Abstract
The present paper studies uniqueness properties of the solution of the inverse problem for the Sturm-Liouville equation with discontinuous leading coefficient and the separated boundary conditions. It is proved that the considered boundary-value is uniquely reconstructed, i.e. the potential function of the equation and the constants in the boundary conditions are uniquely determined by given Weyl function or by the given spectral data.
Key words
Asymptotic formulas for eigenvalues; boundary value problems; inverse problems; spectral; analysis of ordinary differential operators; Sturm-Liouville theory; transformation operator
INTRODUCTION
This paper is concerned with the uniqueness theorems for the solution of some inverse spectral problems for the boundary value problem
where is real-valued function in is a complex parameter, , are real numbers,
with
Inverse spectral problems consist in recovering differential operators from their spectral characteristics (see Marchenko 201120 MARCHENKO VA. 2011. Sturm-Liouville operators and applications, AMS Chelsea Publishing Volume 373, 393 p., Levitan 198714 LEVITAN BM. 1987. Inverse Surm-Liouville problems, VNU Sci. Press, Utrecht, 240 p.). Such problems arise in many areas of science and engineering (see Hald 19809 HALD O. 1980. Inverse eigenvalue problems for the mantle. Geophys J R Astron Soc 62: 41-48., Krueger 198213 KRUEGER R. 1982. Inverse problems for nonabsorbing media with discontinuous material properties. J Math Phys 23: 396-404. , Willis 198419 WILLIS C. 1984. Inverse problems for torsional modes. Geophys J R Astron Soc 78: 847-853.). The goal of this work is to prove the uniqueness theorems for the solution of the inverse problem which determines the potential function and the constants by the Weyl function or by the spectral data of the boundary value problem .
In the classical case the direct and inverse problems for the Sturm-Liouville operators have been completely studied (see Marchenko 2011, Levitan 1987, Levitan and Gasymov 196415 LEVITAN BM AND GASYMOV MG. 1964. Determination of a differential equation by two spectra. Uspehi Mat Nauk 19: 3-63., Freiling and Yurko 20016 FREILING G AND YURKO V. 2001. Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington: NY, 305 p., Hryniv and Mykytyuk 200311 HRYNIV RO AND MYKYTYUK YAV. 2003. Inverse spectral problems for Sturm-Liouville operators with singular potentials. Inverse Probl 19: 665-684. , 200412 HRYNIV RO AND MYKYTYUK YAV. 2004. Half-inverse spectral problems for Sturm-Liouville operators with singular potentials. Inverse Probl 20: 1423-1444. , and the references therein). Direct and inverse problems for the discontinuous Sturm-Liouville boundary-value problems in different settings have been studied in Hald (198410 HALD O. 1984. Discontinuous inverse eigenvalue problems. Commun Pure Appl Math 37: 539-577. ), Andersson (19884 ANDERSSON L. 1988. Inverse eigenvalue problems for a Sturm-Liouville equation in impedance form. Inverse Probl 4: 929-971.), Guseinov and Pashaev (20028 GUSEINOV IM AND PASHAEV RT. 2002. On an inverse problem for a second-order differential equation. Russian Math Surveys 57: 597-598.), Carlson (19945 CARLSON R. 1994. An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients. Proc Amer Math Soc 120: 475-484.), Yurko (200021 YURKO VA. 2000. On boundary value problems with discontinuity conditions inside an interval. Differ Equ 36: 1266-1269.), Gasymov (19777 GASYMOV MG. 1977. The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient. Non-Classical Methods in Geophysics, 37-44, Nauka, Novosibirsk, Russia.), Amirov (20063 AMIROV RKH. 2006. On Sturm-Liouville operators with discontinuity conditions inside an interval. J Math Analysis Appl 317: 163-176.), Mamedov (200616 MAMEDOV KHR. 2006. Uniqueness of the solution of the inverse problem of scattering theory for Sturm-Liouville operator with discontinuous coefficient. Proc Inst Math Mech Natl Acad Sci Azerb 24: 163-172. , 201017 MAMEDOV KHR. 2010. On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition. Bound Value Probl ID 171967, 17 p.), Mamedov and Palamut (200918 MAMEDOV KHR AND PALAMUT N. 2009. On a direct problem of scattering theory for a class of Sturm-Liouville operator with discontinuous coefficient. Proc Jangjeon Math Soc 12: 243-251. )) and other works. Note that, the direct and inverse spectral problem for the equation with simple boundary conditions on the interval recently has been investigated in Akhmedova and Huseynov (201020 AKHMEDOVA EN AND HUSEYNOV HM. 2010. On inverse problem for Sturm-Liouville operator with discontinuous coefficients. Proc of Saratov University, New ser. Ser Math Mech and Inf 10: 3-9.) by using a new integral representations of the special solutions of Eq.
