R E S E A R C H
Open Access
Eigenparameter dependent inverse
boundary value problem for a class of
Sturm-Liouville operator
Khanlar R Mamedov and F Ayca Cetinkaya
**Correspondence:
[email protected] Science and Letters Faculty, Department of Mathematics, Mersin University, Mersin, 33343, Turkey
Abstract
In this work a Sturm-Liouville operator with piecewise continuous coefficient and spectral parameter in the boundary conditions is considered. The eigenvalue problem is investigated; it is shown that the eigenfunctions form a complete system and an expansion formula with respect to the eigenfunctions is obtained. Uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data are proved.
MSC: 34L10; 34L40; 34A55
Keywords: Sturm-Liouville operator; expansion formula; inverse problem; Weyl function
1 Introduction
We consider the boundary value problem
–y+q(x)y=λρ(x)y, ≤x≤π, ()
U(y) :=y() +α–λα
y() = , ()
V(y) :=λβy(π) +βy(π)
–βy(π) –βy(π) = , ()
whereq(x)∈L(,π) is a real valued function,λis a complex parameter,αi,βj,i= , , j= , are positive real numbers and
ρ(x) =
⎧ ⎨ ⎩
, ≤x<a,
γ, a<x≤π,
where <γ= .
Physical applications of the eigenparameter dependent Sturm-Liouville problems,i.e. the eigenparameter appears not only in the differential equation of the Sturm-Liouville problem but also in the boundary conditions, are given in [–]. Spectral analyses of these problems are examined as regards different aspects (eigenvalue problems, expansion problems with respect to eigenvalues,etc.) in [–]. Similar problems for discontinuous Sturm-Liouville problems are examined in [–].
Inverse problems for differential operators with boundary conditions dependent on the spectral parameter on a finite interval have been studied in [–]. In particular, such problems with discontinuous coefficient are studied in [–].
We investigate a Sturm-Liouville operator with discontinuous coefficient and a spec-tral parameter in boundary conditions. The theoretic formulation of the operator for the problem is given in a suitable Hilbert space in Section . In Section , an asymptotic for-mula for the eigenvalues is given. In Section , an expansion forfor-mula with respect to the eigenfunctions is obtained and Section contains uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data.
2 Operator formulation
Letϕ(x,λ) andψ(x,λ) be the solutions of () satisfying the initial conditions
ϕ(,λ) = , ϕ(,λ) =λα–α, ()
ψ(π,λ) =β–λβ, ψ(π,λ) =λβ–β. ()
For the solution of (), the following integral representation asμ±(x) =±xρ(x) +a(∓
ρ(x)) is obtained similar to [] for allλ:
e(x,λ) =
+
ρ(x) e
iλμ+(x)+
–
ρ(x) e iλμ–(x)+
μ+(x)
–μ+(x)K(x,t)e iλtdt,
where K(x,·)∈L(–μ+(x),μ+(x)). The following properties hold for the kernel K(x,t) which has the partial derivative Kx belonging to the space L(–μ+(x),μ+(x)) for every x∈[,π]:
d dxK
x,μ+(x)= ρ(x)
+
ρ(x) q(x), ()
d dxK
x,μ–(x) + – d dxK
x,μ–(x) – = ρ(x)
–
ρ(x) q(x). ()
We obtain the integral representation of the solutionϕ(x,λ):
ϕ(x,λ) =ϕ(x,λ) +
μ+(x)
A(x,t)cosλt dt+λα–α
μ+(x)
˜
A(x,t)sinλt
λ dt, ()
where
A(x,t) =K(x,t) –K(x, –t), A(x,˜ t) =K(x,t) +K(x, –t)
satisfying (), (). Let us define
which is independent fromx∈[,π]. Substitutingx= andx=πinto () we get
(λ) = –U(ψ) =V(ϕ).
The function(λ) is entire and has zeros at the eigenvalues of the problem ()-(). In the Hilbert spaceHρ=L,ρ(,π)⊕Clet an inner product be defined by
(f,g) :=
π
f(x)g(x)ρ(x)dx+ fg
α +fg
δ ,
where
f =
⎛ ⎜ ⎝
f(x) f f
⎞ ⎟
⎠∈Hρ, g=
⎛ ⎜ ⎝
g(x) g g
⎞ ⎟
⎠∈Hρ, δ:=ββ–ββ> .
