R E S E A R C H
Open Access
Inverse spectral problem for
Sturm-Liouville operator with discontinuous
coefficient and cubic polynomials of spectral
parameter in boundary condition
Aynur Çöl
**Correspondence: [email protected] Department of Mathematics Education, Sinop University, Sinop, 57100, Turkey
Abstract
In this paper, the inverse scattering problem for the Sturm-Liouville operator with discontinuous coefficient and cubic polynomials of the spectral parameter in the boundary condition is considered. The scattering data of the problem is defined, and its properties are investigated. The modified Marchenko main equation is obtained and it is shown that the potential is uniquely recovered by the scattering data.
MSC: 34A55; 34B40; 34B07; 34L25; 81U40
Keywords: inverse problem; scattering data; Sturm-Liouville operator; spectral parameter; discontinuous coefficient
1 Introduction
Consider the boundary value problem generated by the differential equation
–y+q(x)y=λρ(x)y ( <x<∞), () with the boundary condition
α+iαλ–αλ–iαλ
y() –β+iβλ–βλ–iβλ
y() = , () whereλis a spectral parameter,q(x) is real valued function with the condition
∞
( +x)q(x)dx<∞, ()
andρ(x) is a piecewise constant function in the form
ρ(x) =
α, ≤x<a,
, a≤x<∞, () =α> . Herepj(λ) (j= , ) is a polynomial
p(λ)≡α+iαλ–αλ–iαλ, p(λ)≡β+iβλ–βλ–iβλ
with the relations
αi+βi–αiβi+> , αi+βi–αiβi+< , αi+βi–αiβi+= ()
forαi,βi∈R(i= , ).
Inverse spectral problems of spectral analysis often appear in mathematics, mechanics, physics and other branches of natural sciences. The direct scattering problem consists of the determination the collection {S(λ),{λj}Nj=,{mj}N
j=} whenq(x) is known. The inverse
scattering problem deals with the construction ofq(x) in terms of the scattering data. In this paper in the case of discontinuous coefficient (), we consider the inverse scattering problem for Sturm-Liouville operator with cubic polynomials of spectral parameter in boundary condition ().
Discontinuous inverse problems appear in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics and in geophysi-cal models for oscillations of the earth (see [] and the references therein). In the case thatρ(x)≡ and the boundary condition does not contain a spectral parameter, the in-verse scattering problem for () was solved by Marchenko [, ] and Levitan [, ]. The inverse scattering problem of the Sturm-Liouville operator with discontinuous coefficient was studied in [–]. The problem was examined in [] by using a new integral represen-tation of the Jost solution of ().
An important case in spectral theory is that containing the spectral parameter in equations and boundary conditions. Sturm-Liouville problems with spectral parameter-dependent boundary conditions arise in studies of heat conduction problems and vibrat-ing strvibrat-ing problems. Fulton and Pruess showed a kind of heat conduction problems in []. Problems with the dependence on spectral parameter can be found in [–]. The inverse scattering problem for () with a linear spectral parameter in the boundary condition was solved in [].
The main result of this paper is that the potentialq(x) can be uniquely recovered from the given scattering data. The Marchenko method is applied to solving the boundary value problem when the boundary conditions depend on spectral parameter as nonlinear.
Ifq(x) = , the result is obtained that the function
f(x,λ) =
+
ρ(x) e
iλμ+(x)+
–
ρ(x) e
iλμ–(x)
is a solution of (), whereμ±(x) =±xρ(x) +a(∓ρ(x)).
As proven in [], if the condition () is satisfied, () has a unique solutionf(x,λ), which satisfies the asymptotic behavior
lim
x→+∞e
–iλxf(x,λ) =
forImλ≥ and can be expressed by
f(x,λ) =f(x,λ) +
∞
μ+(x)
which is called the Jost solution. For the kernel functionK(x,t), the inequality
is satisfied. Also, ifq(x) is differentiable, the kernelK(x,t) is twice differentiable and satis-fies both the equation Denote byϕ(λ,x) the solution of () satisfying the conditions
ϕ(,λ) =p(λ), ϕ(,λ) =p(λ).
This solution satisfies the condition (). The function
S(λ) =p(λ)f
(,λ) –p
(λ)f(,λ) p(λ)f(,λ) –p(λ)f(,λ)
()
is the scattering function of the boundary value problem ()-(). It is a meromorphic function in the upper half plane Imλ> with the poles at the zeros of the function
p(λ)f(,λ) –p(λ)f(,λ). These poles are simple and lie on the imaginary axis. The
norm-ing numbers are defined as
m–k ≡
integral equation which is called the main equation,
F(x) =
π
∞
–∞
S∞(λ) –S(λ)eiλxdλ+ N
k=
mke–μkx, λ
k=iμk,
S∞(λ) =
⎧ ⎨ ⎩
e–iλaτe–iλaα–
τ–e–iλaα, αn= , e–iλaαα(–+τe–iλaα)–β(+τe–iλaα)
αα(e–iλaα–τ)–β(e–iλaα+τ) , αn= ,
andτ=α–
α+.