The spectral analysis of the boundary value problem was examined in Adiloglu and Amirov (20131 ADILOGLU NA AND AMIROV RKH. 2013. On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficient. Math Methods Appl Sci 36: 1685-1700.) where useful integral representations for two linearly independent solutions of equation were constructed (see also Akhmedova and Huseynov 2010), the asymptotic formulas for the eigenvalues and eigenfunctions were obtained, completeness and expansion theorems for the system of the eigenfunctions were proved. Using the results of Adiloglu and Amirov (2013), in the present paper, we investigate uniqueness properties of the solution of the inverse problems for the problem and study some other types boundary value problems related with the equation . In section 2 we prove that the boundary-value problem is uniquely reconstructed, i.e. the potential function and the constants are uniquely determined by given Weyl function or by the given spectral data. We also show that in the special case the potential and the coefficient can be determined by the one spectrum only. In section 3 we investigate the properties of the spectral characteristics of two boundary-value problems related to Eq. .
UNIQUENESS OF THE SOLUTION OF THE INVERSE PROBLEM
Let , be solutions of Eq. with initial conditions
and , be solutions of under conditions at
Then and are solutions of with initial conditions
and
respectively (see Adiloglu and Amirov 2013).
Denote by the solution of equation satisfying conditions
We set and consider the linearly independent solutions and of equation We have
where
Therefore,
Then Eq. can be written as
Additionally we see that the solution also satisfies the second one of the conditions . Consequently we obtain
We also have
The functions and are called the Weyl solution and the Weyl function of the boundary value problem respectively. From equation we have that the Weyl function is a meromorphic function with simple poles at the points where are eigenvalues of the boundary value problem (see Adiloglu and Amirov 2013).
We recall that the normalized numbers (see Adiloglu and Amirov 2013) of the boundary value problem are defined as
the set is the spectral data of the problem Note that there exists the sequence such that
(see Adiloglu and Amirov 2013).
The following theorem shows that the given spectral data uniquely determines the boundary- value problem
Theorem 1.
The following formula holds
Proof.
Since it follows from (Adiloglu and Amirov (2013)) (see formula Adiloglu and Amirov 2013) that , where Since where and for some we have
Further using the Lemma 1 (Adiloglu and Amirov (2013)) (see also the formula there) we find
Let
Then by virtue of we have On the other hand by the residue theorem
which gives the desired results as Theorem is proved. ∎
Now let the Weyl function of the boundary value problem is given. In the following theorem we prove that the boundary problem is uniquely reconstructed, i.e. the potential function and the constants are uniquely determined by given Weyl function.
Let us denote the boundary value problem by and the similar boundary value problem with the potential and boundary constants by . Then the following theorem is satisfied.
Theorem 2.
Let the Weyl functions of the boundary value problem and are and respectively. If then
Proof.
