On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions
We study properties and the asymptotic behavior of spectral characteristics for a class of singular Sturm-Liouville differential operators with discontinuity conditions and an eigenparameter in boundary conditions. We also determine the Weyl function for this problem and prove uniqueness theorems...
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| Дата: | 2012 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508633819774976 |
|---|---|
| author | Amirov, R. Kh. Güldü, Y. Topsakal, N. Аміров, Р. Х. Гюль, Ю. Топсакал, Н. |
| author_facet | Amirov, R. Kh. Güldü, Y. Topsakal, N. Аміров, Р. Х. Гюль, Ю. Топсакал, Н. |
| author_sort | Amirov, R. Kh. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:32:55Z |
| description | We study properties and the asymptotic behavior of spectral characteristics for a class of singular
Sturm-Liouville differential operators with discontinuity conditions and an eigenparameter in boundary conditions.
We also determine the Weyl function for this problem and prove uniqueness theorems for a solution of the inverse problem corresponding to this function and spectral data. |
| first_indexed | 2026-03-24T02:28:19Z |
| format | Article |
| fulltext |
UDC 517.5
Y. Guldu, R. Kh. Amirov, N. Topsakal (Cumhuriyet Univ., Turkey)
ON IMPULSIVE STURM – LIOUVILLE OPERATORS
WITH SINGULARITY AND SPECTRAL PARAMETER
IN BOUNDARY CONDITIONS
ПРО IМПУЛЬСНI ОПЕРАТОРИ ШТУРМА – ЛIУВIЛЛЯ
IЗ СИНГУЛЯРНIСТЮ ТА СПЕКТРАЛЬНИМ ПАРАМЕТРОМ
У ГРАНИЧНИХ УМОВАХ
We study properties and the asymptotic behavior of spectral characteristics for a class of singular Sturm – Liouville
differential operators with discontinuity conditions and an eigenparameter in boundary conditions. We also determine the
Weyl function for this problem and prove uniqueness theorems for a solution of the inverse problem corresponding to this
function and spectral data.
Дослiджено властивостi та асимптотичну поведiнку спектральних характеристик для класу сингулярних диферен-
цiальних операторiв Штурма – Лiувiлля з розривними умовами та власним параметром у граничних умовах. Визна-
чено функцiю Вейля для цiєї задачi та доведено теореми про єдинiсть розв’язку оберненої задачi, що вiдповiдає цiй
функцiї та спектральним даним.
1. Introduction. In spectral theory, the inverse problem is the usual name for any problem in which
it is required to ascertain the spectral data that will determine a differential operator uniquely and a
method of construction of this operator from the data. This kind of problem was first formulated and
investigated by Ambartsumyan in 1929 [1]. Since 1946, various forms of the inverse problem have
been considered by N. Levinson [2], B. M. Levitan [3], G. Borg [4], and now there exists an extensive
literature on the [5]. Later, the inverse problems having specified singularities were considered in [6].
Spectral functions are important for determining the operators, that is, for solving the inverse
problem for differential operators. However, in finite intervals, the integral representations for the
solution of differential equations which generate the operator with initial conditions are more useful
for investigating the spectral properties of the operator.
In case of q(x) ≡ 0, since this operator is the singular Sturm – Liouville operator, linearly inde-
pendent solutions of this kind of differential equation could be given with hypergeometric functions
and this integral representation is also a representation for hypergeometric functions. For this rea-
son, obtaining this kind of integral representation is so important. Therefore, when it is obtained,
these integral representations can be used for asymptotic behaviours of hypergeometric functions as
x→ +∞.
In interval (a, b), i.e., when the given interval is finite, Sturm – Liouville operator which is gener-
ated by the differential expression `(y) := −y′′(x)+q(x)y(x) satisfies the condition q(x) ∈ L1 (a, b)
in general. In singular case, i.e., when interval (a, b) is infinite or the function q(x) has nonintegrable
singularity in extremity points of interval, the condition of q(x) ∈ L1,loc (a, b) is given.
When q(x) is a first order singular generalized function, singular Sturm – Liouville operator which
has a potential as q = u′ by using concept of generalized derivative such that u ∈ L2 (0, 1) has been
defined in [7, 8].
On the other hand, one-dimensional Schrödinger operators S = −d2/dx2 + q with real-valued
distributional potentials q in W−1
2,loc(R) are studied in [9]. The operator S can then be rigorously
c© Y. GULDU, R. KH. AMİROV, N. TOPSAKAL, 2012
1610 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1611
defined e.g. by the so-called regularization method that was used in [10] in the particular case
q(x) = 1/x and then developed for generic distributional potentials in W−1
2,loc(R) by Savchuk and
Shkalikov [8, 11]; see also recent extensions to more general differential expressions in [12, 13].
Moreover self-adjoint extensions of differential operators generated by differential expression
`(y) which has a potential q(x) = u′(x) such that u ∈ L2 (0, 1) are studied in [8]. When a 6=
6= 2, 4, 6, . . . , generalized functions can be corresponded to the functions |x|−a sgnx by using the
method of canonical regularization [14]. When a <
3
2
, generalized functions which are obtained
by this way can be shown as generalized derivative of functions from the space L2 and therefore
Sturm – Liouville operator which is given by the differential expression `(y) is defined such that
it has a potential like q(x) = |x|−a sgnx. In [15], when q(x) = Cx−a and a <
3
2
, C ∈ R, a
regularization of constructing boundary-value problem for Sturm – Liouville equation which has this
type of potential has been given.
As in this studies of [16] and [17], when q(x) = Cx−a and a ∈ [1, 2), all self-adjoint extensions
of operators generated by the differential expression `(y) which has this type of potential according
to boundary conditions have been given and therefore when a ∈ [1, 2), regularization of constructing
boundary-value problems for Sturm – Liouville equation which has this type of potential has been
investigated. Regularization in the [8] and [16] coincides only when a <
3
2
.
Let’s consider the differential expression
`(y) := −y′′(x) +
C
xa
y(x) + q(x)y(x), 0 < x < π, (1.1)
where C is a real number, q(x) is a real valued bounded function.
We shall define an operator L′0 : L′0y = `(y), on the q set of D′0 = C∞0 (0, π). It is obvious
that the operator L′0 is symmetric in the space of L2[0, π]. We say that the operator L0 which is the
closure of L′0 is the minimal operator generated by the differential expression (1.1). The conjugate L∗0
of the operator L0 is said to be the maximal operator generated by the differential expression (1.1).