Obviously, from the given scattering data {S(λ),{λj}Nj=,{mj}Nj=}, the functionF(x,y) is found by () which is then introduced in (). If () has a unique solutionK(x,y), then the functionq(x) can be found from () and (). The solvability of the integral equation () is examined and the algorithm of recovering the potentialq(x) is given.
This paper is organized as follows. In Section , the scattering data to the boundary value problem ()-() is found by using a new integral representation for the solution of (), and its properties are investigated. In Section , the main equation for the boundary value problem ()-() is derived. Finally, the solvability of the main equation is proved and the unique recovery of the potential from the solution of the main equation is shown in Section .
Lety(x,λ) andz(x,λ) be solutions of (). The expression
Wy(x,λ),z(x,λ)=yz–yz
is called the Wronskian of the functionsy(x,λ) andz(x,λ). It is clear that for all realλ= ,
f(x,λ) andf(x,λ) constitute fundamental solutions of (). The Wronksian of these func-tions does not depend onxand equals iλ.
2 Scattering data
Lemma For all realλ= ,the following identity is valid:
iλϕ(x,λ)
E(λ) =f(x,λ) –S(λ)f(x,λ) ()
and S(λ)possesses the following properties:
S(λ) =S(–λ), S(λ)< .
Proof Sincef(x,λ) andf(x,λ) constitute the fundamental solution system of () for real
λ= , we have
ϕ(x,λ) =c(λ)f(x,λ) +c(λ)f(x,λ), ()
wherec(λ) andc(λ) are functions which we have to find. Taking account of the following
equalities:
c(λ) andc(λ) are found and substituted in (). Thus, we obtain
ϕ(x,λ) = –p(λ)f(,λ) –p(λ)f(,λ) iλ f(x,λ) +
p(λ)f(,λ) –p(λ)f(,λ)
iλ f(x,λ). ()
Let
E(λ)≡p(λ)f(,λ) –p(λ)f(,λ).
Now, it is necessary to showE(λ)= for all realλ= . Assume the contrary, then there existsλ∈R,λ= , such that
p(λ)f(,λ) =p(λ)f(,λ).
Also
Wf(,λ),f(,λ)
= iλ
is satisfied. By using these relations, we obtain
|f(,λ)|
|p(λ)|
αβ–αβ+ (αβ–αβ)|λ|+ (αβ–αβ)|λ|
= –
and this is a contradiction since the left-hand side is positive. Thus, by dividing the equality () by iλE(λ), we obtain () whereS(λ) is defined with (). SinceE(λ) =E(–λ) and the numerator has the same property, it is clear that
S(λ) =S(–λ).
Also [p(λ)p(λ) –p(λ)p(λ)][f(,λ)f(,λ) –f(,λ)f(,λ)] < holds for all realλ= , and
so the identity
S(λ)<
is satisfied. Thus, the lemma is proved.
Lemma The function E(λ)has only a finite number of zeros on the half plane(Imλ> ).
All the zeros are simple and lie on the imaginary axis.
Proof By the proof of Lemma we haveE(λ)= for all realλ= , the pointλ= is the pos-sible zero ofE(λ). SinceE(λ) is analytic in the upper planeImλ> andf(,λ) is bounded as|λ| → ∞, it follows that the set of zeros ofE(λ) is bounded and forms at most countable set having as zero the only possible limit point.
Now we show that zeros ofE(λ) lie on the imaginary axis. Assume thatλandλare
zeros ofE(λ). Then they satisfy ():
–f(x,λ) +q(x)f(x,λ) =λρ(x)f(x,λ), ()
We multiply () byf(x,λ) and () byf(x,λ), subtract the latter from the former, and
finally integrate this relation according toxfrom to∞. As a result,
Since we have the condition (), the expression in the parentheses is positive, and it implies thatμ+μ= ,i.e.,μis pure imaginary.
Let us prove that there are only finitely many zeros. Letδdenote the infimum of the dis-tances between two neighboring zeros ofE(λ), and showδ> . Let us assume the contrary and let{iλk}and{iλˆk}be two sequences of zeros of the functionE(λ) such that
ForAlarge enough, the inequality
f(x,iλk) >
On the other hand, the equality () yields
+ (λˆk+λk)
A
f(x,iλˆk)f(x,iλˆk)ρ(x)dx
+ (λˆk+λk)
∞
A
f(x,iλˆk)f(x,iλk)ρ(x)dx
+f(,iλˆk)f(,iλk)
p(iλk)p(iλˆk)
m=
(α+mβm–αmβ+m)(λˆkλk)m
+
m=
(αmβ+m–α+mβm)(λˆkλk)m(λˆk+λk)
and lettingk→ ∞, we get
lim
k→∞
∞
A
f(x,iλˆk)f(x,iλk)ρ(x)dx≤. ()
Since
lim
k→∞
f(x,iλk) –f(x,iλˆk)
=
uniformly with respect tox∈[,A]. Comparing () and () we reach a contradiction. We conclude thatδ> and so the functionE(λ) has only a finite number of zeros.
