On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions
The Sturm–Liouville problem with linear discontinuities is investigated in the case where an eigenparameter appears not only in a differential equation but also in boundary conditions. Properties and the asymptotic behavior of spectral characteristics are studied for the Sturm–Liouville operators wi...
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| 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. |
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| description | The Sturm–Liouville problem with linear discontinuities is investigated in the case where an eigenparameter appears not only in a differential equation but also in boundary conditions. Properties and the asymptotic behavior of spectral characteristics are studied for the Sturm–Liouville operators with Coulomb potential that have discontinuity conditions inside a finite interval. Moreover, the Weyl function for this problem is defined and uniqueness theorems are proved for a solution of the inverse problem with respect to this function. |
| first_indexed | 2026-03-24T02:33:20Z |
| format | Article |
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УДК 512.662.5
R. Kh. Amirov, N. Topsakal, Y. Güldü (Cumhuriyet University, Turkey)
ON IMPULSIVE STURM – LIOUVILLE OPERATORS
WITH COULOMB POTENTIAL AND SPECTRAL PARAMETER
LINEARLY CONTAINED IN BOUNDARY CONDITIONS
ON IMPULSIVE STURM – LIOUVILLE OPERATORS
WITH COULOMB POTENTIAL AND SPECTRAL PARAMETER
LINEARLY CONTAINED IN BOUNDARY CONDITIONS
The Sturm – Liouville problem with linear discontinuities is investigated in the case where an eigenparameter
appears not only in the differential equation but also in the boundary conditions. Properties and asymptotic
behaviours of spectral characteristic are studied for the Sturm – Liouville operators with the Coulomb potential
which have discontinuity conditions inside a finite interval. Moreover, the Weyl function for this problem under
consideration is defined and uniqueness theorems for solution of inverse problem according to this function
are proved.
In this study, Sturm – Liouville problem with discontinuities linearly is investigated when an eigenparameter
appears not only in the differential equation but it also appears in the boundary conditions. Properties and
asymptotic behaviours of spectral characteristic are studied for Sturm – Liouville operators with Coulomb
potential which have discontinuity conditions inside a finite interval. Also Weyl function for this problem
under consideration has been defined and uniqueness theorems for solution of inverse problem according to
this function have been proved.
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 [7]. Since 1946, various forms of the inverse problem have been considered by
numerous authors — G. Borg [15], N. Levinson [8], B. M. Levitan [9], etc. and now
there exists an extensive literature on the [10 – 14]. Later, the inverse problems having
specified singularities were considered by a number of authors [18 – 20].
We consider the boundary-value problem L for the equation:
` (y) := −y′′ + C
x
y + q(x)y = k2y (1.1)
on the interval 0 < x < π with the boundary conditions
U (y) := y(0) = 0, V (y) :=
(
α1k
2 + α2
)
y (π) +
(
β1k
2 + β2
)
y′ (π) = 0 (1.2)
and with the jump conditions
y (d+ 0) = αy (d− 0) ,
y′ (d+ 0) = α−1y′ (d− 0) ,
(1.3)
where k is spectral parameter; C, α, α1, α2, β1, β2 ∈ R, α2β1 − β2α1 > 0, α 6= 1,
α > 0, d ∈
(π
2
, π
)
, q(x) is a real valued bounded function and q(x) ∈ L2(0, π).
c© R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9 1155
1156 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
In case of q(x) ≡ 0, since this operator is the singular Sturm – Liouville operator with
Coulomb potential, linearly independent solutions of this kind of differential equation
could be given with hypergeometric functions and this integral representation is also a
representation for hypergeometric functions.
The boundary-value problems that contain the spectral parameter in boundary conditi-
ons linearly were investigated in [30 – 32]. In [30, 41], an operator-theoretic formulation
of the problems of the form (1.1) – (1.3) has been given. Oscillation and comparison
results have been obtained in [33 – 35]. In case of α1 6= 0, problem (1.1) – (1.3) is associ-
ated 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 [30] and also [37] 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 problem
for a bar of length one takes the form
ut = ρ(x)uxx, (1.4)
ux (0, t) = 0, (1.5)
−kAux
(
π−, t
)
= qM (dv/dt) + k1Bv (t) for all t, (1.6)
u (x, 0) = u0(x) for x ∈ [0, π], (1.7)
v(0) = v0
after factoring out the steady-state solution, where
ρ(x) =
1, 0 < x < d,
α2, d < 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 (1.1) – (1.3) will appear to be consequence of the
above problem. Indeed, the condition (1.2) is obtained from (1.5) and the condition (1.3)
is obtained from (1.6) easily. Here α1 =
c
k
, β2 = −cA+ k1B
qM
and α2 = −k1Bc
qMk
.
