R E S E A R C H
Open Access
A new kind of uniqueness theorems for
inverse Sturm-Liouville problems
Yuri Ashrafyan
**Correspondence:
[email protected] Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, Yerevan, 0025, Armenia
Abstract
We prove Marchenko-type uniqueness theorems for inverse Sturm-Liouville problems. Moreover, we prove a generalization of Ambarzumyan’s theorem.
Keywords: inverse problem; Sturm-Liouville operator; uniqueness theorem; Ambarzumyan theorem
1 Introduction
Let us denote byL(q,α,β) the Sturm-Liouville boundary value problem
–y+q(x)y=μy, x∈(,π),μ∈C, (.) y()cotα+y() = , α∈(,π), (.) y(π)cotβ+y(π) = , β∈(,π), (.)
whereqis a real-valued, summable function, q∈L
R(,π). At the same time,L(q,α,β)
denotes the self-adjoint operator generated by problem (.)-(.) (see, e.g., [–]). It is known that under the above conditions the spectrum of operatorL(q,α,β) is discrete and consists of real, simple eigenvalues (see, e.g., [, ]), which we denote byμn=μn(q,α,β), n≥, emphasizing the dependence ofμnonq,αandβ. We assume that eigenvalues are enumerated in the increasing order, i.e.,
μ(q,α,β) <μ(q,α,β) <· · ·<μn(q,α,β) <· · ·.
Letϕ(x,μ) be a solution of equation (.), which satisfies the initial conditions
ϕ(,μ) = , ϕ(,μ) = –cotα. (.)
The eigenvaluesμn=μn(q,α,β),n≥, ofL(q,α,β) are the solutions of equation
ϕ(π,μ)cotβ+ϕ(π,μ) = .
It is easy to see that the functionsϕ(x,μn),n≥, are the eigenfunctions corresponding to the eigenvalueμn. The squares of theL-norm of these eigenfunctions
an=an(q,α,β) :=
π
ϕ(x,μn)dx, n≥,
are called norming constants. The eigenvalues and norming constants are called spectral data (besides these, there are other quantities, which are also called spectral data). The inverse Sturm-Liouville problem is to reconstruct the quantitiesq,α,β by some spectral data.
LetL=L(q,α,β) andL=L(q,α,β) be two operators. The following assertion is
usu-ally called the uniqueness theorem of Marchenko.a
Theorem .(Marchenko []) Let q∈LR(,π).If
μn(q,α,β) =μn(q,α,β), (.)
an(q,α,β) =an(q,α,β), (.)
for all n≥,thenα=α,β=βand q(x) =q(x)almost everywhere.
One of the results of the present paper is the following theorem which, in some sense, is a generalization of Marchenko’s uniqueness theorem.
Theorem . Let q∈L
R(,π).If
μn(q,α,β) =μn(q,α,β), (.)
an(q,α,β)≥an(q,α,β), (.)
for all n≥,thenβ=βand q(x)≡q(x).
This kind of uniqueness theorem has not been considered before. The main difference between Theorems . and . is that we replace the equality in (.) with the inequality in (.). Note that we assumeq∈L
R(,π) instead of generalq∈LR(,π) since our proof
is based on the results of Jodeit and Levitan (see []). And the parameterαof boundary condition is in advance fixedα=α.
Remark Some analogues of Theorem . will be stated in the Appendix.
Historically, the first work in the theory of inverse spectral problems for Sturm-Liouville operators belongs to Ambarzumyan []. He proved that if the eigenvalues of Sturm-Liouville operator with Neumann boundary conditions aren, then the potentialqis
on [,π]. It is known that the eigenvaluesμn(,π/,π/) of operatorL(,π/,π/) are n,n≥. The classical Ambarzumyan theorem in our notations will be as follows.
This was an exception as in general additional information was needed in order to recon-struct the potentialquniquely. There are many generalizations of Ambarzumyan’s theo-rem in various directions, we mention several of them (see, e.g., [–] and the references therein).
Our generalization of Ambarzumyan’s theorem is as follows.
Theorem . Let q∈L R(,π).
Ifμn(q,α,π–α) =μn(,α,π–α)for all n≥,then q(x)≡.
We think that Theorem . is a natural generalization, because we use only one spectrum to reconstruct the potentialqwithout any additional conditions, as it is in the classical result.
