R E S E A R C H
Open Access
Direct and inverse spectral problems for
discrete Sturm-Liouville problem with
generalized function potential
Bayram Bala
1, Abdullah Kablan
1and Manaf Dzh Manafov
2*In memory of GSh Guseinov (1951-2015)
*Correspondence: [email protected] 2Faculty of Arts and Sciences, Department of Mathematics, Adıyaman University, Adıyaman, 02040, Turkey
Full list of author information is available at the end of the article
Abstract
In this work, we study the inverse problem for difference equations which are constructed by the Sturm-Liouville equations with generalized function potential from the generalized spectral function (GSF). Some formulas are given in order to obtain the matrixJ, which need not be symmetric, by using the GSF and the structure of the GSF is studied.
MSC: Primary 39A12; 34A55; 34L15
Keywords: difference equation; inverse problems; generalized spectral function
1 Introduction
In this paper we deal with theN×Ntridiagonal matrix
J=
The definitions and some properties of GSF are given in [–]. The inverse problem for the infinite Jacobi matrices from the GSF was investigated in [–], see also []. The inverse spectral problem forN×Ntridiagonal symmetric matrix has been studied in [] and the inverse spectral problem with spectral parameter in the initial conditions has been studied in []. The goal of this paper is to study the almost symmetric matrixJof the form (.). Almost symmetric here means that the entries above and below the main diagonal are the same except the entriesaMandcM.
The eigenvalue problem we consider in this paper isJy=λy, wherey={yn}Nn=–is a col-umn vector. There exists a relation between this matrix eigenvalue problem and the second order linear difference equation
an–yn–+bnyn+anyn+=λρnyn, n∈ {, , . . . ,M, . . . ,N– },
a–=cN–= ,
(.)
for{yn}Nn=–, with the boundary conditions
y–=yN = , (.)
whereρnis a constant defined by
ρn=
, ≤n≤M,
α, M<n≤N– , =α> . (.)
These expressions are equivalent. The problem (.), (.) is a discrete form of the Sturm-Liouville operator with discontinuous coefficients
d dx p(x)
d dxy(x)
+q(x)y(x) =λρ(x)y(x), x∈[a,b], (.)
y(a) =y(b) = , (.)
whereρ(x) is a piecewise function defined by
ρ(x) =
, a≤x≤c,
α, c<x≤b, α = ,
[a,b] is a finite interval,α is a real number, andcis a discontinuity point in [a,b]. On eigenvalues and eigenfunctions of such an equation, see [], and the inverse problem for this kind equation has been investigated in [].
2 Generalized spectral function
In this section, we find the characteristic polynomial for the matrixJand then give the existence of linear functional which is defined from the ring of all polynomials in λof degree≤Nwith the complex coefficients toC. Let us denote by{Pn(λ)}Nn=–, the solution of equation (.) together with the initial data
By starting with (.), we can derive from equation (.) iteratively the polynomialsPn(λ) of
subject to the initial condition
P(λ) = . (.)
Lemma The following equality holds:
det(J–λI) = (–)Naa· · ·aMcM+· · ·cN–PN(λ). (.)
Therefore,the roots of the polynomial PN(λ)and the eigenvalues of the matrix J are
coin-cident.
Proof We will consider the proof in three cases. For eachn= ,M, let us define the deter-minantn(λ) as follows: by the elements of the last row, we obtain
By using the same method, we get
n+(λ) = (dn–λ)n(λ) –cn–n–(λ), (.)
and finally, forn=M+ , we find
M+(λ) = (dM+–λ)M+(λ) –aMcMM(λ). (.)
Dividing (.) and (.) by the producta· · ·an–, (.) by the producta· · ·an–cn–, we can easily show that the sequence
h–= , h= , hn= (–)n(a· · ·an–)–n(λ), n= ,M+ ,
hn= (–)n(a· · ·cM+· · ·cn–)–n(λ), n=M+ ,N,
satisfies (.), (.). Thenhnis solution of (.), (.). We can show it byPn(λ) forn= ,N. SinceN(λ) is also equal todet(J–λI) if we combine (.), (.) and (.), we obtain (.), for alln∈ {, , . . . ,M,M+ , . . . ,N}.
Theorem There exists a unique linear functional:CN[λ]→Csuch that the following
relations hold:
Pm(λ)Pn(λ)
=δmn
η , m,n∈ {, , . . . ,M, . . . ,N– }, (.)
