Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition
We investigate a problem for the Dirac differential operators in the case where an eigenparameter not only appears in the differential equation but is also linearly contained in a boundary condition. We prove uniqueness theorems for the inverse spectral problem with known collection of eigenvalues a...
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| author | Amirov, R. Kh. Keskin, B. Özkan, G. Аміров, Р. Х. Кескін, Б. Озкан, Г. |
| author_facet | Amirov, R. Kh. Keskin, B. Özkan, G. Аміров, Р. Х. Кескін, Б. Озкан, Г. |
| author_sort | Amirov, R. Kh. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:45:12Z |
| description | We investigate a problem for the Dirac differential operators in the case where an eigenparameter not only appears in the differential equation but is also linearly contained in a boundary condition. We prove uniqueness theorems for the inverse spectral problem with known collection of eigenvalues and normalizing constants or two spectra. |
| first_indexed | 2026-03-24T02:36:00Z |
| format | Article |
| fulltext |
UDC 517.9
R. Kh. Amirov, B. Keskin, A. S. Ozkan (Cumhuriyet Univ., Turkey)
DIRECT AND INVERSE PROBLEMS
FOR THE DIRAC OPERATOR WITH SPECTRAL PARAMETER
LINEARLY CONTAINED IN BOUNDARY CONDITION
PRQMI TA OBERNENI ZADAÇI DLQ OPERATORA DIRAKA
IZ SPEKTRAL|NYM PARAMETROM,
WO LINIJNO VXODYT| DO HRANYÇNO} UMOVY
We investigate a problem for the Dirac differential operators in the case where an eigenparameter not
only appears in the differential equation but is also linearly contained in the boundary condition. We
prove uniqueness theorems for the inverse spectral problem with a known collection of eigenvalues and
normalizing constants or two spectra.
DoslidΩeno zadaçu dlq dyferencial\nyx operatoriv Diraka u vypadku, koly vlasnyj parametr
ne til\ky prysutnij u dyferencial\nomu rivnqnni, ale j linijno vxodyt\ do hranyçno] umovy.
Dovedeno teoremy [dynosti v obernenij spektral\nij zadaçi z vidomym naborom vlasnyx znaçen\ i
normugçyx stalyx abo dvox spektriv.
Let us consider the canonical system of Dirac differential equations
l y : = By x y′ + Ω( ) = λy , x ∈( , )0 π , (1)
where B =
0 1
1 0−
, Ω( )x =
p x q x
q x p x
( ) ( )
( ) ( )−
, y x( ) =
y x
y x
1
2
( )
( )
, p x( ) and
q x( ) are real valued functions in L2 0( , )π , λ is a spectral parameter.
By L we denote the boundary-value problem generated by equation (1) with the
boundary conditions
U y( ) : = y1 0( ) = 0 , (2)
V y( ) : = λ π π π π( )( ) ( ) ( ) ( )y H y H y H y2 1 1 1 2 2+ − − = 0 , (3)
where H, H1 and H2 are real numbers. We assume that ρ : = HH H1 2− > 0.
Boundary-value problems often appear in mathematics, mechanics, physics, geo-
physics and other branches of natural properties. The inverse problem of reconstruc-
ting the material properties of a medium from data collected outside of the medium is
central importance in disciplins ranging from engineering to the geosciences.
Eigenvalue dependent boundary conditions were examined even before the time of
Sturm and Liouville [1]. Linear conditions like (2) and (3) were investigated in [2, 3].
Direct and inverse problems for Dirac operators are fairly well studied (see [4 – 6] and
references therein).
The inner product in the Hilbert space H = L L2 20 0( , ) ( , )π π� � C is defined
by
〈 〉Y Z, : = ( )( ) ( ) ( ) ( )y x z x y x z x dx y z1 1 2 2
0
3 3
1
+ +∫
π
ρ
© R. KH. AMIROV, B. KESKIN, A. S. OZKAN, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1155
1156 R. KH. AMIROV, B. KESKIN, A. S. OZKAN
for
Y =
y x
y x
y
1
2
3
( )
( )
, Z =
z x
z x
z
1
2
3
( )
( )
∈ H.
Define an operator T (see [3]) acting in H such that
T Y( ) : =
By x x y x
H y H y
′ +
′ +
( ) ( ) ( )
( ) ( )
Ω
1 1 2 1π π
with
D T( ) = Y Y y y y y y ACT∈ = ∈{ H : , , , , [ , ],( )1 2 3 1 2 0 π
lY y y y H y∈ = = + }H, ( ) , ( ) ( ) ( )1 3 2 10 0 0 π π .
It is clear that T is a closed operator in H and the eigenvalue problem of operator T
is adequate problem of (1) – (3).
1. Properties of spectrum. In this section, we investigate some properties of
operator T and its spectrum. We assume that q x( ) = 0 (see [4]).