In [16], all maximal dissipative and accumulative and also self-adjoint extensions of the operator
L0 have been studied according to the domain and boundary conditions of minimal and maximal
operators generated by differential expression (1.1).
We define Γαy by (Γαy) (x) = y′(x)− u(x)y(x), where u(x) = C
x1−α
1− α
.
It has been shown in [16] that if y(x) ∈ D (L∗0) then the function (Γαy)(x) has a limit as
x→ 0+, i.e.,
lim
x→0+
(Γαy)(x) = (Γαy)(0).
Hence the domain D (L0) of minimal operator L0 generated by differential expression (1.1)
contains only functions y(x) ∈ D (L∗0) such that function y(x) satisfies the conditions y(0) =
= y(π) = (Γαy) (0) = y′(π) = 0.
Let us consider the boundary-value problem L for the equation
`(y) := −y′′(x) +
C
xα
y(x) + q(x)y(x) = λy(x), λ = k2, (1.2)
on the interval 0 < x < π with the boundary conditions
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1612 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
U(y) := y(0) = 0, V (y) :=
(
α1k
2 + α2
)
y(π) +
(
β1k
2 + β2
)
y′(π) = 0 (1.3)
and with the jump conditions
y (a+ 0) = βy (a− 0),
y′ (a+ 0) = β−1y′ (a− 0),
where λ is spectral parameter; α ∈ (1, 3/2) , C, β, α1, α2, β1, β2 are real numbers, α2β1−β2α1 > 0
and a ∈
(π
2
, π
)
, β 6= 1, β > 0, q(x) is a real valued bounded function and q(x) ∈ L2(0, π).
The boundary-value problems that contain the spectral parameter in boundary conditions linearly
were investigated in [18 – 20]. In [18, 21], an operator-theoretic formulation of the problems of the
form (1.2) – (1.4) has been given. Oscillation and comparison results have been obtained in [22 – 24].
In case of α1 6= 0, problem (1.2) – (1.4) is associated with the physical problem of cooling a thin
solid bar one end of which is placed in contact with a finite amount of liquid at time zero (see [18]
and also [25] in it). Assuming that heat flows only into the liquid which has un-uniform density
ρ(x) and is convected only form the liquid into the surrounding medium, the initial boundary-value
prolem for a bar of length one takes the form
ut = ρ(x)uxx, (1*)
ux(0, t) = 0, (2*)
−kAux(π−, t) = qM
(
dv
dt
)
+ k1Bv(t) for all t, (3*)
u(x, 0) = u0(x) for x ∈ [0, π], (4*)
v(0) = v0
after factoring out the steady-state solution where
ρ(x) =
1, 0 < x < a,
α2, a < x < π.
Assuming that the rate of heat transfer across the liquid-solid interface is proportional to the
difference in temperature between the end of the bar and the liquid with which it is in contract
(Newton’s law of cooling) and applying Fourier’s law of heat conduction at x = π, we get
v(t) = u(π, t) + kc−1ux(π−1, t) for t > 0,
where c > 0 is the coefficient of heat transfer for the liquid. If we put u(x, t) = y(x) exp(−λt)
then the problem (1.2) – (1.4) will appear to be consequence of the above problem. Indeed, the
condition (1.3) is obtained from (2∗) and the condition (1.4) is obtained from (3∗) easily. Here
α1 =
c
k
, β2 = −cA+ k1B
qM
and α2 = −k1Bc
qMk
. Finally, if we put
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1613
t =
x, 0 < x < a,
αx, a < x < π,
then the discontinuity conditions (1.4) and a particular case of (1.2) will appear. This corresponds to
the case of nonperfect thermal contact. Since the density is changed at one point in interval, both of
the intensity and the instant velocity of heat change at this point. Hence, (1.2) – (1.4) will appear to
be consequence of the above problem.
Boundary-value problems with discontinuities inside the interval often appear in mathematics,
mechanics, physics, geophysics and other branches of natural properties. The inverse problem of
reconstructing the material properties of a medium from data collected outside of the medium is of
central importance in disiplines ranging from engineering to the geo-sciences.
For example, discontinuous inverse problems appear in electronics for constructing parameters of
heterogeneous electronic lines with desirable technical characteristics [26, 27]. After reducing corre-
sponding mathematical model we come to boundary-value problem L where q(x) must be constructed
from the given spectral information which describes desirable amplitude and phase characteristics.
Spectral information can be used to reconstruct the permittivity and conductivity profiles of one-
dimensional discontinuous medium [28]. Boundary-value problems with discontinuties in an interior
point also appear in geophysical models for oscillations of the Earth [29]. Here, the main discon-
tinuity is cased by reflection of the shear waves at the base of the crust. Further, it is known that
inverse spectral problems play an important role for investigating some nonlinear evolution equa-
tions of mathematical physics. Discontinuous inverse problems help to study the blow-up behaviour
of solutions for such nonlinear equations. We also note that inverse problem considered here appears
in mathematics for investigating spectral properties of some classes of differential, integrodifferential
and integral operators.
It must be noted that some special cases of the considered problem (1.2) – (1.4) arise after an
application of the method of seperation of variables to the varied assortment of physical problems. For
example, some boundary-value problems with discontinuity condition arise in heat and mass transfer
problems (see, for example, [31]), in vibrating string problems when the string loaded additionally
with point masses (see, for example, [25]) and in diffraction problems (see, for example, [30]).
Moreover, some of the problems with boundary conditions depend on the spectral parameter occur
in the theory of small vibrations of a damped string and freezing of the liquid (see, for example,
[32, 33, 25]).
Furthermore, representation with transformation operator was shown in [17], as in [34] and [35].
In this study, properties of characteristic function of L0 and asymptotic behaviours of spectral
characteristics of considering operator have been given such that the remaining parts are in the space
`2 as in [35].
Moreover three statements of the inverse problem of the reconstruction of the boundary problem
from the Weyl function, from the spectral data {λn, αn}n≥0 and from two spectra {λn, µn}n≥0 have
been studied. These inverse problems are generalizations of the well known inverse problems for the
Sturm – Liouville operator (see [36, 37]).