We define the operator
L(f) :=
⎛ ⎜ ⎝
–f(x) +q(x)f(x) f() +αf()
βf(π) +βf(π)
⎞ ⎟ ⎠
with
D(L) =f ∈Hρ:f(x),f(x)∈AC[,π],l(f)∈L[,π], f=αf(),f=βf(π) +βf(π)
,
where
l(f) =
ρ(x)
–f+q(x)f
.
The boundary value problem ()-() is equivalent to the equationLY=λY. Whenλ=λ n are the eigenvalues, the eigenfunctions of operatorLare in the form of
(x,λn) =n:=
⎛ ⎜ ⎝
ϕ(x,λn)
αϕ(,λn)
βϕ(π,λn) +βϕ(π,λn)
⎞ ⎟
⎠, n= , .
For any eigenvalueλnthe solutions (), () satisfy the relation
ψ(x,λn) =knϕ(x,λn) ()
and the normalized numbers of the boundary value problem ()-() are given below:
αn:=
π
ϕ(x,λn)ρ(x)dx+αϕ(,λn)
+
δ
βϕ(π,λn) +βϕ(π,λn)
Lemma The eigenvalues of the boundary value problem()-()are simple,i.e.
˙
(λ) = λnknαn. ()
Proof Since
–ϕ(x,λn) +q(x)ϕ(x,λn) =λnρ(x)ϕ(x,λn), –ψ(x,λ) +q(x)ψ(x,λ) =λρ(x)ψ(x,λ),
we get
d dx
ϕ(x,λn)ψ(x,λ) –ϕ(x,λn)ψ(x,λ)=λn–λρ(x)ϕ(x,λn)ψ(x,λ).
With the help of (), () we get
(λn) –(λ) =
λn–λ
π
ϕ(x,λn)ψ(x,λ)ρ(x)dx.
Adding
λn–λαϕ(,λn)ψ(,λ)
+(λ n–λ)
δ
βϕ(π,λn) +βϕ(π,λn)
βψ(π,λ) +βψ(π,λ)
to both sides of the last equation and using the relations (), () we have
(λn) –(λ) = (λn+λ)(λn–λ)knαn.
Takingλ→λn, we find ().
3 Asymptotic formulas of the eigenvalues
The solution of () satisfying the initial conditions () whenq(x)≡ is in the following form:
ϕ(x,λ) =c(x,λ) +
λα–α
s(x,λ)
λ , ()
where
c(x,λ) =
⎧ ⎨ ⎩
cosλx, ≤x<a,
( +
√
ρ(x))cosλμ +(x) +
( –
√
ρ(x))cosλμ
–(x), a<x≤π,
and
s(x,λ) =
⎧ ⎨ ⎩
sinλx
λ , ≤x<a,
( +
√
ρ(x))
sinλμ+(x)
λ +
( –
√
ρ(x))
sinλμ–(x)
The eigenvaluesλ
n(n= ,∓,∓, . . .) of the boundary value problem ()-() whenq(x)≡ can be found by using the equation
(λ) =
λβ–β
ϕ(π,λ) –
β–λβ
ϕ(π,λ) =
and can be represented in the following way:
λn=n+ψ(n), n= ,∓,∓, . . . ,
wheresupn|ψ(n)|< +∞.
Rootsλnof the function(λ) are separated,i.e.,
inf n=k
λn–λk=τ> .
Lemma The eigenvalues of the boundary value problem()-()are in the form of
λn=λn+ dn
λ n
+ηn
n, λn> , ()
where(dn)is a bounded sequence,
dn= λ
n˙(λn)
π
–
ρ(t)
q(t)sin(λ nμ–(π))
ρ(t) dt
– α–α λ
n˙(λn)
π
+
ρ(t)
q(t)cos(λnμ–(π))
ρ(t) dt
and{ηn} ∈l.