Now, let us show that all zeros of the functionE(λ) are simple. By the derivation of the identity
–f(x,λ) +q(x)f(x,λ) =λρ(x)f(x,λ) () with respect toλ, we get
–f˙(x,λ) +q(x)˙f(x,λ) =λρ(x)˙f(x,λ) + λρ(x)f(x,λ), () heref˙ denotes differentiation with respect toλ. Multiplying () byf˙(x,λ) and () by
f(x,λ) and subtracting the second from the first and integrating this relation with respect toxover (,∞), the result is obtained that
λ ∞
f(x,λ)ρ(x)dx+Wf(x,λ),f˙(x,λ)x== .
Letλbe a zero of the functionE(λ). By using the expression for the functionE(λ), it is found that
˙
E(λ)f(,λ)
p(λ)
= λ ∞
f(x,λ)ρ(x)dx+if (,λ) p(λ)
m=
(α+mβm–αmβ+m)(iλ)m
+
m=
(α+mβm–αmβ+m)(iλ)m+
. ()
Substitutingλk=iμk,μk> , in () and multiplying by –ithe result is obtained that the
right side of the equality is positive. ThusE(i˙μk)= ,i.e.the zeros ofE(λ) are simple. The
The numbers
m–k ≡
∞
f(x,iμk)ρ(x)dx+ f(,iμ
k) p(iμk)
m=
(α+mβm–αmβ+m)μkm–
+
m=
(αmβ+m–α+mβm)μkm
are callednorming numbers. The collection{S(λ),{λk}Nk=,{mk}N
k=}is called thescattering datafor the boundary value
problem ()-().
Using () and substituting the related expressions intoS(λ), the following result is ob-tained:
S(λ) =e–iλaτe
–iλaα– e–iλaα–τ +O
λ
ifα= as|λ| → ∞, and
S(λ) =e–iλaαα(– +τe
–iλaα) –β
( +τe–iλaα)
αα(e–iλaα–τ) –β(e–iλaα+τ)
+O
λ
ifα= as|λ| → ∞.
Let
S∞(λ) =
⎧ ⎨ ⎩
e–iλaτe–iλaα–
e–iλaα–τ , α= ,
e–iλaαα(–+τe–iλaα)–β(+τe–iλaα)
αα(e–iλaα–τ)–β(e–iλaα+τ) , α= .
HenceS∞(λ) –S(λ)∈L(–∞,∞) and so the function
FS(x) =
π
∞
–∞
S∞(λ) –S(λ)eiλxdλ
also belongs to the spaceL(–∞,∞).
3 The main equation
The inverse scattering problem consists in recovering the coefficientq(x) from the scatter-ing data. It is clear that in order to determineq(x) it is sufficient to know the kernelK(x,t) of the solution (). To derive the integral equation forK(x,t), we use the equality (), which was obtained in Lemma . Rewriting the identity () we get the following form:
iλϕ(λ,x)
E(λ) –f(x,λ) +S∞(λ)f(x,λ) =
∞
μ+(x)
S∞(λ) –S(λ)K(x,t)eiλtdt+S∞(λ) –S(λ)f(x,λ)
+
∞
μ+(x)K(x,t)e
–iλtdt–
∞
μ+(x)S∞(
Multiplying both sides of this equality by πeiλyand integrating it toλfrom –∞to∞, it
To compute the third term on the right, we need to find new expression forS∞(λ). After some calculations, in the caseα= we have
Hence, the right-hand side of () equals
K(x,y) +FS(x,y) +
Therefore, the right-hand side of () becomes
and we arrive the same result of the right of ().
On the other side, using the residue theorem and Jordan’s lemma we have
Taking () into account we can transform this expression to the form
–
Substituting this value into the left side of (), we obtain (). Thus we arrive at the following theorem.
Theorem For every fixed x≥,the kernel K(x,t)to the special solution()satisfies the integral equation().
The integral equation () is called themain equationfor the boundary value problem ()-(). The main equation does not have same form as the classical Marchenko equation and we call () themodified Marchenko equation.
4 Solvability of the main equation
We construct () only on the basis of the given scattering data. In this equation, we can take kernelK(x,t) as unknown and regard it as a Fredholm-type equation for every fixedx. The main equation is rewritten in the form
Proof Let
f(y) =Kx,y+μ+(x).