Lemma 1. (i) Two eigenfunctions y x( , )λ1 = [ ]( , ), ( , )y x y x T
1 1 2 1λ λ , z x( , )λ2 =
= [ ]( , ), ( , )z x z x T
1 2 2 2λ λ corresponding to different eigenvalues λ1 and λ2 are
orthogonal in the sense of
[ ]( , ) ( , ) ( , ) ( , )y x z x y x z x dx1 1 1 2 2 1 2 2
0
1
λ λ λ λ
ρ
π
+ +∫ yy z3 3 = 0. (4)
(ii) All eigenvalues of the operator T (or problem L) are real numbers and
all eigenfunctions are real valued.
Proof. (i) Since the eigenfunctions y x( , )λ1 and z x( , )λ2 are the solutions of
the system (1), the following equalities hold:
′ + −y x p x y x2 1 1 1 1( , ) { ( ) } ( , )λ λ λ = 0,
′ + +y x p x y x1 1 1 2 1( , ) { ( ) } ( , )λ λ λ = 0,
′ + −z x p x z x2 2 2 1 2( , ) { ( ) } ( , )λ λ λ = 0,
′ + +z x p x z x1 2 2 2 2( , ) { ( ) } ( , )λ λ λ = 0.
If the multiply these equalities by z x1 2( , )λ , – z x2 2( , )λ , – y x1 1( , )λ and y x2 1( , )λ ,
respectively, to get
d
dx
y x z x y x z x2 1 1 2 1 1 2 2( , ) ( , ) ( , ) ( , )λ λ λ λ−{ } =
= ( ) ( , ) ( , ) ( , ) ( , )λ λ λ λ λ λ1 2 1 1 1 2 2 1 2 2− +{ }y x z x y x z x .
Integrate last equality from 0 to π with respect to x to obtain
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
DIRECT AND INVERSE PROBLEMS FOR THE DIRAC OPERATOR … 1157
( ) ( , ) ( , ) ( , ) ( , )λ λ λ λ λ λ1 2 1 1 1 2 2 1 2 2− +{ }y x z x y x z x dxx
0
π
∫ =
= y x z x y x z x1 1 2 2 2 1 1 2( , ) ( , ) ( , ) ( , )λ λ λ λ−{ } .
Since the functions y x( , )λ1 and z x( , )λ2 are eigenfunctions, the equality
( ) ( , ) ( , ) ( , ) ( , )λ λ λ λ λ λ1 2 1 1 1 2 2 1 2 2− +{ }y x z x y x z x dxx
0
π
∫ =
= y z y z1 1 2 2 2 1 1 2( , ) ( , ) ( , ) ( , )π λ π λ π λ π λ− (5)
is valid. On the other hand, from (3) we have
λ π λ π λ1 2 1 1 1y Hy( , ) ( , )+ = H y H y1 1 1 2 2 1( , ) ( , )π λ π λ+ ,
λ π λ π λ2 2 2 1 2z Hz( , ) ( , )+ = H z H z1 1 2 2 2 2( , ) ( , )π λ π λ+ .
Let us multiply these equalities by z Hz2 2 1 2( , ) ( , )π λ π λ+ and y Hy2 1 1 1( , ) ( , )π λ π λ+ ,
respectively, and subtract side by side to get
( ) ( , ) ( , ) ( , ) (λ λ π λ π λ π λ π1 2 2 1 1 1 2 2 1− + +y Hy z Hz ,, )λ2 =
= – ρ π λ π λ π λ π λy z y z1 1 2 2 2 1 1 2( , ) ( , ) ( , ) ( , )−( ) . (6)
The proof is completed by using (5) and (6).
(ii) Let λ λ≠ , then y x( , )λ and y x( , )λ are the different eigenfunctions of
operator T associated with eigenvalues λ and λ , respectively. From (i) we have
y x y x y x y x dx y y1 1 2 2
0
3 3
1
( , ) ( , ) ( , ) ( , )λ λ λ λ
ρ
π
+( ) +∫ = 0,
then
y x y x dx y1
2
2
2
0
3
21
( , ) ( , )λ λ
ρ
π
+{ } +∫ = 0,
hence, y x1( , )λ = y x2( , )λ = y3 = 0. This contradiction gives the proof of (ii).
The lemma is proved.
Let us denote the solutions of (1) by ϕ λ( , )x and ψ λ( , )x satisfying the initial
conditions
ϕ λ( , )0 =
0
1−
, ψ π λ( , ) =
H
H H
2
1
−
−
λ
λ
, (7)
respectively.