Pm(λ)PN(λ)
= , m∈ {, , . . . ,M, . . . ,N}, (.)
whereδmnis the Kronecker delta,ηis defined by
η=
, m,n≤M,
α, m,n>M, (.)
and(P(λ))shows the value ofon any polynomial P(λ).
Proof In order to show the uniqueness of we assume that there exists such a linear functional, satisfying (.) and (.). Let us define the N+ polynomials as follows:
Pn(λ) (n= ,N– ), Pm(λ)PN(λ) (m= ,N). (.)
It is clear that this polynomial set is a basis for the linear spaceCN[λ]. Indeed the polyno-mials defined by (.) are linearly independent and their number is equal to dimension ofCN[λ]. On the other hand, by using (.) and (.), the quantities of the polynomials given in (.) under the functionalcan be found as finite values:
Pn(λ)
=δn
η , n∈ {, , . . . ,M, . . . ,N– }, (.)
Pm(λ)PN(λ)
= , m∈ {, , . . . ,N}. (.)
To show the existence of, let us define it on the polynomials (.) by (.), (.) and then we expandto over the whole spaceCN[λ] by using the linearity of. It can be shown that the functionalsatisfies (.), (.). Denote
Pm(λ)Pn(λ)
=Bmn, m,n∈ {, , . . . ,M, . . . ,N}. (.)
It is clear thatBmn=Bnm, form,n∈ {, , . . . ,N}. From (.) and (.), we get
Bm=Bm=δm, m∈ {, , . . . ,M}, (.)
Bm=Bm=
δm
α , m∈ {M+ , . . . ,N}, (.)
BmN=BNm= , m∈ {, , . . . ,N}. (.)
Since{Pn(λ)}N is the solution of (.), we derive from the first equation of (.), using (.),
λ=b+aP(λ).
Inserting this into the remaining equations in (.), we get
an–Pn–(λ) +bnPn(λ) +anPn+(λ) =bPn(λ) +aP(λ)Pn(λ), n∈ {, , . . . ,M},
cn–Pn–(λ) +dnPn(λ) +cnPn+(λ) =bPn(λ) +aP(λ)Pn(λ), n∈ {M+ , . . . ,N– }.
If we apply the linear functionalto both sides of the last two equations, by taking into account (.), (.), and (.), we get
Bn=Bn=δn, n∈ {, , . . . ,M}, (.)
Bn=Bn=
δn
α , n∈ {M+ , . . . ,N}. (.)
Further, recalling the definition ofρnin (.), we write
am–Pm–(λ) +bmPm(λ) +amPm+(λ) =λρmPm(λ), m∈ {, , . . . ,M, . . . ,N– },
an–Pn–(λ) +bnPn(λ) +anPn+(λ) =λρnPn(λ), n∈ {, , . . . ,M, . . . ,N– }.
If the first equality is multiplied byPn(λ) and the second equality is multiplied byPm(λ), then the second result is subtracted from the first, we obtain:
form,n∈ {, , . . . ,M},
am–Pm–(λ)Pn(λ) +bmPm(λ)Pn(λ) +amPm+(λ)Pn(λ)
=an–Pn–(λ)Pm(λ) +bnPn(λ)Pm(λ) +anPn+(λ)Pm(λ),
form∈ {, , . . . ,M},n∈ {M+ , . . . ,N– },
am–Pm–(λ)Pn(λ) +bmPm(λ)Pn(λ) +amPm+(λ)Pn(λ)
form∈ {M+ , . . . ,N– },n∈ {, , . . . ,M},
cm–Pm–(λ)Pn(λ) +dmPm(λ)Pn(λ) +cmPm+(λ)Pn(λ)
=an–Pn–(λ)Pm(λ) +bnPn(λ)Pm(λ) +anPn+(λ)Pm(λ),
form,n∈ {M+ , . . . ,N– },
cm–Pm–(λ)Pn(λ) +dmPm(λ)Pn(λ) +cmPm+(λ)Pn(λ)
=cn–Pn–(λ)Pm(λ) +dnPn(λ)Pm(λ) +cnPn+(λ)Pm(λ).