Define the matrix by the formula
where and are solutions of the equation of the boundary value problem which is identical to the solutions and Since we have
or
Using the formula in equation we find
Now using the estimations and in (Adiloglu and Amirov (2013)) we have
On the other hand from the equation we obtain
Thus if then the functions and are entire in Together with this gives , Then from we obtain , for all and Since and satisfy the equations
correspondingly, substructing these equations we have
Therefore a.e. on . Further, because of for all and from the conditions we also have that Consequently Theorem is proved. ∎
Theorem 3.
Let and are the spectral data of the problems and respectively. If then
Proof.
It is known (Adiloglu and Amirov (2013)) that the solutions and of equation satisfying the conditions
are expressed as
respectively, where
and for each Using and we have
where
Clearly for all Note that the function also satisfies the conditions (see Adiloglu and Amirov 2013)
Since
where the equation can be written as
where
It is clear that if then
where is continuous kernel. Then we can see the equation as a Volterra integral equation with respect to From the theory of the Volterra integral equations, we know that the equation is then uniquely solvable and the solution is
where is a continuous kernel.
Let now In this case the equation is written as
where Here the kernel has a jump discontinuity at Clearly, and when and therefore takes the form
where
Now using we obtain that
where is a continuous kernel. Therefore the equation is written as
where the kernel is continuous. Hence, we obtain the Volterra integral equation
where
Solving the Volterra integral equation with respect to we find
Taking into account, the expression for we have
where is a kernel with jump at Consequently, we have
where is a kernel with jump at
Let implies that
where Hence for all
where and Using the Parseval’s equality (Adiloglu and Amirov (2013)), we calculate
Now if we consider the operator
we have and Then by we obtain
for any Then is possible if and only if Thus i.e. a.e. on and Theorem is proved. ∎
Let for and . We now show that in this case the potential and coefficient can be determined by the spectrum only.
Theorem 4.
If and then a.e. on and
Proof.
If and is a solution of the Eq. for , then is a solution of for
Indeed, it is easy to check that
is a solution for In particular, if we take the solution
then is the solution of satisfying the initial conditions
Consequently, Now using we have
which implies On the other hand and
Consequently we have hence from the formula we obtain
Since we have Then by the previous theorem a.e. on and Theorem is proved. ∎
BOUNDARY VALUE PROBLEMS AND
(i) Consider the boundary value problem for equation with the boundary conditions
The characteristic function of the problem is and the eigenvalues of are the squares of zeros of the equation
Note that as in the case of the problem we can prove that the eigenvalues of the problem are real and simple.
Since
we have for the eigenvalues the following asymptotic formula:
where
are the roots of the equation and
Further it is easy to show that (see Adiloglu and Amirov 2013 also) the solution satisfies the asymptotic relation
where
Therefore we obtain
where
with
Since , equations and imply that
and consequently
which implies
where
Hence
Moreover if we define the normalized numbers for the problem as
then we have
where
Since is an entire function of order one by the H’Adamard’s theorem the function is uniquely determined up to a multiplicative constant by its zeros:
We also have
Since
we have
therefore
We have proved the following two theorems:
Theorem 5.
The boundary value problem has a countable set of eigenvalues and for sufficiently large values of the asymptotic formula
are satisfied,where
is bounded and Moreover, we have
for the normalized numbers
the problem , where
Theorem 6.
The specification of the spectrum uniquely determines the characteristic function by the formula
(ii) Consider the boundary value problem for equation with the boundary condition
The eigenvalues of the problem are simple and coincide with the squares of zeros of the characteristic function
where is the solution of Eq. with the initial conditions
From the results of Adiloglu and Amirov (2013), we have
Since
We have
where
Therefore
From we obtain the following expression for the roots of the function
where
Since some simple transformations lead to
where Therefore
Moreover we can obtain that
We can formulate the following theorem for the boundary- value problem
Theorem 7.
The boundary value problem has a countable set of eigenvalues and for sufficiently large values of the asymptotic formula is satisfied. Moreover, the characteristic function
is uniquely determined by the spectrum via the formula
Lemma 1.