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1614 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
2. Representation for the solution. We define
y1(x) = y(x), y2(x) = (Γαy) (x) = y′(x)− u (x) y(x), u(x) = C
x1−α
1− α
and let’s write the expression of left-hand side of equation (1.1) as follows:
`(y) = − [(Γαy) (x)]′ − u(x) (Γαy) (x)− u2(x)y + q(x)y = k2y. (2.1)
Then equation (1.1) reduces to the system
y′1 − y2 = u(x)y1,
y′2 + k2y1 = −u(x)y2 − u2(x)y1 + q(x)y1
(2.2)
with the boundary conditions
y1(0) = 0,
(
α1k
2 + α2
)
y1(π) +
(
β1k
2 + β2
)
y2(π) = 0 (2.3)
and with the jump conditions
y1 (a+ 0) = βy1 (a− 0),
y2 (a+ 0) = β−1y2 (a− 0)− 2β−u(a)y1 (a− 0).
(2.4)
Matrix form of system (2.2) (
y1
y2
)′
=
(
u 1
−k2 − u2 + q −u
)(
y1
y2
)
(2.5)
or y′ = Ay such that A =
(
u(x) 1
−k2 − u2(x) + q(x) −u(x)
)
, y =
(
y1
y2
)
. Since x = 0 is a regular
singular end point for equation (2.5), Theorem 2 in [38] (see Remark 1.2, p. 56) extends to interval
[0, π]. For this reason, by [38], there exists only one solution of the system (2.2) which satisfies the
initial condititons y1(ξ) = υ1, y2(ξ) = υ2 for each ξ ∈ [0, π], υ = (υ1, υ2)T ∈ C2, especially the
initial conditions y1(0) = 1, y2(0) = h.
Definition 2.1. The first component of the solution of system (2.2) which satisfies the initial
condititons y1(ξ) = υ1, y2(ξ) = (Γαy) (ξ) = υ2 is called the solution of equation (1.2) which
satisfies the same initial conditions.
It was shown in [17] by the successive approximations method that (see [37]) the following
theorem is true.
Theorem 2.1. Each solution of system (2.2) which satisfying the initial conditions
(
y1
y2
)
(0) =
=
(
1
ik
)
and the jump conditions (2.4), has the form:
for x < a
y1 = eikx +
x∫
−x
K11(x, t)eiktdt,
y2 = ikeikx + b(x)eikx +
x∫
−x
K21 (x, t) eiktdt+ ik
x∫
−x
K22(x, t)eiktdt,
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1615
for x > a
y1 = β+eikx + β−eik(2a−x) +
x∫
−x
K11(x, t)eiktdt,
y2 = ikβ+eikx − ikβ−eik(2a−x) + b (x)
[
β+eikx + β−eik(2a−x)
]
+
+
x∫
−x
K21(x, t)eiktdt+ ik
x∫
−x
K22(x, t)eiktdt,
where
b(x) = −1
2
x∫
0
[
u2(s)− q(s)
]
e−
1
2
∫ x
s u(t)dtds, K(x, t) =
(
K11(x, t) 0
K21(x, t) K22(x, t)
)
,
K11 (x, x) =
β+
2
u(x),
K21 (x, x) = b′(x)− 1
2
β+
x∫
0
[
u2 (s)− q(s)
]
K11 (s, s) ds+
x∫
0
u (s)K11 (s, s) ds
,
K22 (x, x) = −β
+
2
u(x)− β+b(x), β± =
1
2
(
β ± 1
β
)
.
3. Properties of the spectrum. In this section, properties of the spectrum of problem L have
been given.
Let us denote problem L as L0 in the case of C = 0 and q(x) ≡ 0.
When C = 0 and q(x) ≡ 0, it is easily shown that solution ϕ0(x, k) satisfying the initial
conditions ϕ0 (0, k) = 0, (Γαϕ0) (0, k) = k and the jump conditions (2.4), is shown as
ϕ0(x, k) =
sin kx for x < a,
β+ sin kx+ β− sin k (2a− x) for x > a,
(Γαϕ0) (x, k) =
k cos kx for x < a,
kβ+ cos kx− kβ− cos k (2a− x) for x > a.
(3.1)
We denote characteristic function, eigenvalues sequence and normalizing constant sequence by
∆(k), {kn} and {αn}, respectively. Denote
∆(k) = 〈ψ(x, k), ϕ(x, k)〉 , (3.2)
where
〈y(x), z(x)〉 := y (x) (Γαz) (x)− (Γαy) (x)z(x).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1616 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
We define normalizing constants by
αn =
π∫
0
ϕ2(x, kn)dx+
1
ρ
[α1ϕ (π, kn) + β1 (Γαϕ) (π, kn)]2 ,
where ρ = α2β1 − α1β2.
According to the Liouville formula, 〈ψ(x, k), ϕ(x, k)〉 does not depend on x.
We shall assume that ϕ(x, k) and ψ(x, k) are solutions of equation (1.2) under the following
initial conditions:
ϕ (0, k) = 0, (Γαϕ) (0, k) = k, ψ (π, k) = β1k
2 + β2, (Γαψ) (π, k) = −(α1k
2 + α2).
Clearly, for each x, functions 〈ψ(x, k), ϕ(x, k)〉 are entire in k and
∆(k) = V (ϕ) = U(ψ) = (α1k
2 + α2)ϕ (π, k) + (β1k
2 + β2) (Γαϕ) (π, k) = ψ(0, k). (3.3)
By using the representation of the function y(x, k) for the solution ϕ(x, k):
ϕ(x, k) = ϕ0(x, k) +
π∫
0
K̃11 (π, t) sin ktdt (3.4)
is obtained.
Lemma 3.1 (Lagrange’s formula). Let y, z ∈ D (L∗0). Then
(L∗0y, z) =
π∫
0
`(y)zdx = (y, L∗0z) + [y, z]
(∣∣∣a−0
0
+
∣∣∣π
a+0
)
,
where [y, z]
(∣∣a−0
0
+
∣∣π
a+0
)
=
[
(Γαz) (x)y(x)− (Γαy) (x)z(x)
](∣∣a−0
0
+
∣∣π
a+0
)
.
Proof. We have
(L∗0y, z) = −
π∫
0
(
y′ − u y
)′
zdx−
π∫
0
u
(
y′ − u y
)
zdx−
π∫
0
(
u2 − q(x)
)
yzdx =
=
π∫
0
(
y′ − u y
) (
z′ − uz
)
dx−
π∫
0
(
u2 − q(x)
)
yzdx− (Γαy) (x)z(x)
(∣∣∣a−0
0
+
∣∣∣π
a+0
)
=
=
π∫
0
y` (z) dx+ [y, z]
(∣∣∣a−0
0
+
∣∣∣π
a+0
)
= (y, L∗0z) + [y, z]
(∣∣∣a−0
0
+
∣∣∣π
a+0
)
.