2 Preliminaries
Two operatorsL=L(q,α,β) andL=L(q,α,β) are called isospectral if they have the
same spectra, i.e.,μn(q,α,β) =μn(q,α,β),n≥. In what follows, if a certain symbolγ
denotes an object related toL, thenγ(orγdepending on situation) will denote a similar
object related toL.
The problem of describing all the operatorsLisospectral withLfirst was considered
by Trubowitz et al. (see [–]) forq∈L
R(,π). The same problem was considered by
Jodeit and Levitan in [] forqsuch thatq∈L
R(,π). For this aim the Gelfand-Levitan
integral equation and transformation operators were used in []. They constructed the kernelF(x,y) of the integral equation as follows. Letcn,n≥, be arbitrary real numbers converging to zero, asn→ ∞, so rapidly that the function
F(x,y) =
∞
n=
cnϕ
x,μnϕ
y,μn (.)
is continuous and all the second order partial derivatives are also continuous. The integral equation
K(x,y) +F(x,y) + x
K(x,t)F(t,y)dt= , ≤y≤x≤π, (.)
is called Gelfand-Levitan integral equation.b
They proved that if +cnan> for alln≥, then the integral equation (.) has a unique solutionK(x,y) and the function
ϕ(x,μ) =ϕ(x,μ) + x
K(x,t)ϕ(t,μ)dt
is a solution of the differential equation (.), with potential function
q(x) =q(x) +
d
dxK(x,x), (.)
andϕ(x,μ) satisfies the initial conditions
where
cotα=cotα+
∞
n=
cn. (.)
It means that the functionϕ(x,μ) satisfies the boundary condition (.) for allμ∈C. Findβ∈(,π) such thatμn(q,α,β) =μn(q,α,β) for alln≥, i.e.,ϕ(x,μ) should
sat-isfy, at the pointx=π, the boundary condition (.)
ϕπ,μncotβ+ϕπ,μn=
for thisβ∈(,π). Suchβ(in []) is being defined from the following relation
cotβ=cotβ+
∞
n=
cnϕ(π,μn) +cnan
. (.)
Thus Jodeit and Levitan showed that each admissible sequence {cn}∞n= generates an
isospectral operatorL(q,α,β), whereq,αandβare given by formulae (.), (.) and (.), respectively. In this way they obtained all the potentialsq, withq∈L(,π), having a given
spectrumμ
n=μn(q,α,β),n≥.
3 Proof of Theorem 1.2
Consider operatorsL=L(q,α,β) andL=L(q,α,β) with the set of norming constants
an=an(q,α,β) andan=an(q,α,β),n≥, respectively. It is known (see, e.g., []) that
in this case the kernelF(x,y) of the integral equation (.) is
F(x,y) =
∞ n= an – a n ϕ x,μnϕ
y,μn. (.)
Since by the condition of Theorem . the operatorsLandLare isospectral, then
formu-lae (.)-(.) hold. If we compare kernels (.) and (.), we will refer thatcn=an–a
n. So formulae (.) and (.) will become
cotα=cotα+
∞ n= an – a n , (.)
cotβ=cotβ+
∞
n=
an–an
ϕ(π,μn) (a
n)
. (.)
Thus, we have all the operatorsL(q,α,β) isospectral withL(q,α,β).
We supposed thatα=α, then by formula (.) we have
∞ n= an – a n = . (.)
Sincean≥anfor all n≥, thus from equation (.) it refers thatan=anfor all n≥. Thus, from Marchenko’s uniqueness theorem . we obtainq(x)≡q(x) andβ=β.
Remark From equation (.) it follows that the conditionan≥ancan be changed with an≤an. From relation (.) it follows that we can assumeβ=βinstead ofα=αwith
the conditionan≥anoran≤an, and then we will also obtainq(x)≡q(x) andα=α.
4 Proof of Theorem 1.4
Consider an operatorL(q,α,π–α) and an even operatorcL(,α,π–α).
Levinson proved [] (see also []) that an operatorLis even if and only if
ϕ(π,μn) = (–)n, n≥. (.)
The condition of the theorem means that the operatorL(q,α,π–α) is isospectral with L(,α,π–α). Since the method of Jodeit and Levitan has described all the isospectral operators for a potential functionqwithq∈L(,π), then there exists a sequence{c
n}∞n=
such that +cnan> for alln≥,{cn}∞n=has the properties described in Section , and
formulae (.)-(.) hold for operatorsL(q,α,π–α) andL(,α,π–α).
