Direct and inverse spectral problems for discrete Sturm Liouville problem with generalized function potential Bala et al Advances in Difference Equations (2016) 2016 172 DOI 10 1186/s13662 016 0898 z[.]
Trang 1R E S E A R C H Open Access
Direct and inverse spectral problems for
discrete Sturm-Liouville problem with
generalized function potential
Bayram Bala1, Abdullah Kablan1and Manaf Dzh Manafov2*
In memory of GSh Guseinov (1951-2015)
* Correspondence:
mmanafov@adiyaman.edu.tr
2 Faculty 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 matrix J, 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 the N × N tridiagonal matrix
J=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a b a · · · · · ·
. . . . . . .
· · · c M d M+ · · ·
. . . . . . .
· · · · · · d N– c N–
· · · · · · c N– d N– c N–
· · · · · · c N– d N–
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
where a n , b n ∈ C, a n= and
c n = a n /α, n ∈ {M, M + , , N – },
d n = b n /α, n ∈ {M + , M + , , N – }, and α= is a positive real number
© 2016 Bala et al This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
Trang 2The 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 for N × N tridiagonal 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 matrix J of the form
(.) Almost symmetric here means that the entries above and below the main diagonal
are the same except the entries a M and c M
The eigenvalue problem we consider in this paper is Jy = λy, where y = {y n}N–
n= is a col-umn vector There exists a relation between this matrix eigenvalue problem and the second
order linear difference equation
a n–y n–+ b n y n + a n y n+= λρ n y n, n ∈ {, , , M, , N – },
a–= c N–= ,
(.)
for{y n}N
n=–, with the boundary conditions
where ρ nis a constant defined by
ρ n=
, ≤ n ≤ M,
These expressions are equivalent The problem (.), (.) is a discrete form of the
Sturm-Liouville operator with discontinuous coefficients
d dx
p (x) d
dx y (x) + q(x)y(x) = λρ(x)y(x), x ∈ [a, b], (.)
where ρ(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, and c is 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 matrix J and then give the
existence of linear functional which is defined from the ring of all polynomials in λ of
degree≤N with the complex coefficients to C Let us denote by {P n (λ)}N
n=–, the solution
of equation (.) together with the initial data
Trang 3By starting with (.), we can derive from equation (.) iteratively the polynomials P n (λ) of
order n, for n = , N In this way we obtain the unique solution {P n (λ)}N
n=of the following recurrence relations:
bP(λ) + aP(λ) = λP(λ), c N–= ,
a n–P n–(λ) + b n P n (λ) + a n P n+(λ) = λP n (λ), n ∈ {, , , M},
c n–P n–(λ) + d n P n (λ) + c n P n+(λ) = λP n (λ), n ∈ {M + , , N – },
(.)
subject to the initial condition
Lemma The following equality holds:
det(J – λI) = (–) N aa· · · a M c M+· · · c N–P N (λ). (.)
Therefore , the roots of the polynomial P N (λ) and the eigenvalues of the matrix J are
coin-cident
Proof We will consider the proof in three cases For each n = , M, let us define the
deter-minantn (λ) as follows:
n (λ) =
. . . . .
· · · b n–– λ a n–
· · · a n– b n–– λ a n–
Then expandingn (λ) by adding a row and column and finding the determinant ofn+(λ)
by the elements of the last row, we obtain
n+(λ) = (b n – λ)n (λ) – an–n–(λ), n = , M,(λ) = . (.)
Now for n = M + , N , let us definen (λ) as follows:
Trang 4
By using the same method, we get
n+(λ) = (d n – λ)n (λ) – cn–n–(λ), (.)
and finally, for n = M + , we find
M+(λ) = (d M+– λ)M+(λ) – a M c MM (λ). (.)
Dividing (.) and (.) by the product a· · · a n–, (.) by the product a· · · a n–c n–, we can easily show that the sequence
h–= , h= , h n= (–)n (a· · · a n–)–n (λ), n = , M + ,
h n= (–)n (a· · · c M+· · · c n–)–n (λ), n = M + , N,
satisfies (.), (.) Then h n is solution of (.), (.) We can show it by P n (λ) for n = , N
SinceN (λ) is also equal to det(J – λI) if we combine (.), (.) and (.), we obtain (.),
Theorem There exists a unique linear functional :CN [λ] → C such that the following
relations hold:
P m (λ)P n (λ) =δ mn
η , m , n ∈ {, , , M, , N – }, (.)