Lemma 3.2. The zeros {kn} of the characteristic function coincide with the eigenvalues of the
boundary-value problem L. The functions ϕ(x, kn) and ψ(x, kn) are eigenfunctions and there exists
a sequence {γn} such that
ψ(x, kn) = γnϕ(x, kn), γn 6= 0. (3.5)
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1617
Proof. 1. Let k0 be a zero of the function ∆(k). Then by virtue of equation (3.2) and (3.3),
ψ (x, k0) = γ0ϕ (x, k0) and the functions ψ (x, k0) , ϕ (x, k0) satisfy the boundary conditions (1.3).
Hence, k0 is an eigenvalue and ψ (x, k0) , ϕ (x, k0) are eigenfunctions related to k0.
2. Let k0 be an eigenvalue of L, and let y0 be a corresponding eigenfunctions. Then U (y0) =
= V (y0) = 0. Clearly y0(0) = 0. Without loss of generality, we put (Γαy0) (0) = ik. Hence
y0(x) ≡ ϕ (x, k0). Thus, from equation (3.3), ∆ (k0) = V (ϕ (x, k0)) = V (y0(x)) = 0 is obtained.
Lemma 3.3. Eigenvalues of the problem L are simple and separated.
Proof. Since ϕ(x, k) and ψ(x, k) are solutions of equation (1.2), it is obtained that
−ψ′′(x, k) +
[
u′(x) + q(x)
]
ψ(x, k) = k2ψ (x, k),
−ϕ′′(x, kn) +
[
u′ (x) + q(x)
]
ϕ(x, kn) = k2
nϕ(x, kn).
If first equation is multiplied by ϕ(x, kn), second equation is multiplied by ψ(x, k) and substracting
them side by side and finally integrating over the interval [0, π], then the following equality is
obtained:
d
dx
〈ψ(x, k), ϕ (x, kn)〉 =
(
k2 − k2
n
)
ψ(x, k)ϕ(x, kn),
〈ψ(x, k), ϕ(x, kn)〉
[∣∣∣a−0
0
+
∣∣∣π
a+0
]
=
(
k2 − k2
n
) π∫
0
ψ(x, k)ϕ(x, kn)dx.
If jump conditions (1.4) and αn =
π∫
0
ϕ2(x, kn)dx +
1
ρ
[α1ϕ (π, kn) + β1 (Γαϕ) (π, kn)]2 are
considered, it is obtained that
π∫
0
ψ(x, kn)ϕ (x, kn) dx+
1
ρ
[α1ϕ (π, kn) + β1 (Γαϕ) (π, kn)] ,
[
α1ψ (π, kn) + β1 (Γαψ) (π, kn)
]
= −
.
∆ (kn) as k → kn.
From Lemma 3.2, we get that
αnγn = −
.
∆ (kn) . (3.6)
It is obvious that
.
∆ (kn) 6= 0.
Since the function ∆(k) is an entire function ok k, the zeros of ∆(k) are separated.
Lemma 3.3 is proved.
Now, let problems be
L :
−y′′ + [u′(x) + q(x)] y = λy,
(Γαy) (0)− hy(0) = 0,
(β1λ+ β2) (Γαy) (π) + (α1λ+ α2)y(π) = 0,y (a+ 0) = βy (a− 0) ,
(Γαy) (a+ 0) = β−1 (Γαy) (a− 0)− 2β−u(a)y(a− 0),
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1618 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
L̃ :
−y′′ + [u′(x) + q(x)] y = µy,
(Γαy) (0)− hy(0) = 0,
(β̃1µ+ β̃2) (Γαy) (π) + (α̃1µ+ α̃2)y(π) = 0,y (a+ 0) = βy (a− 0),
(Γαy) (a+ 0) = β−1 (Γαy) (a− 0),
where α1β̃1 = α̃2β2, α1β̃2 = α̃2β1, α2β̃1 = α̃1β2. Let {λn}n≥0 and {µn}n≥0 be the eigenvalues of
the problems L and L̃ respectively.
Lemma 3.4. The eigenvalues of the problems L and L̃ are interlace, i.e.,
λn < µn < λn+1, if α2β̃2 < α̃2β2 and µn < λn < µn+1, if α2β̃2 > α̃2β2, n ≥ 0,
(3.7)
where α1α̃2 > α̃1α2 and β1β̃2 > β̃1β2.
Proof. As in the proof of Lemma 3.3, we get that
d
dx
〈ϕ (x, λ) , ϕ (x, µ)〉 = (λ− µ)ϕ (x, λ)ϕ (x, µ)
and from here
(λ− µ)
π∫
0
ϕ (x, λ)ϕ (x, µ) dx = 〈ϕ (x, λ) , ϕ (x, µ)〉
[∣∣∣a−0
0
+
∣∣∣π
a+0
]
=
= ϕ (π, λ) (Γαϕ) (π, µ)− (Γαϕ) (π, λ)ϕ (π, µ) =
=
α̃1α2 − α1α̃2
α2β̃2 − α̃2β2
(λ− µ)ϕ(π, λ)ϕ(π, µ)+
+
β̃1β2 − β1β̃2
α2β̃2 − α̃2β2
(λ− µ)(Γαϕ)(π, λ)(Γαϕ)(π, µ)+
+
1
α2β̃2 − α̃2β2
[
∆̃ (λ) ∆ (µ)− ∆̃ (µ) ∆ (λ)
]
.
Hence
(λ− µ)
π∫
0
ϕ (x, λ)ϕ (x, µ) dx =
α̃1α2 − α1α̃2
α2β̃2 − α̃2β2
(λ− µ)ϕ(π, λ)ϕ(π, µ)+
+
β̃1β2 − β1β̃2
α2β̃2 − α̃2β2
(λ− µ)(Γαϕ)(π, λ)(Γαϕ)(π, µ)+
+
1
α2β̃2 − α̃2β2
[
∆̃ (λ)− ∆̃ (µ)
λ− µ
∆ (µ)− ∆ (λ)−∆ (µ)
λ− µ
∆̃ (µ)
]
.