Finally, if we put
t =
x, 0 < x < d,
αx, d < x < π,
then the discontinuity conditions (1.3) and a particular case of (1.1) 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, (1.1) – (1.3) will appear to be consequence of the above problem.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1157
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 constructi-
ng parameters of heterogeneous electronic lines with desirable technical characteristics
[21, 22]. After reducing corresponding 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 a one-dimensional disconti-
nuous medium [23, 24]. Boundary-value problems with discontinuties in an interiorpoint
also appear in geophysical models for oscillations of the Earth [25, 26]. Here, the main
disconutinuity 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 equations 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 investigati-
ng spectral properties of some classes of differential, integrodifferential and integral
operators.
It must be noted that some special cases of the considered problem (1.1) – (1.3) 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 transmission
condition arise in heat and mass transfer problems (see, for example, [40]), in vibrati-
ng string problems when the string loaded additionally with point masses (see, for
example, [37]) and in diffraction problems (see, for example, [39]). 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,
[36 – 38]).
In the study of [29], there isn’t existed spectral parameter in boundary conditions in
the point x = π. However some certain physical problems are reduced to boundary-value
problems which contain spectral parameter in boundary conditions, so they are reduced
to investigate the type of problems (1.1) – (1.3).
In this study, representation with transformation operator has been obtained as in
[28, 29].
Moreover, 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 [29].
2. Representation for the solution. We define y1(x) = y(x), y2(x) = (Γy)(x) =
= y′(x) − u(x)y(x), u(x) = C lnx and let’s write the expression of left-hand side of
equation (1.1) as follows
` (y) = − [(Γy)(x)]
′ − u(x)(Γy)(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)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1158 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
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 (d+ 0) = αy1 (d− 0),
y2 (d+ 0) = α−1y2 (d− 0).
(2.4)
Matrix form of system (2.2)(
y1
y2
)′
=
(
u 1
−k2 − u2 + q −u
)(
[c]cy1
y2
)
(2.5)
or y′ = Ay such that A =
(
u(x) 1
−k2 − u2(x) + q(x) −u(x)
)
,
(
y1
y2
)
.
x = 0 is a regular-singular end point for equation (2.5) and Theorem 2 in [1, p. 56]
(see Remark 1-2) extends to interval [0, π] . For this reason, by [1], there exists only one
solution of the system (2.2) which satisfies the initial conditions 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) = ik.
Definition 2.1. The first component of the solution of system (2.2) which satisfies
the initial condititons y1 (ξ) = υ1, y2 (ξ) = (Γy) (ξ) = υ2 is called the solution of
equation (1.1) which satisfies these same initial conditions.
It was showed in [29] by the successive approximations method that (see [16]) the
following theorem is true.
Theorem 2.1. For each solution of system (2.2) which satisfying the initial condi-
tions
(
y1
y2
)
(0) =
(
1
ik
)
and the jump conditions (2.4), the following expression is
true:
y1 = eikx + intx−xK11(x, t)eiktdt,
y2 = ikeikx + b(x)eikx + intx−xK21(x, t)eiktdt+ ikintx−xK22(x, t)eiktdt,
x < d,
y1 = α+eikx + α−eik(2d−x) + intx−xK11(x, t)eiktdt,
y2 = ik
(
α+eikx − α−eik(2d−x)
)
+ b(x)
[
α+eikx + α−eik(2d−x)
]
,
+ intx−xK21(x, t)eiktdt+ ikintx−xK22(x, t)eiktdt,
x > d,
where
b(x) = −1
2
intx0
[
u2 (s)− q (s)
]
e−
1
2 intxsu(t)dtds,
K11 (x, x) =
α+
2
u(x),
K21 (x, x) = b′(x)− 1
2
intx0
[
u2 (s)− q (s)
]
K11 (s, s) ds− 1
2
intx0u (s)K21 (s, s) ds,
K22 (x, x) = −α
+
2
[u(x) + 2b(x)] ,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1159
K11 (x, 2d− x+ 0)−K11 (x, 2d− x− 0) =
α−
2
u(x),
∂Kij (x, .)
∂x
,
∂Kij (x, .)
∂t
∈ L2(0, π), i, j = 1, 2.
3. Properties of the spectrum. In this section, properties of the spectrum of problem
L will be learned. 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, (Γϕ0)(0, k) = k and the jump conditions (2.4) is shown
as
ϕ0(x, k) =
sin kx, for x < d,
α+ sin kx+ α− sin k(2d− x), for x > d,
(Γϕ0)(x, k) =
k cos kx, for x < d,
kα+ cos kx− kα− cos k(2d− x), for x > d.
(3.1)
We denote characteristic function, eigenvalues sequence and normalizing constant
sequence by ∆(k), {kn} and {an} respectively. Denote
∆(k) = 〈ψ(x, k), ϕ(x, k)〉 , (3.2)
where
〈y(x), z(x)〉 := y(x)(Γz)(x)− (Γy)(x)z(x).