Proof From (), it follows that
ϕ(π,λ) =ϕ(π,λ) +
μ+(π)
A(π,t)cosλt dt
+λα–α
μ+(π)
˜
A(π,t)sinλt
λ dt. ()
The expressions of(λ) and(λ) let us calculate(λ) –(λ):
(λ) –(λ) = –λA˜
π,μ+(π)α+a+π– α sinλμ
+(π)
+
α+a+π–
α A
π,μ+(π)cosλμ+(π) +I(λ)λ,
where
I(λ) =αβ
μ+(π)
∂
∂xA(˜ π,t)sinλt dt+O
e|Imλ|μ+(π)
Therefore, for sufficiently largen, on the contours
n=
λ:|λ|=λn+τ
,
we have
(λ) –(λ)<(λ).
By the Rouche theorem, we obtain the result that the number of zeros of the function
(λ) –(λ)
+(λ) =(λ)
inside the contourncoincides with the number of zeros of the function(λ). Moreover, applying the Rouche theorem to the circleγn(δ) ={λ:|λ–λn| ≤δ}we find, for sufficiently largen, that there exists one zeroλnof the function(λ) inγn(δ). Owing to the arbitrari-ness ofδ> we have
λn=λn+n, n=o(), n→ ∞. ()
Substituting () into (), asn→ ∞taking into account the equality(λn) = and the relationssinnμ+(π)≈nμ+(π),cosnμ+(π)≈, integrating by parts and using the properties of the kernelsA(x,t) andA(x,˜ t) we have
n≈ dn
λ n+n
+ηn
λ n ,
where
ηn=
μ+(π)
At(π,t)sinλnt dt+ (α–α)
μ+(π)
At(π,t)cosλnt dt.
Let us show thatηn∈l. It is obvious thatηncan be reduced to the integral
μ+(π)
–μ+(π)R(t)e iλtdt,
whereR(t)∈L(–μ+(π),μ+(π)). Now, take
ζ(λ) :=
μ+(π)
–μ+(π)R(t)e iλtdt.
It is clear from [] (p.) that{ζn}=ζ(λn)∈l. By virtue of this we have{ηn} ∈l. The
lemma is proved.
4 Expansion formula with respect to eigenfunctions
Denote
G(x,t;λ) := –
(λ)
⎧ ⎨ ⎩
ϕ(t,λ)ψ(x,λ), t≤x,
and consider the function
y(x,λ) :=
π
G(x,t;λ)f(t)ρ(t)dt– f
(λ)ψ(x,λ) + f
(λ)ϕ(x,λ). ()
Theorem The eigenfunctions(x,λn)of the boundary value problem()-()form a com-plete system in L,ρ(,π)⊕C.
Proof With the help of () and (), we can write
ψ(x,λn) = ˙(λn) λnαn
ϕ(x,λn). ()
Using () and () we get
Resλ=λny(x,λ) = – λnαn
ϕ(x,λn)
π
ϕ(t,λn)f(t)ρ(t)dt
–
λnαn
ϕ(x,λn)
f– f kn
. ()
Now letf(x)∈L,ρ(,π)⊕Cand assume
(x,λn),f(x)
=
π
ϕ(x,λn)f(x)ρ(x)dx+ϕ(,λn)f
+(βϕ
(π,λn) +β
ϕ(π,λn))f
δ
= . ()
Then from (), we haveResλ=λny(x,λ) = . Consequently, for fixedx∈[,π] the func-tiony(x,λ) is entire with respect toλ. Let us denote
Gδ:=
λ:λ–λn≥δ,n= ,∓,∓, . . .,
whereδis sufficiently small positive number. It is clear that the relation below holds:
(λ)≥C|λ|e|Imλ|μ+(π), λ∈Gδ,C= cons. ()
From () it follows that for fixedδ> and sufficiently largeλ∗> we have
y(x,λ)≤ C
|λ|, λ∈Gδ,|λ| ≥λ
∗,C= cons.
Using maximum principle for module of analytic functions and Liouville theorem, we gety(x,λ)≡. From this we obtainf(x)≡ a.e. on [,π]. Thus we conclude the
com-pleteness of the eigenfunctions(x,λn) inL,ρ(,π)⊕C.