If the functionalis applied to both sides of these equations, and using (.)-(.), we obtain forBmnthe following boundary value problems:
form,n∈ {, , . . . ,M},
am–Bm–,n+bmBmn+amBm+,n=an–Bn–,m+bnBnm+anBn+,m, (.)
form∈ {, , . . . ,M},n∈ {M+ , . . . ,N– },
am–Bm–,n+bmBmn+amBm+,n=cn–Bn–,m+dnBnm+cnBn+,m, (.)
form∈ {M+ , . . . ,N– },n∈ {, , . . . ,M},
cm–Bm–,n+dmBmn+cmBm+,n=an–Bn–,m+bnBnm+anBn+,m, (.)
form,n∈ {M+ , . . . ,N– },
cm–Bm–,n+dmBmn+cmBm+,n=cn–Bn–,m+dnBnm+cnBn+,m, (.)
forn∈ {, , . . . ,M},
Bn=Bn=δn, Bn=Bn=δn, BNn=BnN= , (.)
forn∈ {M+ , . . . ,N},
Bn=Bn=
δn
α , Bn=Bn= δn
α , BNn=BnN= . (.)
After starting with boundary values (.), (.) and using equations (.)-(.), we can find allBmnuniquely as follows:
Bmn=δmn, m,n∈ {, , . . . ,M},
Bmn=
δmn
α , m,n∈ {M+ , . . . ,N– },
BmN= , m∈ {, , . . . ,M,M+ , . . . ,N}.
3 Inverse problem from the generalized spectral function
In this section, we solve the inverse spectral problem of reconstructing the matrixJby its GSF and we give the structure of GSF. The inverse spectral problem may be stated as follows: determine the reconstruction procedure to construct the matrixJfrom a given GSF and find the necessary and sufficient conditions for a linear functionalonCN[λ], to be the GSF for some matrixJof the form (.). For the investigation of necessary and sufficient conditions for a given linear functional to be the GSF, we will refer to Theorems and in []. In this paper, we only find the formulas to construct the matrixJ.
whereγnandχnk are constants. Inserting (.) in (.) and using the equality of the poly-nomials, we can find the following equalities between the coefficientsan,bn,cn,dnand the quantitiesγn,χnk:
It is easily shown that there exists an equivalence between (.), (.), and
λmPn(λ)
respectively. Indeed, from (.), we can write
Pm(λ)Pn(λ)
Writing the expansion (.) in (.) and (.) instead ofPn(λ) andPN(λ), respectively, and using the notation in (.), we get
tn+m+
As a result of all discussions above, we write the procedure to construct the matrix in (.). In turn, in order to find the entriesan,bn,cn,dnof the required matrixJ, it suffices to know only the quantitiesγn,χnk. Given the linear functionalwhich satisfies the con-ditions of Theorem in [] onCN[λ], we can use (.) to find the quantitiestland write down the inhomogeneous system of linear algebraic equations (.) with the unknowns
χn,χn, . . . ,χn,n–, for every fixedn∈ {, , . . . ,N}. After solving this system uniquely and using (.), we find the quantitiesγnand so we obtainan,bn,cn,dn, recalling (.), (.). Therefore, we can construct the matrixJ.
Using the numberstldefined in (.), let us present the determinants
Dn=
From the definition of determinants in (.), it can be shown that the determinant of system (.) isDn–. Then, solving system (.) by using Cramer’s rule, we obtain
cn=±
√
Dn–Dn+
Dn
(M<n≤N– ), (.)
bn=n
Dn –n–
Dn–
(≤n≤M),–= , (.)
dn=n
Dn –n–
Dn–
(M<n≤N– ),=t. (.)
Hence , ifwhich satisfies the conditions of Theorem in [] is given, then the valuesan,
bn,cn,dnof the matrixJ are obtained by equations (.)-(.), whereDnis defined by (.) and (.).
In the following theorem, we will show that the GSF ofJhas a special form and we will give a structure of the GSF. LetJbe a matrix which has the form (.) andbe the GSF ofJ. Here we characterize the structure of.
Theorem Letλ, . . . ,λpbe all the eigenvalues with the multiplicities m, . . . ,mp,
respec-tively,of the matrix J.These are also the roots of the polynomial(.).Then there exist numbersβkj(j= ,mk,k= ,p)uniquely determined by the matrix J such that for any
poly-nomial P(λ)∈CN[λ]the following formula holds:
P(λ)= p
k= mk
j=
βkj (j– )!P
(j–)(λ
k), (.)
where P(j–)(λ)denotes the(j– )th derivative of P(λ)with respect toλ.