Also we defined normalizing constants by
an := intπ0ϕ
2 (x, kn) dx+
1
ρ
[
α1ϕ(π, kn) + β1(Γϕ)(π, kn)
]2
, (3.3)
where ρ = α2β1 − β2α1. According to the Liouville formula, 〈ψ(x, k), ϕ(x, k)〉 is not
depend on x.
We shall assume that ϕ(x, k) and ψ(x, k) are solutions of equation (1.1) under the
following initial conditions:
ϕ(0, k) = 0, (Γϕ)(0, k) = k, ψ(π, k) =
(
β1k
2 + β2
)
, (Γψ) (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
)
ϕ(π, k) = ψ(0, k). (3.4)
By using the representation of the function y(x, k) for the solution ϕ(x, k) :
ϕ(x, k) = ϕ0(x, k) + intπ0 K̃11 (π, t) sin ktdt (3.5)
is obtained.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1160 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
Lemma 3.1 (Lagrange’s fomula). Let y, z ∈ D (L∗0). Then
(L∗0y, z) = intπ0 ` (y) zdx = (y, L∗0z) + [y, z]
(
|d−0
0 + |πd+0
)
,
where [y, z]
(
|d−0
0 + |πd+0
)
=
[
(Γz) (x)y(x)− (Γy)(x)z(x)
] (
|d−0
0 + |πd+0
)
.
Proof. We have
(L∗0y, z) = −intπ0 (y′ − u y)
′
zdx− intπ0u (y′ − u y) zdx− intπ0
(
u2 − q(x)
)
yzdx =
= intπ0 (y′ − u y) (z′ − uz) dx− intπ0
(
u2 − q(x)
)
yzdx− (Γy)(x)z(x)
(
|d−0
0 + |πd+0
)
=
= intπ0y` (z) dx+ [y, z]
(
|d−0
0 + |πd+0
)
= (y, L∗0z) + [y, z]
(
|d−0
0 + |πd+0
)
.
Lemma 3.1 is proved.
Lemma 3.2. The zeros {kn} of the characteristic function coincide with the eigen-
values 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. (3.6)
Proof. 1) Let k0 be a zero of the function ∆(k). Then by virtue of equation (3.2) and
(3.4), ψ (x, k0) = γ0ϕ (x, k0) and the functions ψ (x, k0) , ϕ (x, k0) satisfy the boundary
conditions (1.2). 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, y0 be a corresponding eigenfunctions. Then U(y0) =
= V (y0) = 0. Clearly y0(0) = 0. Without loss of generality we put (Γy0)(0) =
= ik. Hence y0(x) ≡ ϕ (x, k0) . Thus, from equation (3.4), ∆ (k0) = V (ϕ (x, k0)) =
= V (y0(x)) = 0 is obtained.
Lemma 3.2 is proved.
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 (1.1),
−ψ′′(x, k) + [u′(x) + q(x)]ψ(x, k) = kψ(x, k)−
−ϕ′′ (x, kn) + [u′(x) + q(x)]ϕ (x, kn) = knϕ (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], the equality
〈ψ(x, k), ϕ (x, kn)〉
[
|d−0
0 + |πd+0
]
= (k − kn) intπ0ψ(x, k)ϕ (x, kn) dx (3.7)
is obtained.
If jump conditions (1.3) and equation (3.3) are considered, then
intπ0 ψ (x, kn)ϕ (x, kn) dx+
+
1
ρ
[
α1ψ(π, kn) + β1 (Γψ) (π, kn)
][
α1ϕ(π, kn) + β1(Γϕ)(π, kn)
]
= −
.
∆ (kn)
as k → kn is obtained. From Lemma 3.2, we get that
anγn = −
.
∆ (kn) . (3.8)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1161
It is obvious that
.
∆ (kn) 6= 0.
Since the function ∆(k) is an entire function of k, the zeros of ∆(k) are separated.
Lemma 3.3 is proved.
Now, let problems be
L :
−y′′ + [u′(x) + q(x)] y = λy,
(Γy)(0)− hy(0) = 0,
(β1λ+ β2) (Γy) (π) + (α1λ+ α2) y (π) = 0,
y (d+ 0) = αy (d− 0) ,
(Γy) (d+ 0) = α−1 (Γy) (d− 0)
and
L̃ :
−y′′ + [u′(x) + q(x)] y = µy,
(Γy)(0)− hy(0) = 0,(
β̃1λ+ β̃2
)
(Γy) (π) + (α̃1λ+ α̃2) y (π) = 0,
y (d+ 0) = αy (d− 0) ,
(Γy) (d+ 0) = α−1 (Γy) (d− 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,
µn < λn < µn+1, if α2β̃2 > α̃2β2, n ≥ 0.