Theorem If f(x)∈D(L),then the expansion formula
f(x) =
∞
n=
is valid,where
an= αn
π
ϕ(t,λn)f(t)ρ(t)dt,
and the series converges uniformly with respect to x∈[,π].For f(x)∈L,ρ(,π),the series
converges in L,ρ(,π),moreover,the Parseval equality holds:
π
f(x)ρ(x)dx=
∞
n=
αn|an|.
Proof Sinceϕ(x,λ) andψ(x,λ) are the solutions of the boundary value problem ()-(), we have
y(x,λ) = –ψ(x,λ)
(λ)
π
[–ϕ(t,λ) +q(t)ϕ(t,λ)]f(t)
λ dt
–ϕ(x,λ)
(λ)
x
π
[–ψ(t,λ) +q(t)ψ(t,λ)]f(t)
λ dt
– f
(λ)ψ(x,λ) + f
(λ)ϕ(x,λ). ()
Integrating by parts and taking into account the boundary conditions (), () we obtain
y(x,λ) = –
λf(x) –
λ
Z(x,λ) +Z(x,λ)
– f
(λ)ψ(x,λ) + f
(λ)ϕ(x,λ), ()
where
Z(x,λ) =
(λ)ψ(x,λ)
x
ϕ(t,λ)f(t)dt+
(λ)ϕ(x,λ)
π
x
ψ(t,λ)f(t)dt,
Z(x,λ) =
(λ)
λα–α
ψ(x,λ)f()–
(λ)
λβ–β
ϕ(x,λ)f(π)
+
(λ)ψ(x,λ)
x
ϕ(t,λ)q(t)f(t)dt+
(λ)ϕ(x,λ)
π
x
ψ(t,λ)q(t)f(t)dt.
If we consider the following contour integral wherenis a counter-clockwise oriented contour:
In(x) = πi
n
λy(x,λ)dλ,
and then taking into consideration () we get
In(x) =
∞
n=
Resλ=λn
λy(x,λ)
=
∞
n=
anϕ(x,λn) +
∞
n=
λnf
˙
(λn)ψ(x,λn) –
∞
n=
λnf
˙
where
an=
αn
π
ϕ(t,λn)f(t)ρ(t)dt.
On the other hand, with the help of () we get
In(x) =f(x) – πi
n
Z(x,λ) +Z(x,λ)
dλ+
∞
n=
λnf
˙
(λn)ψ(x,λn)
–
∞
n=
λnf
˙
(λn)ϕ(x,λn). ()
Comparing () and () we obtain
∞
n=
anϕ(x,λn) =f(x) +n(x),
where
n(x) = – πi
n
Z(x,λ) +Z(x,λ)
dλ.
The relations below hold for sufficiently largeλ∗>
max x∈[,π]
Z(x,λ)≤ C
|λ|, λ∈Gδ,|λ| ≤λ
∗, ()
max x∈[,π]
Z(x,λ)≤ C
|λ|, λ∈Gδ,|λ| ≤λ
∗. ()
The validity of
lim n→∞xmax∈[,π]
n(x)=
can easily be seen from () and (). The last equation gives us the expansion formula
f(x) =
∞
n=
anϕ(x,λn).
Since the system of(x,λn) is complete and orthogonal inL,ρ(,π)⊕C, the Parseval
equality
π
f(x)ρ(x)dx=
∞
n=
αn|an|
5 Uniqueness theorems
We consider the statement of the inverse problem of the reconstruction of the boundary value problem ()-() from the Weyl function.
Let the functionsc(x,λ) ands(x,λ) denote the solutions of () satisfying the conditions c(,λ) = ,c(,λ) = ,s(,λ) = ands(,λ) = , respectively, andϕ(x,λ) andψ(x,λ) be the solutions of () under the initial conditions (), ().
Further, let the function(x,λ) be the solution of () satisfyingU() = andV() = . We set
M(λ) :=ψ(,λ)
(λ) .