P m (λ)P N (λ) = , m ∈ {, , , M, , N}, (.)
where δ mn is the Kronecker delta , η is defined by
η=
, m , n ≤ M,
and (P(λ)) shows the value of on 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:
P n (λ) (n = , N – ), P m (λ)P N (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 functional can be found as finite values:
P n (λ) =δ n
η , n ∈ {, , , M, , N – }, (.)
P m (λ)P N (λ) = , m ∈ {, , , N}. (.)
Therefore, by linearity, the functional defined onC [λ] is unique.
Trang 5To show the existence of , let us define it on the polynomials (.) by (.), (.) and then we expand to over the whole spaceCN [λ] by using the linearity of It can be
shown that the functional satisfies (.), (.) Denote
P m (λ)P n (λ) = B mn, m , n ∈ {, , , M, , N}. (.)
It is clear that B mn = B nm , for m, n ∈ {, , , N} From (.) and (.), we get
B m= B m = δ m, m ∈ {, , , M}, (.)
B m= B m=δ m
Since{P n (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
a n–P n–(λ) + b n P n (λ) + a n P n+(λ) = bP n (λ) + aP(λ)P n (λ), n ∈ {, , , M},
c n–P n–(λ) + d n P n (λ) + c n P n+(λ) = bP n (λ) + aP(λ)P n (λ), n ∈ {M + , , N – }.
If we apply the linear functional to both sides of the last two equations, by taking into
account (.), (.), and (.), we get
B n= B n = δ n, n ∈ {, , , M}, (.)
B n= B n=δ n
Further, recalling the definition of ρ nin (.), we write
a m–P m–(λ) + b m P m (λ) + a m P m+(λ) = λρ m P m (λ), m ∈ {, , , M, , N – },
a n–P n–(λ) + b n P n (λ) + a n P n+(λ) = λρ n P n (λ), n ∈ {, , , M, , N – }.
If the first equality is multiplied by P n (λ) and the second equality is multiplied by P m (λ),
then the second result is subtracted from the first, we obtain:
for m, n ∈ {, , , M},
a m–P m–(λ)P n (λ) + b m P m (λ)P n (λ) + a m P m+(λ)P n (λ)
= a n–P n–(λ)P m (λ) + b n P n (λ)P m (λ) + a n P n+(λ)P m (λ),
for m ∈ {, , , M}, n ∈ {M + , , N – },
a m–P m–(λ)P n (λ) + b m P m (λ)P n (λ) + a m P m+(λ)P n (λ)
= c P (λ)P (λ) + d P (λ)P (λ) + c P (λ)P (λ),
Trang 6for m ∈ {M + , , N – }, n ∈ {, , , M},
c m–P m–(λ)P n (λ) + d m P m (λ)P n (λ) + c m P m+(λ)P n (λ)
= a n–P n–(λ)P m (λ) + b n P n (λ)P m (λ) + a n P n+(λ)P m (λ),
for m, n ∈ {M + , , N – },
c m–P m–(λ)P n (λ) + d m P m (λ)P n (λ) + c m P m+(λ)P n (λ)
= c n–P n–(λ)P m (λ) + d n P n (λ)P m (λ) + c n P n+(λ)P m (λ).