(3.9)
where α1α̃2 > α̃1α2 and β1β̃2 > β̃1β2.
Proof. As in the proof of Lemma 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) intπ0ϕ(x, λ)ϕ(x, µ)dx = 〈ϕ(x, λ), ϕ(x, µ)〉
[
|d−0
0 + |πd+0
]
=
= ϕ(π, λ)(Γϕ) (π, µ)− (Γϕ)(π, λ)ϕ(π, µ) =
=
α̃1α2 − α1α̃2
α2β̃2 − α̃2β2
(λ− µ)ϕ(π, λ) (ϕ) (π, µ)+
+
β̃1β2 − β1β̃2
α2β̃2 − α̃2β2
(λ− µ) (Γϕ)(π, λ)(Γϕ) (π, µ) +
+
1
α2β̃2 − α̃2β2
[
∆̃(λ)∆ (µ)− ∆̃ (µ) ∆(λ)
]
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1162 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
Hence
(λ− µ) intπ0ϕ(x, λ)ϕ(x, µ)dx =
=
α̃1α2 − α1α̃2
α2β̃2 − α̃2β2
(λ− µ)ϕ(π, λ) (ϕ) (π, µ)+
+
β̃1β2 − β1β̃2
α2β̃2 − α̃2β2
(λ− µ) (Γϕ)(π, λ)(Γϕ)(π, µ)+
+
1
α2β̃2 − α̃2β2
[
∆̃(λ)− ∆̃ (µ)
λ− µ
∆ (µ)− ∆(λ)−∆ (µ)
λ− µ
∆̃ (µ)
]
.
As µ→ λ
intπ0ϕ
2(x, λ)dx =
1
α2β̃2 − α̃2β2
×
×
[
(α̃1α2 − α1α̃2)ϕ2(π, λ) +
(
β̃1β2 − β1β̃2
)
(Γϕ)
2
(π, λ)+
+
.
∆̃(λ)∆(λ)−
.
∆ (λ) ∆̃(λ)
]
, (3.10)
where
.
∆(λ) =
d
dλ
∆(λ),
.
∆̃(λ) =
d
dλ
∆̃(λ). From equation (3.10), if ∆̃(λ) 6= 0
1
∆̃2(λ)
[
intπ0ϕ
2(x, λ)dx−
(α̃1α2 − α1α̃2)ϕ2 (π, λ) +
(
β̃1β2 − β1β̃2
)
(Γϕ)2 (π, λ)
α̃2β2 − α2β̃2
]
=
= − 1(
α̃2β2 − α2β̃2
) d
dλ
(
∆ (λ)
∆̃(λ)
)
, −∞ < λ <∞,
is obtained.
If α2β̃2 < α̃2β2, then
∆(λ)
∆̃(λ)
is monotonically decreasing in the set ofR\{µn, n ≥ 0} .
Thus it is obvious that lim
λ→µ±0
n
∆ (λ)
∆̃(λ)
= ±∞.
When α2β̃2 > α̃2β2, if we write the equality (3.10) as
1
∆2(λ)
intπ0ϕ
2(x, λ)dx−
(α̃1α2 − α1α̃2)ϕ2 (π, λ) +
(
β̃1β2 − β1β̃2
)
(Γϕ)2 (π, λ)
α2β̃2 − α̃2β2
=
= − 1(
α2β̃2 − α̃2β2
) d
dλ
(
∆̃(λ)
∆(λ)
)
, −∞ < λ <∞, ∆(λ) 6= 0,
we get that the function
∆̃ (λ)
∆(λ)
is monotonically decreasing in R \ {λn, n ≥ 0} and it
is clear that limλ→λ±0
n
∆̃(λ)
∆(λ)
= ±∞. From here (3.9) is obtained.
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1163
Theorem 3.1. The eigenvalues kn, eigenfunctions ϕ(x, kn) and the normalizing
numbers αn of problem L have the following asymptotic behaviour
kn = k0
n +
dn
k0
n
+
δn
k0
n
, (3.11)
ϕ (x, kn) = α+ sin
(
k0
n + ε
)
x+ α− sin
(
k0
n + ε
)
(2d− x) +
sn
k0
n
+
bn
k0
n
, (3.12)
an =
[(
α+
)2
+
(
α−
)2](π − d
2
)
+
d
2
− α+α− cos 2k0
nd+ γn +
ξn
n
, (3.13)
where δn, sn, ξn ∈ `2, bn, dn, γn ∈ `∞ and k0
n are roots of ∆0(k) := k3
[
α+ cos kπ −
− α− cos k (2d− π)
]
and k0
n = n+ hn, hn ∈ `∞.