The functions(x,λ) andM(λ) are called the Weyl solution and the Weyl function for the boundary value problem ()-(), respectively. The Weyl function is a meromorphic function having simple poles at pointsλn, eigenvalues of the boundary value problem of ()-(). The Wronskian
W(x) :=ϕ(x,λ),(x,λ)
does not depend onx. Takingx= , we get
W() =ϕ(,λ)(,λ) –ϕ(,λ)(,λ) = . Hence,
W(x) =ϕ(x,λ),(x,λ)= . ()
In view of () and (), we get forλ=λn
(x,λ) =ψ(x,λ)
(λ) . ()
Using () we obtain
M(λ) = – (λ)
(λ),
where(λ) = –ψ(,λ) is the characteristic function of the boundary value problemL:
ly=λy, ≤x≤π,
y() = , V(y) = .
It is clear that
(x,λ) =s(x,λ) +M(λ)ϕ(x,λ). ()
Proof Let us denote the matrixP(x,λ) = [Pjk(x,λ)]j,k=,as
P(x,λ)
˜
ϕ(x,λ) ˜(x,λ)
˜
ϕ(x,λ) ˜(x,λ)
=
ϕ(x,λ) (x,λ)
ϕ(x,λ) (x,λ)
. ()
Then we have
ϕ(x,λ) =P(x,λ)ϕ˜(x,λ) +P(x,λ)ϕ˜(x,λ),
(x,λ) =P(x,λ)˜(x,λ) +P(x,λ)˜(x,λ)
()
or
P(x,λ) =ϕ(x,λ)˜(x,λ) –ϕ˜(x,λ)(x,λ), P(x,λ) =ϕ˜(x,λ)(x,λ) –ϕ(x,λ)˜(x,λ).
()
Taking () into consideration in () we get
P(x,λ) = +
(λ)ψ(x,λ)
ϕ(x,λ) –ϕ˜(x,λ)+
(λ)ϕ(x,λ)
˜
ψ(x,λ) –ψ(x,λ),
P(x,λ) =
(λ)
˜
ϕ(x,λ)ψ(x,λ) –ϕ(x,λ)ψ˜(x,λ).
()
From the estimates as|λ| → ∞
ϕ(x,λ) –(λϕ)˜(x,λ)=O
|λ|e
|Imλ|μ+(x) ,
ψ˜(x,λ) –(λψ)(x,λ)=O
|λ|e
|Imλ|(μ+(π)–μ+(x)) ,
we have from ()
lim
|λ|→∞xmax∈[,π]
P(x,λ) – = lim
|λ|→∞xmax∈[,π]
P(x,λ)= ()
forλ∈Gδ.
Now, if we take into consideration () and (), we have
P(x,λ) =ϕ(x,λ)˜s(x,λ) –ϕ˜(x,λ)s(x,λ) +ϕ˜(x,λ)ϕ(x,λ)
˜
M(λ) –M(λ),
P(x,λ) =ϕ˜(x,λ)s(x,λ) –ϕ(x,λ)s(x,˜ λ) +ϕ(x,λ)ϕ˜(x,λ)
M(λ) –M(˜ λ).
Therefore ifM(λ) =M(˜ λ), one has
P(x,λ) =ϕ(x,λ)˜s(x,λ) –s(x,λ)ϕ˜(x,λ), P(x,λ) =ϕ(x,λ)s(x,˜ λ) –s(x,λ)ϕ˜(x,λ).
Thus, for every fixedxfunctionsP(x,λ) andP(x,λ) are entire functions forλ. It can easily be seen from () thatP(x,λ) = andP(x,λ) = . Consequently, we getϕ(x,λ)≡
˜
The validity of the equation below can be seen analogously to []:
M(λ) =M() +
∞
n=
λ
αnλn(λ–λn)
. ()
Theorem The spectral data identically define the boundary value problem()-().
Proof From (), it is clear that the functionM(λ) can be constructed byλn. Sinceλ˜n=λn for everyn∈N, we can say thatM(λ) =M(˜ λ). Then from Theorem , it is obvious that
L=L.˜
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
This work is supported by The Scientific and Technological Research Council of Turkey (TÜB˙ITAK).
Received: 10 March 2014 Accepted: 29 July 2014
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Cite this article as:Mamedov and Cetinkaya:Eigenparameter dependent inverse boundary value problem for a class