If the functional is applied to both sides of these equations, and using (.)-(.),
we obtain for B mnthe following boundary value problems:
for m, n ∈ {, , , M},
a m–B m –,n + b m B mn + a m B m +,n = a n–B n –,m + b n B nm + a n B n +,m, (.)
for m ∈ {, , , M}, n ∈ {M + , , N – },
a m–B m –,n + b m B mn + a m B m +,n = c n–B n –,m + d n B nm + c n B n +,m, (.)
for m ∈ {M + , , N – }, n ∈ {, , , M},
c m–B m –,n + d m B mn + c m B m +,n = a n–B n –,m + b n B nm + a n B n +,m, (.)
for m, n ∈ {M + , , N – },
c m–B m –,n + d m B mn + c m B m +,n = c n–B n –,m + d n B nm + c n B n +,m, (.)
for n ∈ {, , , M},
B n = B n = δ n , B n= B n = δ n, B Nn = B nN= , (.)
for n ∈ {M + , , N},
B n = B n=δ n
α , B n= B n=δ n
After starting with boundary values (.), (.) and using equations (.)-(.), we
can find all B mnuniquely as follows:
B mn = δ mn, m , n ∈ {, , , M},
B mn=δ mn
α , m , n ∈ {M + , , N – },
B mN= , m ∈ {, , , M, M + , , N}.
Definition The linear functional defined by Theorem is called the GSF of the
ma-trix J given in (.).
Trang 73 Inverse problem from the generalized spectral function
In this section, we solve the inverse spectral problem of reconstructing the matrix J by
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 matrix J from a given
GSF and find the necessary and sufficient conditions for a linear functional onCN [λ],
to be the GSF for some matrix J of 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 matrix J.
Recall that P n (λ) is a polynomial of degree n, so it can be expressed as
P n (λ) = γ n
λ n+
n–
k=
χ nk λ k
, n ∈ {, , , M, , N}. (.)
where γ n and χ nk are constants Inserting (.) in (.) and using the equality of the
poly-nomials, we can find the following equalities between the coefficients a n , b n , c n , d nand
the quantities γ n , χ nk:
a n= γ n
γ n+ (≤ n ≤ M), γ= ,
c n= γ n
γ n+ (M < n ≤ N – ), c M= γ M
αγ M+,
(.)
b n = χ n ,n– – χ n +,n (≤ n ≤ M), χ,–= ,
d n = χ n ,n– – χ n +,n (M < n ≤ N – ). (.)
It is easily shown that there exists an equivalence between (.), (.), and
λ m P n (λ) =δ mn
ηγ n
, m = , n, n ∈ {, , , M, , N – }, (.)
respectively Indeed, from (.), we can write
P m (λ)P n (λ) = γ m
λ m P n (λ) + γ m
m–
j=
χ mj
Then, since
λ j=
j
i=
c (j) i P i (λ), j ∈ {, , , N},
it follows from (.) that (.), (.) hold if we have (.), (.) and conversely if (.), (.)
hold, then (.), (.) can be obtained from (.) and (.)
Now, let us introduce
t l =
which are called ‘power moments’ of the functional .
Trang 8Writing the expansion (.) in (.) and (.) instead of P n (λ) and P N (λ), respectively,
and using the notation in (.), we get
t n +m+
n–
k=
χ nk t k +m= , m = , n – , n ∈ {, , , N}, (.)
t N+
N–
k=
t n+
n–
k=
χ nk t k +n=
ηγ
n
where η is defined in (.).
As a result of all discussions above, we write the procedure to construct the matrix in
(.) In turn, in order to find the entries a n , b n , c n , d n of the required matrix J, it suffices
to know only the quantities γ n , χ nk Given the linear functional which satisfies the
con-ditions of Theorem in [] onCN [λ], we can use (.) to find the quantities t land write
down the inhomogeneous system of linear algebraic equations (.) with the unknowns
χ n , χ n, , χ n ,n– , for every fixed n ∈ {, , , N} After solving this system uniquely and
using (.), we find the quantities γ n and so we obtain a n , b n , c n , d n, recalling (.), (.)
Therefore, we can construct the matrix J.
Using the numbers t ldefined in (.), let us present the determinants
D n=
t t · · · t n
t t · · · t n+
. .
t n t n+ · · · t n
From the definition of determinants in (.), it can be shown that the determinant of
system (.) is D n– Then, solving system (.) by using Cramer’s rule, we obtain
χ nk= –D
(k)
n–
D n–
where D (k) m (k = , m) is the determinant formed by exchanging in D m the (k + )th column
by the vector (t m+, t m+, , t m+)T Next, substituting equation (.) of χ nkinto the
left-hand side of (.), we find
γ n–= ηD n
D n–
where η is defined in (.) Now if we set D (m) m =m, then we obtain from (.), (.), by
using (.), (.),
a n=±
√
D n–D n+
D n
a M=±
√
αD M–D M+
√
D M–D M+
√
Trang 9c n=±
√
D n–D n+
D n
b n=n
D n–
n–
d n=n
D n
–n–
D n–
(M < n ≤ N – ), = t (.)