It is shown in [7] that every solution of equations (1) satisfying the above initial
conditions, has a representation as follows:
ϕ λ( , )x =
sin
cos
( , )
sin
cos
λ
λ
λ
λ
x
x
K x t
t
t
dt
−
+
−
00
π
∫ , (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1158 R. KH. AMIROV, B. KESKIN, A. S. OZKAN
where K x t( , ) =
K x t K x t
K x t K x t
11 12
21 22
( , ) ( , )
( , ) ( , )
−
−
and K x L x xij ( , .) ( , )∈ −2 , i , j = 1, 2, for
every fixed x ∈[ , ]0 π .
One can easily check that the following asymptotic formulae hold for sufficiently
large λ :
ϕ λj x( , ) = O xexp τ( ) , j = 1, 2, 0 < x < π , (9)
′ϕ λj x( , ) = O xλ τexp( ) , j = 1, 2, 0 < x < π , (10)
ψ λj x( , ) = O xλ τ πexp ( )−( ) , j = 1, 2, 0 < x < π , (11)
where τ = Im λ .
The characteristic function ∆( )λ of the problem L is defined as follows:
∆( )λ = λ ϕ π λ ϕ π λ ϕ π λ ϕ π λ2 1 1 1 2 2( , ) ( , ) ( , ) ( , )+( ) − +H H H (12)
and zeros of ∆( )λ coincide with the eigenvalues of problem L .
We define norming constants by
αn : = ϕ λ ϕ λ
ρ
ϕ
π
1
2
2
2
0
3
21
( , ) ( , )x x dxn n+( ) +∫ . (13)
Lemma 2. The eigenvalues of the problem L are simple and separated.
Proof. Let us write the following equations:
′ + −ψ λ λ ψ λ2 1( , ) { ( ) } ( , )x p x x = 0,
′ + +ψ λ λ ψ λ1 2( , ) { ( ) } ( , )x p x x = 0,
′ + −ϕ λ λ ϕ λ2 1( , ) { ( ) } ( , )x p x xn n n = 0,
′ + +ϕ λ λ ϕ λ1 2( , ) { ( ) } ( , )x p x xn n n = 0.
Multiply these equalities by ϕ λ1( , )x n , – ϕ λ2( , )x n , – ψ λ1( , )x and ψ λ2( , )x , re-
spectively, to get
d
dx
x x x xn nϕ λ ψ λ ϕ λ ψ λ1 2 2 1( , ) ( , ) ( , ) ( , )−{ } =
= ( ) ( , ) ( , ) ( , ) ( , )λ λ ϕ λ ψ λ ϕ λ ψ λn n nx x x x− +{ }1 1 2 2 .
After integrating last equalities from 0 to π with respect to x , we obtain
ϕ λ ψ λ ϕ λ ψ λ1 2 2 1( , ) ( , ) ( , ) ( , )x x x xn n− =
= ( ) ( , ) ( , ) ( , ) ( , )λ λ ϕ λ ψ λ ϕ λ ψ λ
π
n n nx x x x dx− +{ }1 1 2 2
0
∫∫ .
Let the functions ϕ λ( , )x n be an eigenfunction. Use (2) to get
ϕ λ ψ λ ϕ λ ψ λ
π
1 1 2 2
0
( , ) ( , ) ( , ) ( , )x x x x dxn n+{ }∫ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
DIRECT AND INVERSE PROBLEMS FOR THE DIRAC OPERATOR … 1159
=
1
01 2 2 1 1λ λ
ϕ π λ ψ π λ ϕ π λ ψ π λ ψ λ
n
n n−
− −( , ) ( , ) ( , ) ( , ) ( , )){ } =
= –
1
2 1 2 1ρ
ϕ π λ ϕ π λ ψ π λ ψ π λ( , ) ( , ) ( , ) ( , )
(
n nH H+{ } +{ } + ∆ λλ
λ λ
)
− n
.
If we pass throuth the limit as λ λ→ n and use the equality ψ λ( , )x n = β ϕ λn nx( , ) ,
then
β ϕ λ ϕ λ
ρ
ϕ
π
n n nx x dx1
2
2
2
0
3
21
( , ) ( , )+{ } +
∫
= ′∆ ( )λn .
Hence, ′∆ ( )λn = β αn n , where βn = – ψ λ2 0( , )n . It is obvious that ′∆ ( )λn ≠ 0.
So, eigenvalues of the problem L is simple.
Since the function ∆ ( )λ is an entire function of λ , the zeros of ∆ ( )λ are sepa-
rated.
The lemma is proved.
Theorem 1. For the eigenvalues λn and the normalizing numbers αn of the
problem (1) – (3), the following asymptotic formulae hold:
λn = λ εn n
0 + , (14)
αn = π γ+ n , (15)
where εn , γ n l∈ 2 and λn
0 are the zeros of ∆0( )λ : = – cos sinλπ λπ+ H ,
i. e., λn
0 = n
H
+ 1 1
π
arctan .