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1619
As µ→ λ
π∫
0
ϕ2 (x, λ) dx =
1
α2β̃2 − α̃2β2
×
×
[
(α̃1α2 − α1α̃2)ϕ2 (π, λ) +
(
β̃1β2 − β1β̃2
)
(Γϕ)2 (π, λ) +
+
.
∆̃ (λ) ∆ (λ)−
.
∆ (λ) ∆̃ (λ)
]
, (3.8)
where
.
∆ (λ) =
d
dλ
∆ (λ),
.
∆̃ (λ) =
d
dλ
∆̃ (λ) . From equation (3.8), for −∞ < λ <∞, if ∆̃ (λ) 6= 0,
1
∆̃2 (λ)
π∫
0
ϕ2 (x, λ) dx− (α̃1α2 − α1α̃2)ϕ2 (π, λ) + (β̃1β2 − β1β̃2)(Γαϕ)2(π, λ)
α̃2β2 − α2β̃2
=
= − 1
(α̃2β2 − α2β̃2)
d
dλ
(
∆ (λ)
∆̃ (λ)
)
is obtained.
If α2β̃2 < α̃2β2 then
∆ (λ)
∆̃ (λ)
is monotonically decreasing in the set of R \ {µn, n ≥ 0}. Thus it is
obvious that lim
λ→µ±0
n
∆ (λ)
∆̃ (λ)
= ±∞ .
When α2β̃2 > α̃2β2, if we write the equality (3.8) as
1
∆2 (λ)
π∫
0
ϕ2 (x, λ) dx− (α̃1α2 − α1α̃2)ϕ2 (π, λ) + (β̃1β2 − β1β̃2)(Γαϕ)2(π, λ)
α2β̃2 − α̃2β2
=
= − 1
α2β̃2 − α̃2β2
d
dλ
(
∆̃ (λ)
∆ (λ)
)
,
for −∞ < λ < ∞, ∆ (λ) 6= 0, we get that the function
∆̃ (λ)
∆ (λ)
is monotonically decreasing in
R \ {λn, n ≥ 0} and it is clear that lim
λ→λ±0
n
∆̃ (λ)
∆ (λ)
= ±∞. From here, we obtain (3.7).
Theorem 3.1. The eigenvalues kn, eigenfunctions ϕ(x, kn) and the normalizing numbers αn
of problem L have the following asymptotic behaviour:√
λn = kn = k0
n +
dn
k0
n
+
δn
k0
n
, (3.9)
ϕ(x, kn) = β+ sin k0
nx+ β− sin k0
n(2a− x) +
sn
k0
n
+
bn
k0
n
, (3.10)
αn =
a
2
+
[(
β+
)2
+
(
β−
)2](π − a
2
)
−
(
β+
) (
β−
)
cos 2k0
na+
γn
n
+
ξn
n
, (3.11)
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1620 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
where δn, sn, ξn ∈ `2, bn, dn, γn ∈ `∞ and k0
n are roots of ∆0(k) := k3
[
β+ cos kπ− β− cos k(2a−
− π)
]
and k0
n = n+ hn, hn ∈ `∞.
Proof. Using (3.1), (3.3) and (3.4), we get
∆(k) = (α1k
2 + α2)ϕ0 (π, k) + (β1k
2 + β2) (Γαϕ0) (π, k) +
+(α1k
2 + α2)
π∫
0
K̃11 (π, t) sin ktdt+
+(β1k
2 + β2)
π∫
0
K̃21 (π, t) sin ktdt+
π∫
0
K̃22 (π, t) cos ktdt
=
= (α1k
2 + α2)
(
β+ sin kπ + β− sin k (2a− π)
)
+
+(β1k
2 + β2)
(
kβ+ cos kπ − kβ− cos k (2a− π)
)
+ k3O
(
exp |Im k|π
|k|
)
=
= β1∆0(k) + (α1k
2 + α2)
(
β+ sin kπ + β− sin k (2a− π)
)
+
+β2k
(
β+ cos kπ − β− cos k (2a− π)
)
+ k3O
(
exp |Im k|π
|k|
)
.
Denote
Gn =
{
k : |k| =
∣∣k0
n
∣∣+
σ
2
, n = 0,±1,±2, . . .
}
,
Gδ =
{
k :
∣∣k − k0
n
∣∣ ≥ δ, n = 0,±1,±2, . . . , δ > 0
}
,
where δ is sufficiently small positive number
(
δ � σ
2
)
.
Since |∆0(k)| ≥ k3Cδe
|Im k|π for k ∈ Gδ and |∆(k)−∆0(k)| < Cδ
2
|k|3 e|Im k|π for sufficiently
large values of n and k ∈ Gn, we get
|∆0(k)| > Cδk
3e|Im k|π > |∆ (k)−∆0(k)|.
It follows from that for sufficiently large values of n, functions ∆0(k) and ∆0(k) +
(
∆(k) −
−∆0(k)
)
= ∆(k) have the same number of zeros counting multiplicities inside contour Gn, accord-
ing to Rouche’s theorem. That is, they have the (n+ 1) number of zeros: k0, k1, . . . , kn.
Analogously, it is shown by Rouche’s theorem that for sufficiently large values of n, function
∆(k) has a unique of zero inside each circle
∣∣k − k0
n
∣∣ < δ.
Since δ is sufficiently small number, representing of kn = k0
n + εn is acquired, where
limn→∞ εn = 0.
Since numbers kn are zeros of characteristic function ∆(k),
∆ (kn) = (α1k
2
n + α2)
(
β+ sin knπ + β− sin kn (2a− π)
)
+
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1621
+(β1k
2 + β2)
(
knβ
+ cos knπ − knβ− cos kn (2a− π)
)
+O
(
k2
n
)
.
From the last equality, we get
β+ cos knπ − β− cos kn (2a− π) +
α1
β1k2
n
[
β+ sin knπ + β− sin kn (2a− π)
]
+
+
α2
β1k3
n
[
β+ sin knπ + β− sin kn (2a− π)
]
+
+
β2
β1k2
n
[β+ sin knπ − β− cos kn (2a− π)] +O
(
1
kn
)
= 0.
If we write k0
n+εn instead of kn and use ∆0
(
k0
n + εn
)
=
.
∆0
(
k0
n
)
εn+o (εn) and also the study
[39] ( see also [40]) is used then we get that k0
n = n+ hn, where supn |hn| < M. Therefore,
εn =
dn
n
+
δn
n
, δn ∈ `2, dn ∈ `∞.