Proof. Using (3.1), (3.2) and (3.5), we get
∆(k) =
(
α1k
2 + α2
)
ϕ0(π, k) +
(
β1k
2 + β2
)
(Γϕ0)(π, k)+
+
(
α1k
2 + α2
)
intπ0 K̃11 (π, t) sin ktdt+
+β1k
2 + β2
[
intπ0 K̃21 (π, t) sin ktdt+ intπ0 K̃22 (π, t) cos ktdt
]
=
=
(
α1k
2 + α2
) (
α+ sin kπ + α− sin k (2d− π)
)
+
+
(
β1k
2 + β2
) (
kα+ cos kπ − kα− cos k (2d− π)
)
+ k3O
(
exp | Im k|π
|k|
)
=
= β1∆0(k) +
(
α1k
2 + α2
) (
α+ sin kπ + α− sin k (2d− π)
)
+
+β2k
(
α+ cos kπ − α− cos k (2d− 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 ∈ Gn |∆(k) − ∆0(k)| <
<
Cδ
2
|k|3e| Im k|π for sufficiently large values of n , 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, according 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
lim
n→∞
εn = 0.
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1164 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
Since numbers kn are zeros of characteristic function ∆ (k) ,
∆ (kn) =
(
α1k
2
n + α2
) (
α+ sin knπ + α− sin kn (2d− π)
)
+
+
(
β1k
2 + β2
) (
knα
+ cos knπ − knα− cos kn (2d− π)
)
+O
(
k2
n
)
.
From the last equality, we get
α+ cos knπ − α− cos kn (2d− π) +
α1
β1k2
n
[
α+ sin knπ + α− sin kn (2d− π)
]
+
+
α2
β1k3
n
[
α+ sin knπ + α− sin kn (2d− π)
]
+
+
β2
β1k2
n
[
α+ cos knπ − α− cos kn (2d− 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 [5] ( see also [6]) is used then we get that k0
n = n + hn where
sup
n
|hn| < M. Therefore
εn =
dn
n
+
δn
n
, δn ∈ `2, dn ∈ `∞.
Hence for the eigenvalues kn of the problem L, asymptotic formula (3.10) is true. Now,
let’s try to find the asymptotic formula for the eigenfunction:
ϕ (x, kn) = α+ sin knx+ α− sin kn(2d− x) + intx0K̃11(x, t) sin kntdt =
= α+ sin
(
k0
n + εn
)
x+ α− sin
(
k0
n + εn
)
(2d− x) +
− 1
k0
n + εn
intx0K̃11(x, t)d
(
cos
(
k0
n + εn
))
t dt =
= α+ sin k0
nx+ α− sin k0
n(2d− x)−
− 1
k0
n + εn
[
K̃11(x, t) cos k0
nt
] (
|2d−x−0
0 + |x2d−x+0
)
+
+
1
k0
n + εn
intx0K̃
′
11t(x, t) cos k0
nt dt.
Since
K̃11 (x, x) =
α+
2
u(x), K̃11 (x, 2d− x+ 0)− K̃11 (x, 2d− x− 0) =
α−
2
u(x),
and
intx0K̃
′
11t(x, t) cos k0
ntdt ∈ `2
it is obtained that
ϕ (x, kn) = α+ sin k0
nx+ α− sin k0
n(2d− x)+
+
α− cos k0
n(2d− x)− α+ cos k0
nx
2k0
n
u(x) +
bn
n
+
sn
n
, sn ∈ `2, bn ∈ `∞.
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1165
Then we get the asymptotic formula (3.12). Finally, in order to show (3.13) is true,
using (3.1) and (3.5), we get
an = intπ0ϕ
2 (x, kn) dx+
1
ρ
[α1ϕ(π, kn) + β1(Γϕ)(π, kn)]
2
=
= intd0
[
sin2 knxdx+
(
intx0K̃11(x, t) sin kntdt
)2
]
dx+
+2intd0 sin knxintx0K̃11(x, t) sin kntdtdx+
+intπd
[(
α+
)2
sin2 knx+
(
α−
)2
sin2 kn(2d− x) +
(
intxdK̃11(x, t) sin kntdt
)2
]
dx+
+2α+α−intπd sin knx sin kn(2d− x)dx+
+2α+intπd sin knxintxdK̃11(x, t) sin kntdtdx+
+2α−intπd sin kn(2d− x)intxdK̃11(x, t) sin kntdtdx+
+
1
ρ
[α1ϕ(π, kn) + β1 (Γϕ) (π, kn)]
2
=
=
π − d
2
[(
α+
)2
+
(
α−
)2]
+
d
2
− α+α− cos 2k0
nd+ γn +
ξn
n
.
Theorem 3.1 is proved.
4. Inverse problem. In the present section, we study the inverse problems recovering
the boundary value problem L from the spectral data. We consider three statements of
the inverse problem of the reconstruction of the boundary-value problem L from the
Weyl function, from the spectral data {kn, an}n≥0 and from two spectra {kn, µn}n≥0 .