Therefore, taking into account thatq(x)≡,α=α,β=β=π–αand (.), relations
(.)-(.), which connect these two operators, will become
q(x) = d
dxK(x,x), (.)
∞
n=
cn= , (.)
∞
n=
cn +cnan
= . (.)
If we subtract (.) from (.), we will obtain
∞
n=
cnan +cnan
= . (.)
Since +cnan> andan> for alln≥, then from equation (.) we obtain thatcn= ,n≥. Thus, from equations (.), (.) and (.) it follows thatq(x)≡.
Remark We will get the classical Ambarzumyan theorem if we takeα=π/.
Appendix: Analogues of Theorem 1.2
Consider theL(q,α,β) problem. Letψ(x,μ) be a solution of equation (.), which satisfies the initial conditions
ψ(π,μ) = , ψ(π,μ) = –cotβ. (A.)
The eigenvaluesμn=μn(q,α,β),n≥, are the solutions of the equation
or of the equation
(μ) :=ψ(,μ)cotα+ψ(,μ) = .
(μ) and (μ) are called characteristic functions for the operatorL(q,α,β). In [] it is proved that characteristic functions and their derivatives are uniquely determined only from their zeros, i.e., from eigenvalues{μn}∞n=. It is easy to see that the functionsψ(x,μn), n≥, are the eigenfunctions corresponding to the eigenvalueμn. The squares of theL -norm of these eigenfunctions
bn=bn(q,α,β) :=
π
ψ(x,μn)dx, n≥,
are called norming constants.
Since all the eigenvalues of L(q,α,β) are simple, then there exist constants κn =
κn(q,α,β),n≥, such that
ϕ(x,μn) =κnψ(x,μn). (A.)
The theorem of uniqueness of Harutyunyan (see []) states the following.
Theorem A. If
μn(q,α,β) =μn(q,α,β),
κn(q,α,β) =κn(q,α,β),
for all n≥,thenα=α,β=βand q(x) =q(x)almost everywhere.
From (.), (A.) and (A.) it follows
κn=ϕ(π,μn) =ψ–(,μn). (A.)
There is a relationship between norming constants and characteristic functions (see, e.g., [, ])
an=ϕ(π,μn)˙(μn), (A.)
bn=ψ(,μn)˙(μn), (A.)
where the dot over(or over ) denotes the derivative of(μ) with respect toμ. From equations (A.) and (A.) we obtain
an=|κn|˙(μn). (A.)
Consider two isospectral operatorsL(q,α,β) andL(q,α,β). The following formulae,
analogues to (.) and (.), can be obtained forκn:
cotα=cotα+
∞
n=
| ˙(μ
n)|
|κn|
– |κ
n|
cotβ=cotβ+
∞
n=
|κn|–|κn| | ˙(μ
n)|
. (A.)
From Theorem A. and formulae (.), (A.), (A.), a new statement, similar to Theo-rem ., can be proven forκnas follows.
Theorem A. Let q∈L
R(,π).If
μn(q,α,β) =μn(q,α,β), κn(q,α,β)≥κn(q,α,β),
for all n≥,thenβ=βand q(x)≡q(x).
Remark Instead ofα=α, we can fixβ=βand/or replace the inequality sign (‘≥’) with less than or equal sign (‘≤’). Even so, the result is valid. Similar theorems can be proven forϕ(π,μn).
Remark Since the uniqueness theorem of Marchenko is also true for norming constants bn, taking into consideration relations (A.), (A.) and (A.), analogues to Theorem . can be proven forψ(,μn) andbn.
Acknowledgements
The author would like to thank the referees for their helpful comments and suggestions. The author is also grateful to professor TN Harutyunyan for valuable remarks and discussions.
Funding
This work was supported by the RA MES State Committee of Science, in the frames of the research project No.15T-1A392.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Endnotes
a The theorem of Marchenko is more general, see e.g. [5, 23–25].
b HereF(x,y) is a kernel of integral equation (2.2), wherexis a parameter,F(x,y) is a known function andK(x,y) is an
unknown function, as functions ofy.
c A problemL(q,α,β) is said to be even ifq(x) =q(π–x) andα+β=π.
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Received: 25 February 2017 Accepted: 18 May 2017
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