Hence , if which satisfies the conditions of Theorem in [] is given, then the values a n,
b n , c n , d n of the matrix J are obtained by equations (.)-(.), where D nis defined by
(.) and (.)
In the following theorem, we will show that the GSF of Jhas a special form and we will give a structure of the GSF Let J be a matrix which has the form (.) and be the GSF
of J Here we characterize the structure of .
Theorem Let λ, , λ p be all the eigenvalues with the multiplicities m, , m p ,
respec-tively , of the matrix J These are also the roots of the polynomial (.) Then there exist
numbers β kj (j = , m k , 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=
m k
j=
β kj
(j – )! P
where P (j–) (λ) denotes the (j – )th derivative of P(λ) with respect to λ.
Proof Let J be a matrix which has the form (.) Take into consideration the difference
equation (.)
a n–y n–+ b n y n + a n y n+= λρ n y n, n ∈ {, , , N – }, a–= c N–= , (.) where{y n}N
n=–is desired solution and
ρ n=
, ≤ n ≤ M,
α, M < n ≤ N – .
Denote by{P n (λ)}N
n=–and{Q n (λ)}N
n=–the solutions of (.) satisfying the initial condi-tions
For each n ≥ , the degree of polynomial P n (λ) is n and the degree of polynomial Q n (λ) is
n – It is clear that the entries R nm (λ) of the resolvent matrix R(λ) = (J – λI)–are of the
form
R nm (λ) =
ρ n P n (λ)[Q m (λ) + M(λ)P m (λ)], ≤ n ≤ m ≤ N – ,
ρ n P m (λ)[Q n (λ) + M(λ)P n (λ)], ≤ m ≤ n ≤ N – , (.)
Trang 10M (λ) = – Q N (λ)
and ρ n is defined in (.) Let f = (f, f, , f N–)T∈ CN be an arbitrary vector Since
R (λ)f = – f
λ + O
λ
,
as|λ| → ∞, we get for each n ∈ {, , , N –} and for a sufficiently large positive number r
f n= –
π i
r
N–
m=
R nm (λ)f m
dλ+
r
O
λ
where r is the circle in the λ-plane of radius r centered at the origin.
Let all the distinct zeros of P N (λ) in (.) be λ, , λ p with multiplicities m, , m p, respectively Then
P N (λ) = c(λ – λ)m· · · (λ – λ N)m p, (.)
where c is a constant We have ≤ p ≤ N and m+· · · + m p = N By (.), we can write
Q N (λ)
P N (λ) as the sum of partial fractions:
Q N (λ)
P N (λ)=
p
k=
m k
j=
β kj
where β kj are some uniquely determined complex numbers which depend on the matrix J.
Inserting (.) in (.) and using (.), (.) we get, by the residue theorem and
pass-ing then to the limit r→ ∞,
f n=
p
k=
m k
j=
β kj
(j – )!
d j–
dλ j–
ρ n F (λ)P n (λ)
λ =λ k
, n ∈ {, , , N – }, (.)
where
F (λ) =
N–
m=
Now define the functional onCN [λ] by the formula
P (λ) =
p
k=
m k
j=
β kj
(j – )! P
Thus, (.) can be written as follows:
f n
ρ =
F (λ)P n (λ) , n ∈ {, , , N – }. (.)
... data-page="7">3 Inverse problem from the generalized spectral function< /b>
In this section, we solve the inverse spectral problem of reconstructing the matrix J by
its GSF and we give... the functional is applied to both sides of these equations, and using (.)-(.),
we obtain for B mnthe following boundary value problems:
for. .. sufficient conditions for a linear functional onCN [λ],
to be the GSF for some matrix J of the form (.) For the investigation of necessary and< /i>
sufficient