For the proof of the solvability of the given main equation, it is enough to show that the homogeneous equation
f(y) +υKx, a–y–μ+(x)+
∞
f(t)F
t+y+ μ+(x)dt= ()
has no nontrivial solution in the corresponding space.
Iff(y)∈L(,∞) is a solution of (), then bothf(y)∈L∞(,∞) andf(y)∈L(,∞).
Hence f(y)∈L(,∞)∩L∞(,∞)⊂L(,∞) and it is sufficient to investigate () in L(,∞). Equation () has the same properties as the fundamental equation of the
prob-lem in []. The proof of this fact is analog to Lemma . and Corollary . in []. Multiplying () byf(y) and integrating it from –∞to∞with respect toy, we obtain
π
∞
∞
f(λ)dλ+ π
∞
∞ υe
iλμ+(x)–iλa
f(–λ)f(λ)dλ+
n
k=
mke–λkμ+(x)f(–iλ
k)
+ π
∞
∞
S∞(λ) –S(λ)eiλμ+(x)f(–λ)f(λ)dλ=
and hence π
∞
∞
f(λ)dλ= π
∞
∞ S(λ)e
iλμ+(x)
f(–λ)f(λ)dλ–
n
k=
mke–λkμ+(x)f(–iλ
k)
≤
π ∞
∞
S(λ)eiλμ+(x)f(–λ)f(λ)dλ≤
π ∞
∞
S(λ)f(λ)dλ,
i.e.,
∞
∞
–S(λ)f(λ)dλ≤.
Since –|S(λ)|> for allλ= , this implies thatf(λ)≡. Therefore, the homogeneous equation () has only the null solution, and this proves the theorem.
Theorem The scattering data of the boundary value problem()-()determines the po-tential q(x)in()uniquely.
Proof Obviously, to form the main equation () it is sufficient to know the matrix func-tion F(x,y) and in its turn, to find F(x,y) it is sufficient to know the scattering data
{S(λ),{λj}Nj=,{mj}Nj=}. It is seen in Theorem that the main equation (), constructed only on the basis of the scattering data, has a unique solutionK(x,y). Then the functionq(x) in () can be uniquely found according to () and (). Equation () is constructed by the given algorithm. The theorem is proved.
Competing interests
Author’s contributions
The author carried out the research of this paper, and she read and approved the final version of the manuscript.
Acknowledgements
This work is supported by The Scientific and Technological Research Council of Turkey (TÜB˙ITAK).
Received: 9 December 2014 Accepted: 17 April 2015 References
1. Shieh, CT, Yurko, VA: Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J. Math. Anal. Appl.347, 266-272 (2008)
2. Marchenko, VA: Sturm-Liouville Operators and Applications. Birkhäuser, Basel (1986)
3. Marchenko, VA: On reconstruction of the potential energy from phases of the scattered waves. Dokl. Akad. Nauk SSSR
104, 695-698 (1955)
4. Levitan, BM: The inverse scattering problem of quantum theory. Mat. Zametki17(4), 611-624 (1975) 5. Levitan, BM: Inverse Sturm-Liouville Problems. VSP, Zeist (1987). Translated from the Russian by O Efimov 6. Gasymov, MG: Direct and inverse problems of spectral analysis for a class of equations with discontinuous
coefficients. In: Proceedings of the International Conference on Non-Classical Methods in Geophysics, Novosibirsk, pp. 37-44 (1977) (in Russian)
7. Darwish, AA: The inverse scattering problem for a singular boundary value problem. N.Z. J. Math.22, 37-56 (1993) 8. Darwish, AA: On a direct and inverse scattering problem for a boundary value problem with discontinuous
coefficient. N.Z. J. Math.25, 1-14 (1996)
9. Guseinov, IM, Pashaev, RT: On an inverse problem for a second-order differential equation. Russ. Math. Surv.57(3), 597-598 (2002)
10. Fulton, CT, Pruess, S: Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions. J. Math. Anal. Appl.71, 431-462 (1979)
11. Walter, J: Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z.133, 301-312 (1973)
12. Fulton, CT: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb., Sect. A77, 293-308 (1977)
13. Binding, PA, Browne, PJ, Watson, BA: Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. J. Lond. Math. Soc.62, 161-182 (2000)
14. Chugunova, MV: Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions. Oper. Theory, Adv. Appl.123, 187-194 (2001)
15. Chernozhukova, A, Freiling, G: A uniqueness theorem for the boundary value problems with non-linear dependence on the spectral parameter in the boundary conditions. Inverse Probl. Sci. Eng.17(6), 777-785 (2009)
16. Yurko, VA: An inverse problem for pencils of differential operators. Mat. Sb.191(10), 137-160 (2000)
17. Mamedov, KR, Kosar, NP: Inverse scattering problem for Sturm-Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition. Math. Methods Appl. Sci.34, 231-241 (2011)