The following relation holds
i.e. the eigenvalues of two boundary problems and are alternating.
Proof.
Consider the characteristic functions and of the boundary value problems and respectively. Let . Then
Since
for we obtain
where
and
From we have for
and
where is real and
Thus the function is monotonically decreasing when and
Then from asymptotic formulas for and we arrive at ∎
ACKNOWLEDGMENTS
We thank the referees for their encouraging remarks and insightful comments. Also, this work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) with Project 113F366.
REFERENCES
-
1ADILOGLU NA AND AMIROV RKH. 2013. On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficient. Math Methods Appl Sci 36: 1685-1700.
-
20AKHMEDOVA EN AND HUSEYNOV HM. 2010. On inverse problem for Sturm-Liouville operator with discontinuous coefficients. Proc of Saratov University, New ser. Ser Math Mech and Inf 10: 3-9.
-
3AMIROV RKH. 2006. On Sturm-Liouville operators with discontinuity conditions inside an interval. J Math Analysis Appl 317: 163-176.
-
4ANDERSSON L. 1988. Inverse eigenvalue problems for a Sturm-Liouville equation in impedance form. Inverse Probl 4: 929-971.
-
5CARLSON R. 1994. An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients. Proc Amer Math Soc 120: 475-484.
-
6FREILING G AND YURKO V. 2001. Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington: NY, 305 p.
-
7GASYMOV MG. 1977. The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient. Non-Classical Methods in Geophysics, 37-44, Nauka, Novosibirsk, Russia.
-
8GUSEINOV IM AND PASHAEV RT. 2002. On an inverse problem for a second-order differential equation. Russian Math Surveys 57: 597-598.
-
9HALD O. 1980. Inverse eigenvalue problems for the mantle. Geophys J R Astron Soc 62: 41-48.
-
10HALD O. 1984. Discontinuous inverse eigenvalue problems. Commun Pure Appl Math 37: 539-577.
-
11HRYNIV RO AND MYKYTYUK YAV. 2003. Inverse spectral problems for Sturm-Liouville operators with singular potentials. Inverse Probl 19: 665-684.
-
12HRYNIV RO AND MYKYTYUK YAV. 2004. Half-inverse spectral problems for Sturm-Liouville operators with singular potentials. Inverse Probl 20: 1423-1444.
-
13KRUEGER R. 1982. Inverse problems for nonabsorbing media with discontinuous material properties. J Math Phys 23: 396-404.
-
14LEVITAN BM. 1987. Inverse Surm-Liouville problems, VNU Sci. Press, Utrecht, 240 p.
-
15LEVITAN BM AND GASYMOV MG. 1964. Determination of a differential equation by two spectra. Uspehi Mat Nauk 19: 3-63.
-
16MAMEDOV KHR. 2006. Uniqueness of the solution of the inverse problem of scattering theory for Sturm-Liouville operator with discontinuous coefficient. Proc Inst Math Mech Natl Acad Sci Azerb 24: 163-172.
-
17MAMEDOV KHR. 2010. On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition. Bound Value Probl ID 171967, 17 p.
-
18MAMEDOV KHR AND PALAMUT N. 2009. On a direct problem of scattering theory for a class of Sturm-Liouville operator with discontinuous coefficient. Proc Jangjeon Math Soc 12: 243-251.
-
19WILLIS C. 1984. Inverse problems for torsional modes. Geophys J R Astron Soc 78: 847-853.
-
20MARCHENKO VA. 2011. Sturm-Liouville operators and applications, AMS Chelsea Publishing Volume 373, 393 p.
-
21YURKO VA. 2000. On boundary value problems with discontinuity conditions inside an interval. Differ Equ 36: 1266-1269.
Publication Dates
-
Publication in this collection
Dec 2017
History
-
Received
05 Feb 2016 -
Accepted
27 Jan 2017