These inverse problems are generalizations of the well-known inverse problems for
Sturm – Liouville operator [22, 42].
Let Φ(x, k) be solution of (2.2) under the condititons U (Φ) = Φ(0, k) = 1 and
V (Φ) =
(
α1k
2 + α2
)
Φ(π, k) +
(
β1k
2 + β2
)
(ΓΦ) (π, k) = 0 and under the jump
condititons (2.4). Also C(x, k) be solution of (2.2) under the condititons C(0, k) = 1
and (ΓC) (0, k) = 0 and under the jump condititons (2.4). Then function ψ(x, k) can be
represented as follows:
ψ(x, k) =
1
k
(Γψ) (0, k)ϕ(x, k) + ∆(k)C(x, k)
or
1
∆(k)
ψ(x, k) =
(Γψ) (0, k)
k∆(k)
ϕ(x, k) + C(x, k). (4.1)
Denote
M(k) :=
(Γψ) (0, k)
k∆(k)
. (4.2)
It is clear that
Φ(x, k) = M(k)ϕ(x, k) + C(x, k). (4.3)
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1166 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
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 (ΓΦ) (0, k) =
(Γψ) (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
a0 (k − k0)
+
∞∑
n=1
′
{
1
an (k − kn)
+
1
a0
nk
0
n
}
. (4.6)
Proof. Let’s write a representation solution of ψ(x, k) = −(β1k
2 + β2) ×
×C(x, k) +
(
α1k
2 + α2
)
S(x, k) as a representation solution of ϕ(x, k) :
for x > d
ψ(x, k) = −
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x) +
+ intπ−x0 Ñ11(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt,
(Γψ) (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)
]
+
+ intπ−x0 Ñ21(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt+
+ kintπ−x0 Ñ22(x, t)
[(
β1k
2 + β2
)
sin kt+
(
α1k
2 + α2
)
cos kt
]
dt;
for x < d
ψ(x, k) = α+
[
−
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x)
]
+
+ α−
[
−
(
β1k
2 + β2
)
cos k (π − 2d+ x) +
(
α1k
2 + α2
)
sin k (π − 2d+ x)
]
+
+ intπ−x0 Ñ11(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt,
(Γψ) (x, k) = −kα+
[(
β1k
2 + β2
)
sin k (π − x) +
(
α1k
2 + α2
)
cos k (π − x)
]
+
+ kα−
[(
β1k
2 + β2
)
cos k (π − 2d+ x)−
(
α1k
2 + α2
)
sin k (π − 2d+ x)
]
+
+ b(x)α+
[
−
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x)
]
+
+ b(x)α−
[(
β1k
2 + β2
)
cos k (π − 2d+ x)−
(
α1k
2 + α2
)
sin k (π − 2d+ x)
]
+
+ intπ−x0 Ñ21(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt+
+ kintπ−x0 Ñ22(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
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1167
ψ0(x, k) = Ψ01(x, k) + f1,
(Γψ0) (x, k) = Ψ02(x, k) + f2,
where
f1 = intπ−x0 Ñ11(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt,
f2 = b(x)α+
[
−
(
β1k
2 + β2
)
cos k (π − x) +
(
α1k
2 + α2
)
sin k (π − x)
]
+
+b(x)α−
[
−
(
β1k
2 + β2
)
cos k (π − 2d+ x) +
(
α1k
2 + α2
)
sin k (π − 2d+ x)
]
+
+ intπ−x0 Ñ21(x, t)
[
−
(
β1k
2 + β2
)
cos kt+
(
α1k
2 + α2
)
sin kt
]
dt+
+kintπ−x0 Ñ22(x, t)
[(
β1k
2 + β2
)
sin kt+
(
α1k
2 + α2
)
cos kt
]
dt.
On the other hand, we can write
M(k)−M0(k) =
(Γψ) (0, k)
kψ(0, k)
− (Γψ0) (0, k)
kψ0(0, k)
=
f2
k∆(k)
− f1
∆(k)
M0(k).
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 (3.4), (4.1) and
Lemma 3.2, we calculate that
Re s
k=kn
M(k) =
(Γψ) (0, kn)
kn
.
∆ (kn)
= − 1
an
,
Re s
k=k0n
M0(k) =
(Γψ0)
(
0, k0
n
)
k0
n
.
∆ (k0
n)
= − 1
a0
n
.
(4.8)
Consider the contour integral
In(k) =
1
2πi
intΓn
M (µ)−M0 (µ)
k − µ
dµ, k ∈ int Γn.
By virtue of (4.7) , we have lim
n→∞
In (k) = 0. On the other hand, the residue theorem
and (4.8) yield
In(k) = −M(k) +M0(k) +
∑
kn∈intΓn
1
an (k − kn)
−
∑
k0n∈intΓn
1
a0
n (k − k0
n)
.