Proof LetJbe a matrix which has the form (.). Take into consideration the difference equation (.)
an–yn–+bnyn+anyn+=λρnyn, n∈ {, , . . . ,N– },a–=cN–= , (.)
where{yn}Nn=–is desired solution and
ρn=
, ≤n≤M,
α, M<n≤N– .
Denote by{Pn(λ)}Nn=–and{Qn(λ)}Nn=–the solutions of (.) satisfying the initial condi-tions
P–(λ) = , P(λ) = , (.)
Q–(λ) = –, Q(λ) = . (.)
For eachn≥, the degree of polynomialPn(λ) isnand the degree of polynomialQn(λ) is
n– . It is clear that the entriesRnm(λ) of the resolvent matrixR(λ) = (J–λI)–are of the form
Rnm(λ) =
ρnPn(λ)[Qm(λ) +M(λ)Pm(λ)], ≤n≤m≤N– ,
ρnPm(λ)[Qn(λ) +M(λ)Pn(λ)], ≤m≤n≤N– ,
where
whereris the circle in theλ-plane of radiusrcentered at the origin.
Let all the distinct zeros of PN(λ) in (.) beλ, . . . ,λp with multiplicitiesm, . . . ,mp,
whereβkjare some uniquely determined complex numbers which depend on the matrixJ. Inserting (.) in (.) and using (.), (.) we get, by the residue theorem and pass-ing then to the limitr→ ∞,
Thus, (.) can be written as follows:
fn
ρn
=F(λ)Pn(λ)
Now by using (.) in (.) and the arbitrariness of{fm}Nm–=, we see that the first relation in Theorem ,
Pm(λ)Pn(λ)
=δmn, m,n∈ {, , . . . ,M}, (.)
Pm(λ)Pn(λ)
=δmn
α , m,n∈ {M+ , . . . ,N– }, (.)
holds. Moreover, from (.) and (.), we have also the second relation in Theorem ,
Pm(λ)PN(λ)
= , m∈ {, , . . . ,N}. (.)
These mean that the GSF of the matrixJhas the form (.).
Now, we shall work out two examples to illustrate our formulas. In the first example, in order to determineχn,kandγn, we will use (.)-(.).
Example Take into consideration the caseN= ,M= , and the functionaldescribed by the formula
P(λ)=
P() +P() +P().
It is clear that the functional defined above has the structure given in Theorem and satisfies the conditions of Theorem in []. So it can be chosen as a GSF. From (.) we calculate alltlas follows:
t= , t= , t= ,
t= , t=
, t= , t=
.
(.)
Then solving the system of equation (.) by using the values in (.), we get
χ,= –, χ,=
, χ,= –,
χ,= , χ,= , χ,= –.
(.)
Now inserting the quantities in (.) and (.) into equation (.), we obtain
γ= , γ=±
, γ=± √
α. (.)
Now it follows from (.) and (.) that
a=±
, a=±
α
, c=±
α,
where (.) and (.) are used. Consequently, we find the four matricesJ± foras
The characteristic polynomials which are determined by the matricesJ±are obtained:
det(J±–λI) =λ(λ– )(λ– ).
In the following example, by using Theorem in [], it can be shown that the necessary and sufficient conditions for a given linear functionalto be the GSF hold and the matrixJ
can be constructed from (.)-(.).
Example Let us consider the functionaldefined by the formula forN= andM=
P(μ)=P(μ) + P(μ) + P(μ),
whereμis any number. From (.), we obtain
=
where (.)-(.) are used. Consequently, we find the four matricesJ±foras follows:
J±=
The characteristic polynomials which are determined by the matricesJ±are obtained:
det(J±–λI) = (μ–λ).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Author details
1Faculty of Arts and Sciences, Department of Mathematics, Gaziantep University, Gaziantep, 27310, Turkey.2Faculty of Arts and Sciences, Department of Mathematics, Adıyaman University, Adıyaman, 02040, Turkey.
Acknowledgements
The first author is thankful to The Scientific and Technological Research Council of Turkey (TUBITAK) for their support with the Ph.D. scholarship. We thank both referees for their useful suggestions.
Received: 14 March 2016 Accepted: 14 June 2016 References
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