Proof. Using (8), we get
∆ ( )λ = λ λπ λπ λπ λπ( cos sin ) sin cos− + − +H H H1 2 +
+ λ π π λ λ π
π
K t H K t t dt K t H21 11
0
22( , ) ( , ) sin ( , )+( ) + +∫ KK t t dt12
0
( , ) cosπ λ
π
( )∫ –
– H K t H K t t dt H K t1 11 2 21
0
1 12( , ) ( , ) sin ( ,π π λ π
π
+( ) −∫ )) ( , ) cos+( )∫ H K t t dt2 22
0
π λ
π
.
Since the eigenvalues are the zeros of ∆ ( )λ , we can write the following equation
for them:
– cos sin sin cosλπ λπ
λ
λπ
λ
λπ+ − +H
H H1 2 +
+ K t H K t t dt K t H K21 11
0
22 1( , ) ( , ) sin ( , )π π λ π
π
+( ) + +∫ 22
0
( , ) cosπ λ
π
t t dt( )∫ –
–
1
1 11 2 21
0
λ
π π λ
π
H K t H K t t dt( , ) ( , ) sin+( )∫ –
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1160 R. KH. AMIROV, B. KESKIN, A. S. OZKAN
–
1
1 12 2 22
0
λ
π π λ
π
H K t H K t t dt( , ) ( , ) cos+( )∫ = 0.
Denote
Gn = λ λ λ
β
∈ = +
C n: 0
2
, n = ± ± …0 1 2, , , ,
Gδ = λ λ λ δ: , , , ,− ≥ = ± ± …{ }n n0 0 1 2 , ˆ ( )
( )
∆
∆
λ
λ
λ
= ,
where δ is a sufficiently small number.
Since ∆0( ) expλ τ πδ≥ ( )C for λ δ∈G and ˆ ( ) ( )∆ ∆λ λ− 0 < Cδ τ πexp
for sufficiently large values n and λ ∈Gn , we have
∆0( )λ ≥ Cδ τ πexp ( ) > ˆ ( ) ( )∆ ∆λ λ− 0 .
Using the Rouche theorem, we establish that, for sufficiently large n , the functions
∆0( )λ and ∆ ∆ ∆0 0( ) ˆ ( ) ( )λ λ λ+ −{ } = �∆( )λ have the same number of zeros inside
the contour Gn . That is, they have ( )2 1n + numbers of zeros: λ λ λ− … …n n, , , ,0 .
Thus, the eigenvalues { }λn n ≥ 0 are of the form λn = λ εn n
0 + , where lim
n
n→∞
ε =
= 0.
In last equality, if we put λ εn n
0 + instead of λn and use ∆0
0( )λ εn n+ =
= ′ +∆0
0( ) ( )λ ε εn n no , we get εn l∈ 2 .
Hence, the asymptotic formula (14) for the eigenvalues λn of the problem (1) – (3)
is true.
Finally, to prove (15), we can write the following equalities from (8):
ϕ λ1( , )x n = sin ( )λn nx f x+ , ϕ λ2( , )x n = − +cos ( )λn nx g x ,
where
f xn ( ) = K x t t dt K x t t dtn
x
n
x
11
0
12
0
( , ) sin ( , ) cosλ λ∫ ∫+ ,
g xn ( ) = K x t t dt K x t t dtn
x
n
x
21
0
22
0
( , ) sin ( , ) cosλ λ∫ ∫+ ,
and f xn ( ) , g x ln ( ) ∈ 2 for all x ∈( , )0 π . Using (3) and (13), we get
αn : = ϕ λ ϕ λ
ρλ
ϕ π λ
π
1
2
2
2
0
2 1 1
1
( , ) ( , ) ( , )x x dx Hn n
n
n+( ) + +∫ HH n2 2
2ϕ π λ( , )( ) =
= dx n
0
π
γ∫ + .
Hence, αn = π γ+ n , γ n l∈ 2 .
The theorem is proved.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
DIRECT AND INVERSE PROBLEMS FOR THE DIRAC OPERATOR … 1161
Remark 1. If the function Ω( )x is differentiable, eigenvalues and normalizing
numbers of the problem (1) – (3) satisfy the following asymptotic estimates:
γ n = λ
δ ζ
n
n n
n n
0 + + ,
αn = n
n n
n n+ +
γ ξ
,
where ζn , ξn l∈ 2 , δn , γ n l∈ ∞ .
2. Weyl solution, Weyl function. Assume that a vector function Φ( , )x λ =
=
Φ
Φ
1
2
( , )
( , )
x
x
λ
λ
is a solution of the system (4) that satisfies the conditions Φ1 0( , )λ =
= 1 and λ π π( )( ) ( )Φ Φ2 1+ H = H H1 1 2 2Φ Φ( ) ( )π π+ . The function Φ( , )x λ is
called the Weyl solution of the boundary-value problem L .