Therefore as n→∞ we get
M(k) = M0(k) +
+∞∑
n=−∞
1
an (k − kn)
+
+∞∑
n=−∞
1
a0
n (k − k0
n)
.
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1168 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
It follows from the form of the function M0(k) that
M0(k) =
1
a0
nk
+
+∞∑
n=−∞
′ 1
a0
n
(
1
k − k0
n
+
1
k0
n
)
.
From the last two equalities yield (4.6).
Theorem 4.6 is proved.
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, 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.
Proof. Let us define the matrix P (x, k) = [Pjk(x, k)]j,k=1,2 by the formula
P (x, k)
(
ϕ̃ Φ̃
Γϕ̃ ΓΦ̃
)
=
(
ϕ Φ
Γϕ ΓΦ
)
. (4.9)
Using (4.9) and (4.5) we calculate
P11(x, k) = −1
k
[
ϕ(x, k)
(
ΓΦ̃
)
(x, k)− Φ(x, k) (Γϕ̃) (x, k)
]
,
P12(x, k) = −1
k
[
Φ(x, k)ϕ̃(x, k)− ϕ(x, k)Φ̃(x, k)
]
,
P21(x, k) = −1
k
[
(Γϕ) (x, k)
(
ΓΦ̃
)
(x, k)− (ΓΦ) (x, k)
(
Γϕ̃
)
(x, k)
]
,
P22(x, k) = −1
k
[
(ΓΦ) (x, k)ϕ̃(x, k)− (Γϕ)(x, k)Φ̃(x, k)
]
(4.10)
and
ϕ(x, k) = P11(x, k)ϕ̃(x, k) + P12(x, k) (Γϕ̃) (x, k),
(Γϕ)(x, k) = P21(x, k)ϕ̃(x, k) + P22(x, k) (Γϕ̃) (x, k),
Φ(x, k) = P11(x, k)Φ̃(x, k) + P12(x, k)
(
ΓΦ̃
)
(x, k),
(ΓΦ) (x, k) = P21(x, k)Φ̃(x, k) + P22(x, k)
(
ΓΦ̃
)
(x, k).
(4.11)
It follows from (4.10), (4.2) and (4.5)
P11(x, k) = 1 +
1
k∆(k)
[
ϕ(x, k)
((
ΓΨ̃
)
(x, k)− (ΓΨ) (x, k)
)
−
−Ψ(x, k)
((
Γϕ̃
)
(x, k)− (Γϕ)(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)
[
(Γϕ)(x, k)
(
ΓΨ̃
)
(x, k)− (ΓΨ) (x, k) (Γϕ̃) (x, k)
]
,
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ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1169
P22(x, k) = 1 +
1
k∆(k)
[(
ΓΨ
)
(x, k)
(
ϕ̃(x, k)− ϕ(x, k)
)
−
−
(
Γϕ
)
(x, k)
(
Ψ̃(x, k)−Ψ(x, k)
)]
.
According to (4.10) and (4.2), for each fixed x, the functions Pjk(x, k) are meromor-
phic 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
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) and (4.3) we have
P11(x, k) = −1
k
[
ϕ(x, k)
(
ΓC̃
)
(x, k)− C(x, k)
(
Γϕ̃
)
(x, k)+
+
(
M̃(k)−M(k)
)
ϕ(x, k)
(
Γϕ̃
)
(x, k)
]
,
P12(x, k) = −1
k
[
ϕ̃(x, k)C(x, k)− C̃(x, k)ϕ(x, k)+
+
(
M(k)− M̃k)
)
ϕ(x, k)ϕ̃(x, k)
]
,
P21(x, k) = −1
k
[(
Γϕ
)
(x, k)
(
ΓC̃
)
(x, k)− (ΓC)(x, k)
(
Γϕ̃
)
(x, k)
]
−
− 1
k
[(
M̃(k)−M(k)
)
(Γϕ)(x, k)
(
Γϕ̃
)
(x, k)
]
,
P22(x, k) = −1
k
[
ϕ̃(x, k)(ΓC)(x, k)− C̃(x, k)(Γϕ)(x, k)+
+
(
M(k)− M̃(k)
)
(Γϕ)(x, k)ϕ̃(x, k)
]
.
(4.13)
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), (Γϕ)(x, k) ≡ (Γϕ̃) (x, k),
Φ(x, k) ≡ Φ̃(x, k), (ΓΦ) (x, k) ≡
(
ΓΦ̃
)
(x, k)
for all x and k. Consequently L = L̃.
Theorem 4.2 is proved.
