\lambda _{n})_{n=1}^{\infty }\) and \((\mu _{n})_{n=1}^{\infty }\). \(\square \)
3.2 Nonlinear Case, \(\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) \)
In the case when \(\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) \) we have \(\pi \in (2\tau ,3\tau ]\), and using (1.3) and (1.4) from Eq. (2.3), we get
The functions \(\Delta _{0}(\lambda ), \Delta _{1}(\lambda )\) are entire in \(\lambda \) of order 1/2. It is clear that the set of zeros of functions \(\Delta _{0}(\lambda ), \Delta _{1}(\lambda )\) is equivalent to the spectrum of boundary spectral problems \(L_{0}, L_{1}\), respectively. Like in previous case specification of the spectra uniquely determines functions \(\Delta _{0}(\lambda ), \Delta _{1}(\lambda )\). We introduce notation
and
Similar to the first case we have that delay \(\tau \) and the integral \(I_{1}\) are uniquely ordered from the spectrum \((\lambda _{n})_{n=1}^{\infty }\)(see [1]).
Lemma 3.3
The spectra \((\mu _{n})_{n=1}^{\infty }\) and \((\lambda _{n})_{n=1}^{\infty }\) of boundary value problem \(L_{1}\) and \(L_{0}\) uniquely determine polynomial \(P(\lambda )\).
Proof
Similar like in first case. \(\square \)
Now we transform the products of trigonometric functions into sums/differences and we have
We define functions \(K:[0,\pi ]\rightarrow R\), \(\widetilde{q}:[0,\pi ]\rightarrow R\)
It is obvious that \(K(t)\in L^{2}[0,\pi ]\) and \(\widetilde{q}(t)\in L^{2}[0,\pi ]\).
Using the following formulas as well as analogous formulas with cosine function function
and notation
we have
One can easily show that \(\int _{\tau }^{\pi -\tau }K(t){\text{ d }}t=-I_{2}\). Using this formula and integration by parts from (3.5) and (3.6), we have
and
where is
Using \(\int _{\tau }^{\pi -\tau }K(t){\text{ d }}t=-I_{2}\), we have
and
where is
Now we transform (3.7) and (3.8) and we have
and
We define function A(z)
Function A(z) is determined by spectrum \((\lambda _{n})_{n=1}^{\infty }\) and from (3.10), we have
Also, we define function \(B_{1}(z)\)
this function is ordered by spectrum \((\mu _{n})_{n=1}^{\infty }\) and we have
Finally, we define function
which is determined by spectra \((\lambda _{n})_{n=1}^{\infty }\) and \((\mu _{n})_{n=1}^{\infty }\), and
We define function \(K^{*}:[0,\pi ]\rightarrow R\)
Put \(z = m, m \in N\) into (3.11) and (3.12), we have
From (3.11), we have
We multiply Eq. (3.13) by \(\frac{1}{\pi }{\text{ e }}^{2imt} \), Eq. (3.14) by \(\frac{-i}{\pi }{\text{ e }}^{2imt} \) and Eq. (3.15) by \( \frac{1}{\pi } \) and then sum them, using definition of function \(\widetilde{q}(t)\) and \(K^{*}(t)\) we get the integral equation
where
Theorem 3.4
Let \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\) be the spectra of boundary spectral problems \(L_{k}, k=0,1\), respectively, then potential q is uniquely determined by \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\) if \(\frac{2\pi }{5} \leqslant \tau < \frac{\pi }{2}\).
Proof
The potential q satisfies integral equation (3.16), we will show uniqueness of solution of this equation.
-
For \(t\in \left( \pi -\tau , \pi -\frac{\tau }{2}]\right) \), since \(K^{*}(t)=0,t\in \left( \pi -\tau , \pi -\frac{\tau }{2}\right] \), integral equation (3.16) have a form:
$$\begin{aligned} \widetilde{q}(t)=f(t) \end{aligned}$$Function f is determined by \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\), then potential q(x) is determined for \(x\in (\pi -\frac{\tau }{2}, \pi ]\).
-
For \(t\in \left( \frac{\tau }{2}, \tau \right] \), since \(K^{*}(t)=0,t\in \left( \frac{\tau }{2},\tau \right] \), integral equation (3.16) have a form:
$$\begin{aligned} \widetilde{q}(t)=f(t). \end{aligned}$$Function f is determined by \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\), then potential q is determined for \(x\in [\tau , \frac{3\tau }{2}]\).
-
For \(t\in (\tau , \pi - \tau ]\), from (3.16), we have equation
$$\begin{aligned} \widetilde{q}(t) - \int \limits _{t}^{\pi -\tau }K(s){\text{ d }}s=f(t). \end{aligned}$$One can easily show that arguments of the potential q appearing in the function
$$\begin{aligned} \int \limits _{t}^{\pi -\tau }K(s){\text{ d }}s = \int \limits _{t}^{\pi -\tau }\left( q(s+\tau )\int \limits _{\tau }^{s}q(u){\text{ d }}u-q(s)\int \limits _{s+\tau }^{\pi }q(u){\text{ d }}u-\int \limits _{s+\tau }^{\pi }q(u-s)q(u){\text{ d }}u\right) {\text{ d }}s \end{aligned}$$belong to the intervals \([2\tau , \pi ] \subset [\pi -\frac{\tau }{2}, \pi ]\) and \( [\tau , \pi - \tau ] \subset [\tau , \frac{3\tau }{2}] \). Then the function \( \int \limits _{t}^{\pi -\tau }K(s){\text{ d }}s \) is known. Therefore from (3.16) for \(t\in (\tau , \pi - \tau ]\), we get
$$\begin{aligned} \widetilde{q}(t) = \int \limits _{t}^{\pi -\tau }K(s){\text{ d }}s+f(t). \end{aligned}$$Function f is determined by \((\lambda _{n}) _{n=1}^{\infty }\) and \((\mu _{n}) _{n=1}^{\infty }\), then potential q(x) is determined for \(x\in \left( \frac{3\tau }{2}, \pi -\frac{\tau }{2}\right] \).
\(\square \)
When \(\frac{\pi }{3} \leqslant \tau < \frac{\pi }{2}\), we have same integral equation like 3.13, but in the case when \(\frac{\pi }{3} \leqslant \tau < \frac{2\pi }{5}\) not satisfied \([2\tau , \pi ] \subset [\pi -\frac{\tau }{2}, \pi ]\) and \( [\tau , \pi - \tau ] \subset [\tau , \frac{3\tau }{2}] \), and we cannot prove Theorem 3.4 on this way. Moreover, in the case when \(\frac{\pi }{3} \leqslant \tau < \frac{2\pi }{5}\) theorem of uniqueness not true. For this conclusion, the main arguments are the results published in the papers [4] and [5]. In the case when \(\frac{\pi }{3}\leqslant \tau < \frac{2\pi }{5}\) then critical interval \(\left( \frac{3\tau }{2},\pi -\tau \right) \) not equal to \(\varnothing \).
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Communicated by Majid Gazor.
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Vladičić, V., Bošković, M. & Vojvodić, B. Inverse Problems for Sturm–Liouville-Type Differential Equation with a Constant Delay Under Dirichlet/Polynomial Boundary Conditions. Bull. Iran. Math. Soc. 48, 1829–1843 (2022). https://doi.org/10.1007/s41980-021-00616-5
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DOI: https://doi.org/10.1007/s41980-021-00616-5
Keywords
- Differential operators with delay
- Inverse problems
- Fourier trigonometric coefficients
- Integral equations