Let C x( , )λ =
C x
C x
1
2
( , )
( , )
λ
λ
denote solutions of system (1) that satisfy the initial
conditions C ( , )0 λ =
1
0
. It is clear that the functions ψ λ( , )x and C x( , )λ are
entire with respect to λ . Then the function ψ λ( , )x can be represented as follows:
ψ λ( , )x = ψ λ ϕ λ λ λ2 0( , ) ( , ) ( ) ( , )x C x+ ∆
or
ψ λ
λ
( , )
( )
x
∆
= C x x( , )
( , )
( )
( , )λ
ψ λ
λ
ϕ λ− 2 0
∆
. (16)
Denote
M( )λ = –
ψ λ
λ
2 0( , )
( )∆
. (17)
It is clear that
Φ( , )x λ = C x M x( , ) ( ) ( , )λ λ ϕ λ+ . (18)
The function M( )λ = – Φ2 0( , )λ is called the Weyl function for the problem L .
The Weyl solution and Weyl function are meromorphic functions with respect to λ
having poles in the spectrum of the problem L . Relations (17) and (18) yield
Φ( , )x λ =
ψ λ
λ
( , )
( )
x
∆
. (19)
The lemma is proved.
Theorem 2. The following representation is true:
M( )λ =
1 1 1
0 0
0 0
1α λ λ α λ λ α λ( ) ( )−
+
−
+
=
∞
∑
n n n nn
. (20)
Proof. Note that ψ λ1( , )x = ψ λ01 1( , )x f+ , ψ λ2( , )x = ψ λ02 2( , )x f+ , where
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1162 R. KH. AMIROV, B. KESKIN, A. S. OZKAN
f x1( , )λ = sin ( ) ( , ) ( ) cos ( ) ( , )λ ψ λ λ ψ λ
π
t x t p t dt t x t p
x
− + −∫ 1 2 (( )t dt
x
π
∫ ,
f x2( , )λ = cos ( ) ( , ) ( ) sin ( ) ( , )λ ψ λ λ ψ λ
π
t x t p t dt t x t p
x
− − −∫ 1 2 (( )t dt
x
π
∫ .
If we use these equalities, we get
M M( ) ( )λ λ− 0 =
ψ λ
ψ λ
ψ λ
ψ λ
2
1
02
01
0
0
0
0
( , )
( , )
( , )
( , )
− =
ψ λ
ψ λ
ψ λ
ψ λ
02 2
01 1
02
01
0
0
0
0
( , )
( , )
( , )
( , )
+
+
−
f
f
=
=
( ) ( )( , ) ( , ) ( , ) ( , )ψ λ ψ λ ψ λ ψ λ02 2 01 01 1 020 0 0 0+ − +f f
(( )( , ) ( , )ψ λ ψ λ01 1 010 0+ f
=
=
f f
M2 1
0∆ ∆( ) ( )
( )
λ λ
λ− .
Since ∆( )λ > C eδ
λ πλ Im , we have lim
( )
( )
Im
λ
λ π λ
λ→∞
−e
fi
∆
= 0 and, for
λ δ∈G , the equality
M M( ) ( )λ λ− 0 =
f f
M2 1
0∆ ∆( ) ( )
( )
λ λ
λ−
yields
lim sup ( ) ( )
λ λ
ε λ λ
δ
→ ∞ ∈
−n
G
M M0 = 0. (21)
The vector functions ϕ λ ϕ λ( , ) ,( ( ))x xn n0
0 and ψ λ ψ λ( , ) ,( ( ))x xn n0
0 are eigenfunc-
tions of the problem L L( )0 . Therefore, there exists constants β βn n( )0 such that
ψ λ( , )x n = β ϕ λn nx( , ) ( ( ) ( )), ,ψ λ β ϕ λ0
0 0
0
0x xn n n= .
Since ψ λ2 0( , )n = β ϕ λn n2 0( , ) , we get the following equalities:
βn = ψ λ2 0( , )n , βn
0 = ψ λ02
00( ), n .
Since ψ λ2 0( , ) and ∆( )λ are the analytic functions at the point λ = λn and
ψ λ2 0( , )n ≠ 0, ∆( )λn = 0, ′∆ ( )λn ≠ 0, we have that the functions M( )λ and
M0( )λ have simple poles at these points. Hence, using the equalities α ϕ λn n2 0( , ) =
= ′∆ ( )λn and ϕ λ α02
0 00( ), n n = – ′∆0
0( )λn , we get
Res ( ),( = )M nλ λ λ =
ψ λ
λ
2 0( , )
( )
n
n′∆
=
1
αn
,
(22)
Res ( ),( = )M n0 λ λ λ =
ψ λ
λ
2
0
0( , )
( )
n
n′∆
=
1
0αn
.