Thus, asymptotic formula (3.9) is true for the eigenvalues kn of the problem L. Now, let’s try to find
the asymptotic formula for the eigenfunction
ϕ(x, kn) = β+ sin knx+ β− sin kn (2a− x) +
x∫
0
K̃11 (x, t) sin knt dt =
= β+ sin
(
k0
n + εn
)
x+ β− sin
(
k0
n + εn
)
(2a− x)−
− 1
k0
n + εn
x∫
0
K̃11(x, t)d
[
cos
(
k0
n + εn
)
t
]
dt =
= β+ sin k0
nx+ β− sin k0
n (2a− x)−
− 1
k0
n + εn
[
K̃11(x, t) cos k0
nt
](∣∣∣2a−x−0
0
+
∣∣∣x
2a−x+0
)
+
+
1
k0
n + εn
x∫
0
K̃ ′11t(x, t) cos k0
nt dt.
Since
K̃11 (x, x) =
β+
2
u(x), K̃11 (x, 2a− x+ 0)− K̃11 (x, 2a− x− 0) =
β−
2
u(x),
x∫
0
K̃ ′11t (x, t) cos k0
ntdt ∈ `2.
It is obtained that
ϕ(x, kn) = β+ sin k0
nx+ β− sin k0
n (2a− x) +
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1622 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
+
β− cos k0
n(2a− x)− β+ cos k0
nx
2k0
n
u(x) +
bn
n
+
sn
n
, sn ∈ `2 and bn ∈ `∞.
Then we get the asymptotic formula (3.10). Finally, in order to show that (3.11) is true, using
(3.1) and (3.4), we obtain that
αn =
π∫
0
ϕ2(x, kn)dx+
1
ρ
[α1ϕ (π, kn) + β1 (Γαϕ) (π, kn)]2 =
=
a∫
0
sin2 knxdx+
x∫
0
K̃11(x, t) sin kntdt
2 +
+2
a∫
0
sin knx
x∫
0
K̃11(x, t) sin kntdtdx+
+
π∫
a
(β+
)2
sin2 knx+
(
β−
)2
sin2 kn (2a− x) +
π∫
a
x∫
a
K̃11 (x, t) sin kntdt
2 dx+
+2β+β−
π∫
a
sin knx sin kn(2a− x)dx+ 2β+
π∫
a
sin knx
x∫
a
K̃11(x, t) sin kntdtdx+
+2β−
π∫
a
sin kn (2a− x)
x∫
a
K̃11(x, t) sin kntdtdx+
1
ρ
[α1ϕ (π, kn) + β1 (Γαϕ) (π, kn)]2 =
=
[(
β+
)2
+
(
β−
)2](π − a
2
)
+
a
2
− β+β− cos 2kna+
γn
n
+
ξn
n
, γn ∈ `∞, ξn ∈ `2.
4. Inverse problem. Let Φ(x, k) be solution of (1.3) under the conditions
U (Φ) = Φ(0, k) = 1, V (Φ) =
(
α1k
2 + α2
)
Φ (π, k) +
(
β1k
2 + β2
)
(ΓαΦ)(π, k) = 0
and the jump conditions (1.5). C(x, k) be solution of (2.2) with the conditions C (0, k) = 1,
(ΓαC)(0, k) = 0 and the jump conditions (2.4). It is clear that the functions ψ(x, k) and C(x, k) are
entire in k. Then the function ψ(x, k) can be represented as follows:
ψ(x, k) =
1
k
(Γαψ) (0, k)ϕ(x, k) + ∆(k)C (x, k)
or
1
∆(k)
ψ(x, k) =
(Γαψ) (0, k)
k∆(k)
ϕ(x, k) + C(x, k). (4.1)
Denote
M(k) :=
(Γαψ) (0, k)
k∆(k)
. (4.2)
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1623
It is clear that
Φ(x, k) = M(k)ϕ(x, k) + C (x, k). (4.3)
The function Φ(x, k) is called the Weyl solution and the function M(k) is called the Weyl function
for the boundary-value problem L.
The Weyl solution and Weyl function are meromorphic functions with respect to k having poles
in the spectrum of the problem L.
It follows from (4.1) and (4.2) that
Φ(x, k) =
ψ(x, k)
∆ (k)
and (ΓαΦ) (0, k) =
(Γαψ) (0, k)
k∆(k)
= M(k). (4.4)
Note that by virtue of equalities 〈C(x, k), ϕ(x, k)〉 ≡ 1, (4.2) and (4.3) we have
〈Φ(x, k), ϕ(x, k)〉 ≡ k, 〈ψ(x, k), ϕ (x, k)〉 ≡ k∆(k). (4.5)
Theorem 4.1. The following representation holds:
M(k) =
1
α0 (k − k0)
+
∞∑
n=1
{
1
αn (k − kn)
+
1
α0
nk
0
n
}
. (4.6)
Proof. Let’s write a representation solution ψ(x, k) = −(β1k
2+β2)C(x, k)+(α1k
2+α2)S(x, k)
as ϕ (x, k) :
for x > a
ψ(x, k) = −
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x) +
+
π−x∫
0
Ñ11(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt,
(Γαψ) (x, k) = −k
[(
β1k
2 + β2
)
sin k (π − x) +
(
α1k
2 + α2
)
cos k (π − x)
]
−
−b(x)
[(
β1k
2 + β2
)
cos k (π − x)−
(
α1k
2 + α2
)
sin k (π − x)
]
+
+
π−x∫
0
Ñ21(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt+
+
π−x∫
0
kÑ22(x, t)
[(
β1k
2 + β2
)
sin kt+
(
α1k
2 + α2
)
cos kt
]
dt,
for x < a
ψ(x, k) = β+
[
−
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x)
]
+
+β−
[
−
(
β1k
2 + β2
)
cos k (π − 2a+ x) +
(
α1k
2 + α2
)
sin k (π − 2a+ x)
]
+
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1624 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
+
π−x∫
0
Ñ11(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt,
(Γαψ) (x, k) = −kβ+
[(
β1k
2 + β2
)
sin k (π − x) +
(
α1k
2 + α2
)
cos k (π − x)
]
+
+kβ−
[(
β1k
2 + β2
)
cos k (π − 2a+ x)−
(
α1k
2 + α2
)
sin k (π − 2a+ x)
]
+
+b(x)β+
[
−
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x)
]
+
+b(x)β−
[(
β1k
2 + β2
)
cos k (π − 2a+ x)−
(
α1k
2 + α2
)
sin k (π − 2a+ x)
]
+
+
π−x∫
0
Ñ21 (x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt+
+
π−x∫
0
kÑ22 (x, t)
[(
β1k
2 + β2
)
sin kt+
(
α1k
2 + α2
)
cos kt
]
dt,
where Ñij(x, t) = Nij(x, t) − Nij (x,−t), i, j = 1, 2. In the case of C = 0 and q(x) ≡ 0, denote
the solutions with ψ01(x, k) and ψ02(x, k), so we have
ψ(x, k) = Ψ01(x, k) + f1,
(Γαψ)(x, k) = (ΓαΨ02)(x, k) + f2,
where
f1 =
π−x∫
0
Ñ11(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt,
f2 = b(x)
[
β+
[
−
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x)
]
+
+β−
[
−
(
β1k
2 + β2
)
cos k (π − 2a+ x) +
(
α1k
2 + α2
)
sin k (π − 2a+ x)
]]+
+
π−x∫
0
Ñ21(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt+
+
π−x∫
0
kÑ22 (x, t)
[(
β1k
2 + β2
)
sin kt+
(
α1k
2 + α2
)
cos kt
]
dt.