Denote Γn = λ λ λ ε: = +{ }n , where ε is a sufficiently small number. Consi-
der the contour integral
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
DIRECT AND INVERSE PROBLEMS FOR THE DIRAC OPERATOR … 1163
I xn ( ) =
1
2
0
π
µ µ
λ µ
µ
i
M M
d
n
( ) ( )−
−∫
Γ
, λ ∈ int Γn .
By virtue of (21), we have lim ( )
n
nI x
→∞
= 0. On the other hand, according to the theo-
rem on residues, relation (22) yields
I xn ( ) = – M M
n n n nn n n
( ) ( )
( ) ( )int
λ λ
α λ λ α λ λλ λ
+ −
−
+
−∈
∑0 0 0
1 1
Γ 00 ∈
∑
int Γn
.
Hence, as n → ∞ , lim ( )
n
nI x
→∞
= 0 imply
M( )λ = M
n n n nn
0 0 0
0
1 1
( )
( ) ( )
λ
α λ λ α λ λ
+
−
−
−
=
∞
∑ . (23)
It follows from the function M0( )λ that
M0( )λ =
1 1 1 1
0
0 0 0 0
1α λ α λ λ λ
+
−
+
=
∞
∑
n n nn
.
The comparison of the last two equalities yields
M( )λ =
1 1 1 1 1
0
0 0 0 0
1α λ α λ λ λ α λ λ
+
−
+
+ −
−
=
∞
∑
n n nn n n( )
11
0 0
0 α λ λn nn ( )−
=
∞
∑ =
=
1 1 1
0
0
0 0 0
0α λ α λ λ α λ
+
−
−
( )
+
+
1 1 1 1
0 0 0 0 0 0α λ λ α λ α λ λ α λ λn n n n n n n n( ) ( ) ( )−
+ +
−
−
−
=
∞
∑
n 1
.
Hence,
M( )λ =
1 1 1
0 0
0 0
1α λ λ α λ α λ λ( ) ( )−
+ +
−
=
∞
∑
n n n nn
.
The theorem is proved.
3. Inverse problem. In this section, we investigate the inverse problem of the re-
construction of a boundary-value problem L from its spectral characteristics. We con-
sider three statements of the inverse problem of the reconstruction of the boundary-va-
lue problem L from the Weyl function, from the spectral data { },λ αn n n ≥ 0 , and
from two spectra { },λ µn n n ≥ 0 .
Let us formulate a theorem on the uniqueness of a solution of the inverse problem
with the use of the Weyl function. For this purpose, together with L we consider the
boundary-value problem �L of the same form but with potential �Ω( )x . It is assumed
in what follows that if a certain symbol α denotes an object related to the problem L ,
then �α denotes the coresponding object related to te problem �L .
Theorem 3. If M M( ) ( )λ λ= � , then L L= � . Thus, the boundary-value prob-
lem L is uniquely defined by a Weyl function.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1164 R. KH. AMIROV, B. KESKIN, A. S. OZKAN
Proof. We introduce a matrix P x( , )λ = [ ]( , ) , ,P xj jλ λλ =1 2 , by the formula
P x( , )λ
ϕ
ϕ
� �
� �
1 1
2 2
Φ
Φ
=
ϕ
ϕ
1 1
2 2
Φ
Φ
.
For the Wronskian of the solutions
�ϕ =
�
�
ϕ
ϕ
1
2
, �Φ1 =
�
�
Φ
Φ
1
2
,
we have
W x x� �ϕ λ λ( , ), ( , )Φ{ } ≡ � � � �ϕ λ λ ϕ λ λ1 2 2 1( , ) ( , ) ( , ) ( , )x x x xΦ Φ− = 1.
Take this into account and multiply both sides of the following equation from left by
the matrix
� �
� �
Φ Φ2 1
2 1
−
−
ϕ ϕ
,
P x P x
P x P x
11 12
21 22
1( , ) ( , )
( , ) ( , )
λ λ
λ λ
ϕ
� �Φ11
2 2� �ϕ Φ
=
ϕ
ϕ
1 1
2 2
Φ
Φ
,
in order to get
P x P x
P x P x
11 12
21 22
( , ) ( , )
( , ) ( , )
λ λ
λ λ
=
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ
1 2 1 2 1 1 1 1
2 2 2 2 2 1 2
� � � �
� � �
Φ Φ Φ Φ
Φ Φ Φ Φ
− − +
− − + ��ϕ1
or
P x x x x x11 1 2 1 2( , ) ( , ) ( , ) ( , ) ( , )λ ϕ λ λ λ ϕ λ= −� �Φ Φ ,
P x x x x x12 1 1 1 1( , ) ( , ) ( , ) ( , ) ( , )λ ϕ λ λ λ ϕ λ= − +� �Φ Φ ,
(24)
P x x x x x21 2 2 2 2( , ) ( , ) ( , ) ( , ) ( , )λ ϕ λ λ λ ϕ λ= −� �Φ Φ ,
P x x x x x22 2 1 2 1( , ) ( , ) ( , ) ( , ) ( , )λ ϕ λ λ λ ϕ λ= − +� �Φ Φ ,
ϕ λ ϕ λ λ ϕ λ1 11 1 12 2( , ) ( , ) ( , ) ( , )x P x P x x= +� � ,
ϕ λ ϕ λ λ ϕ λ2 21 1 22 2( , ) ( , ) ( , ) ( , )x P x P x x= +� � ,
(25)
Φ Φ Φ1 11 1 12 2( , ) ( , ) ( , ) ( , ) ( , )x P x x P x xλ λ λ λ λ= +� � ,
Φ Φ Φ2 21 1 22 2( , ) ( , ) ( , ) ( , ) ( , )x P x x P x xλ λ λ λ λ= +� � .