On the other hand, we can write
M(k)−M0(k) =
(Γαψ) (0, k)
kψ (0, k)
− (Γαψ0) (0, k)
kψ0 (0, k)
=
f2
k∆(k)
− f1
∆(k)
M0(k).
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1625
Since lim|k|→∞ e
−|Im k|π |fi(k)| = 0 and ∆(k) > Cδe
|Im k|π for k ∈ Gδ, the equality
f2
k∆(k)
− f1
∆(k)
M0(k)
yields
lim sup
|k|→∞ k∈Gδ
|M(k)−M0(k)| = 0. (4.7)
Weyl function M(k) is meromorphic with respect to poles kn. Using (3.3), (4.1) and Lemma 3.2,
we calculate that
Re s
k=kn
M(k) =
(Γαψ) (0, kn)
kn
.
∆ (kn)
= − 1
αn
,
Re s
k=k0n
M0(k) =
(Γαψ0)
(
0, k0
n
)
k0
n
.
∆ (k0
n)
= − 1
α0
n
.
(4.8)
Consider the contour integral
In(k) =
1
2πi
∫
Γn
M (µ)−M0 (µ)
k − µ
dµ, k ∈ int Γn.
By virtue of (4.7) , we have limn→∞ In(k) = 0. On the other hand, the residue theorem and (4.8)
yield
In(k) = −M(k) +M0(k) +
∑
kn∈intΓn
1
αn (k − kn)
−
∑
k0n∈intΓn
1
α0
n (k − k0
n)
.
Therefore, as n→∞, we get
M(k) = M0(k) +
n=+∞∑
n=−∞
1
αn (k − kn)
+
n=+∞∑
n=−∞
1
α0
n (k − k0
n)
.
It follows from the form of the function M0(k) that
M0(k) =
1
α0
nk
+
∞∑
n=−∞
1
α0
n
(
1
k − k0
n
+
1
k0
n
)
.
The composition of the last two equalities yields (4.6).
Theorem 4.1 is proved.
Let us formulate a theorem on the uniqueness of a solution of the inverse problem with the Weyl
function. For this purpose, parallel with L, we consider the boundary-value problem L̃ of the same
form but with different potential q̃(x). It is asumed in what follows that if a certain symbol α denotes
an object related to the problem L, then α̃ denotes the corresponding object related to the problem L̃.
Theorem 4.2. If M(k) = M̃(k) then L = L̃. Thus the specification of the Weyl function
uniquely determines the operator.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1626 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
Proof. Let us define the matrix P (x, k) = [Pjk(x, k)]j,k=1,2 by the formula
P (x, k)
(
ϕ̃ Φ̃
Γαϕ̃ ΓαΦ̃
)
=
(
ϕ Φ
Γαϕ ΓαΦ
)
. (4.9)
Using (4.9) and (4.5) we calculate
P11(x, k) = −1
k
[
ϕ(x, k)
(
ΓαΦ̃
)
(x, k)− Φ(x, k) (Γαϕ̃) (x, k)
]
,
P12(x, k) = −1
k
[
Φ(x, k)ϕ̃(x, k)− ϕ(x, k)Φ̃(x, k)
]
,
(4.10)
P21(x, k) = −1
k
[
(Γαϕ) (x, k)
(
ΓαΦ̃
)
(x, k)− (ΓαΦ) (x, k) (Γαϕ̃) (x, k)
]
,
P22(x, k) = −1
k
[
(ΓαΦ) (x, k)ϕ̃(x, k)− (Γαϕ) (x, k)Φ̃(x, k)
]
and
ϕ(x, k) = P11(x, k)ϕ̃(x, k) + P12(x, k) (Γαϕ̃) (x, k),
(Γαϕ) (x, k) = P21(x, k)ϕ̃(x, k) + P22(x, k) (Γαϕ̃) (x, k),
(4.11)
Φ(x, k) = P11(x, k)Φ̃(x, k) + P12(x, k)
(
ΓαΦ̃
)
(x, k),
(ΓαΦ) (x, k) = P21(x, k)Φ̃(x, k) + P22(x, k)
(
ΓαΦ̃
)
(x, k).
It follows from (4.10), (4.2) and (4.5)
P11(x, k) = 1 +
1
∆(k)
[
ϕ(x, k)
((
ΓαΨ̃
)
(x, k)− (ΓαΨ) (x, k)
)
−
−Ψ(x, k)
(
(Γαϕ̃) (x, k)−
(
Γαϕ
)
(x, k)
)]
,
P12(x, k) =
1
k∆(k)
[
Ψ(x, k)ϕ̃(x, k)− ϕ(x, k)Ψ̃(x, k)
]
,
P21(x, k) =
1
k∆(k)
[
(Γαϕ) (x, k)
(
ΓαΨ̃
)
(x, k)− (ΓαΨ) (x, k) (Γαϕ̃) (x, k)
]
,
P22(x, k) = 1 +
1
k∆(k)
[
(ΓαΨ) (x, k) (ϕ̃(x, k)− ϕ(x, k))− (Γαϕ) (x, k)
(
Ψ̃(x, k)−Ψ(x, k)
)]
.