Relations (16) and (18) yield
P x11 1( , )λ − =
� �
� �
ψ λ ϕ λ ϕ λ
λ
ϕ λ
ψ2 1 1
2
( , ) ( , ) ( , )
( )
( , )
x x x
x
− −
∆
11 1( , )
( )
( , )
( )
x xλ
λ
ψ λ
λ∆ ∆
−
�
� ,
P x12( , )λ =
ψ λ ϕ λ ϕ λ
λ
ϕ λ
ψ1 1 1
1
1( , ) ( , ) ( , )
( )
( , )
(x x x
x
x� − +
∆
,, )
( )
( , )
( )
λ
λ
ψ λ
λ∆ ∆
−
�
�
1 x
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
DIRECT AND INVERSE PROBLEMS FOR THE DIRAC OPERATOR … 1165
P x21( , )λ =
� �
� �
ψ λ ϕ λ ϕ λ
λ
ϕ λ
ψ2 2 2
2
( , ) ( , ) ( , )
( )
( , )
x x x
x
− −
∆
22 2( , )
( )
( , )
( )
x xλ
λ
ψ λ
λ∆ ∆
−
�
� ,
P x22 1( , )λ − =
ψ λ ϕ λ ϕ λ
λ
ϕ λ
ψ2 1 1
2
1( , ) ( , ) ( , )
( )
( , )
(x x x
x
x� − +
∆
,, )
( )
( , )
( )
λ
λ
ψ λ
λ∆ ∆
−
�
�
1 x
.
It follows from the representations of the solutions ϕ λ( , )x and ψ λ( , )x , the inequa-
lity ∆( )λ > C eδ
λ πλ Im , and the Lebesque lemma that
lim max ( , )
λ
λ
π
λ
→∞
∈
≤ ≤
−
R
x
P x
0
11 1 = lim max ( , )
λ
λ
π
λ
→∞
∈
≤ ≤
−
R
x
P x
0
22 1 =
= lim max ( , )
λ
λ
π
λ
→∞
∈
≤ ≤
R
x
P x
0
12 = lim max ( , )
λ
λ
π
λ
→∞
∈
≤ ≤
R
x
P x
0
21 = 0 . (26)
According to (18) and (25), we have
P x x C x C x x11 1 2 1 2( , ) ( , ) ( , ) ( , ) ( , ) (λ ϕ λ λ λ ϕ λ= − +� � �MM M x x( ) ( ) ( , ) ( , ))λ λ ϕ λ ϕ λ− 1 2� ,
P x C x x x C x M12 1 1 1 1( , ) ( , ) ( , ) ( , ) ( , ) (λ λ ϕ λ ϕ λ λ= − +� � (( ) ( ) ( , ) ( , ))λ λ ϕ λ ϕ λ− � �M x x1 1 ,
P x x C x C x x21 2 2 2 2( , ) ( , ) ( , ) ( , ) ( , ) (λ ϕ λ λ λ ϕ λ= − +� � �MM M x x( ) ( ) ( , ) ( , ))λ λ ϕ λ ϕ λ− 2 2� ,
P x C x x C x x M22 2 1 1 2( , ) ( , ) ( , ) ( , ) ( , ) (λ λ ϕ λ λ ϕ λ= − +� � (( ) ( ) ( , ) ( , ))λ λ ϕ λ ϕ λ− � �M x x2 2 .
Thus, the functions P xjλ λ( , ) are entire with respect to λ for every fixed x as
M( )λ = �M( )λ . From (26) we have
P x11( , )λ ≡ 1, P x12( , )λ ≡ 0, P x22( , )λ ≡ 1, P x21( , )λ ≡ 0.
Substituting these relations in (25), we get
ϕ λ1( , )x ≡ �ϕ λ1( , )x , ϕ λ2( , )x ≡ �ϕ λ2( , )x ,
Φ1( , )x λ ≡ �Φ1( , )x λ , Φ2( , )x λ ≡ �Φ2( , )x λ
for all x and λ . Hence, L L= � .
Theorem 4. If λ λn n= � and α αn n= � for all n Z∈ , then L L= � . Thus,
the problem L is uniquelly defined by spectral data.
Proof. Since λ λn n= � , α αn n= � for all n Z∈ and
M( )λ =
1 1 1
0 0
0 0
1α λ λ α λ α λ λ( ) ( )−
+ +
−
=
∞
∑
n n n nn
,
�M( )λ =
1 1 1
0 0
0 0� � � � � � � �α λ λ α λ α λ λ( ) ( )−
+ +
−
= n n n nn 11
∞
∑ ,
we get
M( )λ = �M( )λ .
From Theorem 4 we prove that L L= � .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1166 R. KH. AMIROV, B. KESKIN, A. S. OZKAN
Let us consider the boundary-value problem L1 in which we take the condition
y2 0 0( ), λ = instead of the boundary condition (2) of the problem L . Let { }µn n ≥ 0
be the eigenvalues of the problem L1 .
Theorem 5. If λ λn n= � and µ µn n= � for all n ∈N , then L L= � , i.e., the
problem L is uniquelly determined by the sequences { }λn and { }µn .
Proof. Since λ λn n= � ,
∆
∆
( )
( )
λ
λ� is an entire function in λ . Moreover, ∆( )λ =
= �∆( )λ , since lim
( )
( )λ
λ
λ→∞
∆
∆�
= 1. On the other hand, it is easy to see that ψ λ2 0( ), n =
= �ψ λ2 0( ), n as µ µn n= � . So, from the equality αn = –
′∆ ( )
,( )
λ
ψ λ
n
n2 0
, α αn n= � is
obtained. Thus, the proof is completed by Theorem 5.
1. Poisson S. D. Memoire sur la maniere d’exprimer les fuctions par des series periodiques // J. Ecole
Polytechnique. – 1820. – 18. – P. 417 – 489.
2. Fulton C. T. Two-point boundary value problems with eigenvalue parameter contained in the
boundary conditions // Proc. R. Soc. Edinburgh, A. – 1977. – 77. – P. 293 – 308.
3. Fulton C. T. Singular eigenvalue problems with eigenvalue parameter contained in the boundary
conditions // Ibid. – 1980. – 87. – P. 1 – 34.
4. Levitan B. M., Sargsyan I. S. Sturm – Liouville and Dirac operators (in Russian). – Moscow:
Nauka, 1988.
5. Gasymov M. G. Inverse problem of the scattering theory for Dirac system of order 2n // Tr. Mosk.
Mat. Obshch. – 1968. – 19. – S. 41 – 112.
6. Guseinov I. M. On the representation of Jost solutions of a system of Dirac differential equations
with discontinuous coefficients // Izv. Akad. Nauk Azerb. SSR. – 1999. – # 5. – S. 41 – 45.
7. Amirov R. Kh. On system of Dirac differential equations with discontinuity conditions inside an
interval // Ukr. Math. J. – 2005. – 57, # 5. – P. 712 – 727.
Received 27.01.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
|
| id | umjimathkievua-article-3088 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:00Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/aa/637f17864c95fb91251927006edeecaa.pdf |
| spelling | umjimathkievua-article-30882020-03-18T19:45:12Z Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition Прямі та обернені задачі для оператора Дірака із спектральним параметром, що лінійно входить до граничної умови Amirov, R. Kh. Keskin, B. Özkan, G. Аміров, Р. Х. Кескін, Б. Озкан, Г. We investigate a problem for the Dirac differential operators in the case where an eigenparameter not only appears in the differential equation but is also linearly contained in a boundary condition. We prove uniqueness theorems for the inverse spectral problem with known collection of eigenvalues and normalizing constants or two spectra. Досліджено задачу для диференціальних операторів Дірака у випадку, коли власний параметр не тільки присутній у диференціальному рівнянні, але й лінійно входить до граничної умови. Доведено теореми єдиності в оберненій спектральній задачі з відомим набором власних значень і нормуючих сталих або двох спектрів. Institute of Mathematics, NAS of Ukraine 2009-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3088 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 9 (2009); 1155-1166 Український математичний журнал; Том 61 № 9 (2009); 1155-1166 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3088/2925 https://umj.imath.kiev.ua/index.php/umj/article/view/3088/2926 Copyright (c) 2009 Amirov R. Kh.; Keskin B.; Özkan G. |
| spellingShingle | Amirov, R. Kh. Keskin, B. Özkan, G. Аміров, Р. Х. Кескін, Б. Озкан, Г. Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition |
| title | Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition |
| title_alt | Прямі та обернені задачі для оператора Дірака із спектральним параметром, що лінійно входить до граничної умови |
| title_full | Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition |
| title_fullStr | Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition |
| title_full_unstemmed | Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition |
| title_short | Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition |
| title_sort | direct and inverse problems for the dirac operator with a spectral parameter linearly contained in a boundary condition |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3088 |
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