With respect to (4.10) and (4.2), for each fixed x, the functions Pjk(x, k) are meromorphic in k
with poles in the points kn and k̃n. It follows from the representations of the solutions Ψ(x, k) and
ϕ(x, k) that
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH SINGULARITY AND SPECTRAL PARAMETER . . . 1627
lim
k→∞
k∈Gδ
max
0≤x≤π
|P11(x, k)− 1| = lim
k→∞
k∈Gδ
max
0≤x≤π
|P12 (x, k)| =
= lim
k→∞
k∈Gδ
max
0≤x≤π
|P22(x, k)− 1| = lim
k→∞
k∈Gδ
max
0≤x≤π
|P21 (x, k)| = 0. (4.12)
According to (4.2), (4.3) we have
P11(x, k) = −1
k
[
ϕ(x, k)
(
ΓαC̃
)
(x, k)− C(x, k) (Γαϕ̃) (x, k)+
+
(
M̃(k)−M(k)
)
ϕ(x, k) (Γαϕ̃) (x, k)
]
,
P12(x, k) = −1
k
[
ϕ̃(x, k)C(x, k)− C̃(x, k)ϕ(x, k) +
(
M(k)− M̃(k)
)
ϕ(x, k)ϕ̃(x, k)
]
,
(4.13)
P21(x, k) = −1
k
[
(Γαϕ) (x, k)
(
ΓαC̃
)
(x, k)− (ΓαC) (x, k) (Γαϕ̃) (x, k)
]
−
−1
k
[(
M̃(k)−M(k)
)
(Γαϕ) (x, k) (Γαϕ̃) (x, k)
]
,
P22(x, k) = −1
k
[
ϕ̃(x, k) (ΓαC) (x, k)− C̃(x, k) (Γαϕ) (x, k)+
+
(
M(k)− M̃(k)
)
(Γαϕ) (x, k)ϕ̃(x, k)
]
.
Thus if M(k) = M̃(k) then the functions Pjk(x, k) are entire in k for each fixed x. Together
with (4.12) we get that
P11(x, k) ≡ 1, P12(x, k) ≡ 0, P21(x, k) ≡ 0, P22(x, k) ≡ 1.
Substituting into (4.11), we get
ϕ(x, k) ≡ ϕ̃(x, k), (Γαϕ)(x, k) ≡ (Γαϕ̃ )(x, k),
Φ(x, k) ≡ Φ̃(x, k), (ΓαΦ)(x, k) ≡
(
ΓαΦ̃
)
(x, k)
for all x and k. Consequently L = L̃.
Theorem 4.2 is proved.
Theorem 4.3. If kn = k̃n, αn = α̃n, n ≥ 0, then L = L̃. Thus, the specification of the spectral
data {kn, αn}n≥0 uniquely determines the operator.
Proof. Since
M(k) =
1
α0 (k − k0)
+
∞∑
n=1
′
{
1
αn (k − kn)
+
1
α0
nk
0
n
}
,
M̃(k) =
h̃
α̃0
(
k̃ − k̃0
) +
∞∑
n=1
′
1
α̃n
(
k̃ − k̃n
) +
1
α̃0
nk̃
0
n
(4.14)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1628 Y. GULDU, R. KH. AMİROV, N. TOPSAKAL
under the hypothesis of the theorem and in view of (4.13), we get that M(k) = M̃(k) and conse-
quently by Theorem 4.2, L = L̃ .
Theorem 4.4. If kn = k̃n, µn = µ̃n, n ≥ 0, then L = L̃ .
Proof. In view of properties of functions∆(k) and ∆̃(k), it is clear that limk→∞
∆(k)
∆̃(k)
= 1.
Under the hypothesis kn = k̃n , ∆(k) and ∆̃(k) functions are entire we get that ∆(k) = ∆̃(k).
From Lemma 3.2, we have ψ̃
(
x, k̃n
)
= γ̃nϕ̃
(
x, k̃n
)
= γ̃nϕ̃(x, kn) and Ψ̃
(
x, k̃n
)
= Ψ̃(x, kn) =
= γnϕ̃(x, kn). It follows that γn = γ̃n and so αn = α̃n. Consequently by Theorem 4.3, L = L̃.
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Received 14.09.11,
after revision — 06.11.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
|
| id | umjimathkievua-article-2686 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:19Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f3/401fb42a247dbdbe763fd532e35ccdf3.pdf |
| spelling | umjimathkievua-article-26862020-03-18T19:32:55Z On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions Про iмпульснi оператори Штурма – Лiувiлля iз сингулярнiстю та спектральним параметром у граничних умовах Amirov, R. Kh. Güldü, Y. Topsakal, N. Аміров, Р. Х. Гюль, Ю. Топсакал, Н. We study properties and the asymptotic behavior of spectral characteristics for a class of singular Sturm-Liouville differential operators with discontinuity conditions and an eigenparameter in boundary conditions. We also determine the Weyl function for this problem and prove uniqueness theorems for a solution of the inverse problem corresponding to this function and spectral data. Дослiджено властивостi та асимптотичну поведiнку спектральних характеристик для класу сингулярних диференцiальних операторiв Штурма – Лiувiлля з розривними умовами та власним параметром у граничних умовах. Визначено функцiю Вейля для цiєї задачi та доведено теореми про єдинiсть розв’язку оберненої задачi, що вiдповiдає цiй функцiї та спектральним даним. Institute of Mathematics, NAS of Ukraine 2012-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2686 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 12 (2012); 1610-1629 Український математичний журнал; Том 64 № 12 (2012); 1610-1629 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2686/2130 https://umj.imath.kiev.ua/index.php/umj/article/view/2686/2131 Copyright (c) 2012 Amirov R. Kh.; Güldü Y.; Topsakal N. |
| spellingShingle | Amirov, R. Kh. Güldü, Y. Topsakal, N. Аміров, Р. Х. Гюль, Ю. Топсакал, Н. On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions |
| title | On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions |
| title_alt | Про iмпульснi оператори Штурма – Лiувiлля iз сингулярнiстю та спектральним параметром у граничних умовах |
| title_full | On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions |
| title_fullStr | On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions |
| title_full_unstemmed | On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions |
| title_short | On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions |
| title_sort | on impulsive sturm - liouville operators with singularity and spectral parameter in boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2686 |
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