Theorem 4.3. If kn = k̃n, an = ãn, n ≥ 0 then L = L̃. Thus, the specification of
the spectral data
{
kn, αn
}
n≥0
uniquely determines the operator.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1170 R. KH. AMIROV, N. TOPSAKAL, Y. GÜLDÜ
Proof. We have
M(k) =
1
a0 (k − k0)
+
∞∑
n=1
′
{
1
an (k − kn)
+
1
a0
nk
0
n
}
,
M̃(k) =
1
ã0
(
k̃ − k̃0
) +
∞∑
n=1
′
1
ãn
(
k̃ − k̃n
) +
1
ã0
nk̃
0
n
.
(4.14)
Under the hypothesis of the theorem and in view of (4.13), we get that M(k) = M̃(k)
and consequently by Theorem 3.1, L = L̃.
Theorem 4.4. If kn = k̃n, µn = µ̃n, n ≥ 0, then L = L̃.
Proof. From these properties of functions ∆(k) and ∆̃(k), it is clear that
lim
k→∞
∆(k)
∆̃(k)
= 1, kn = k̃n, and functions of ∆(k), ∆̃(k) are analytic functions. From
the uniqueness theorem of analytic functions, ∆(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 an = ãn. Consequently by Theorem 4.1, L = L̃.
1. Naimark M. A. Linear differential operators (in Russian). – Moscow: Nauka, 1967.
2. Amirov R. Kh. On Sturm – Liouville operators with discotinuity conditions inside an interval // J. Math.
Anal. Appl. – 2006. – 317. – P. 163 – 176.
3. Bellman R., Cooke K. Differential-difference equations. – New York: Acad. Press, 1963.
4. Levin B. Ya. Entire functions. – Moscow: MGU, 1971.
5. Zhdanovich V. F. Formulas for the zeros of dirichlet polynomials and quasipolynomials // Dokl. Akad.
Nauk SSSR. – 1960. – 135, № 8. – P. 1046 – 1049.
6. Krein M. G., Levin B. Ya. On entire almost periodic functions of exponential type // Dokl. Akad. Nauk
SSSR. – 1949. – 64, № 3. – P. 285 – 287.
7. Ambartsumyan V. A. Über eine frage der eigenwerttheorie // Z. für Phys. – 1929. – 53. – P. 690 – 695.
8. Levinson N. The inverse Sturm – Liouville problem // Mathematica Tidsskrift B. – 1949. – 25. – P. 1 – 29.
9. Levitan B. M. On the determination of the Sturm – Liouville operator from one and two spectra // Math.
USSR Izvestija. – 1978. – 12. – P. 179 – 193.
10. Chadan K., Colton D., Paivarinta L., Rundell W. An introduction to inverse scattering and inverse
spectral problems. – Philadelphia: SIAM, 1997.
11. Gesztesy F., SimonB. Inverse spectral analysis with partial information on the potential II, The case of
discrete spectrum // Trans. Amer. Math. Soc. – 2000. – P. 352. – P. 2765 – 2789.
12. Gesztesy F., Kirsch A. One dimensional Schrödinger operators with interactions singular on a discrete
set // J. reine und angew. Math. – 1985. – P. 362. – P. 28 – 50.
13. Gilbert R. P. A method of ascent for solving boundary-value problem // Bull. Amer. Math. Soc. – 1969.
– 75. – P. 1286 – 1289.
14. Pöschel J., Trubowitz E. Inverse spectral theory. – Orlando: Acad. Press, 1987.
15. Borg G. Eine umkehrung der Sturm – Liouvillesehen eigenwertaufgabe // Acta Math. – 1945. – 78. –
P. 1 – 96.
16. Marchenko V. A. Sturm – Liouville operators and their applications. – Kiev: Naukova Dumka, 1986. –
English transl.: Birkhauser, Basel.
17. Carlson R. Inverse spectral theory for some singular Sturm – Liouville problems // J. Different. Equat. –
1993. – 106, № 1. – P. 121 – 140.
18. Carlson R. A Borg – Levinson theorem for Bessel operators // Pacif. J. Math. – 1997. – 177, № 1. –
P. 1 – 26.
19. Hochstadt H. The functions of mathematical physics. – New York: Wiley Intersci., 1971.
20. Marchenko V. A. Certain problems of the theory of one dimensional linear differential operators of the
second order I // Tr. Moskov. Mat. Obshchestva. – 1952. – 1. – P. 327 – 340.
21. Meschanov V. P., Feldstein A. L. Automatic design of directional couplers. – Moscow: Sviaz, 1980.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON IMPULSIVE STURM – LIOUVILLE OPERATORS WITH COULOMB POTENTIAL . . . 1171
22. Litvinenko O. N., Soshnikov V. I. The theory of heterogeneous lines and their applications in radio
engineering (in Russian). – Moscow: Radio, 1964.
23. Krueger R. J. Inverse problems for nonabsorbing media with discontinuous material properties // J.
Math. Phys. – 1982. – 23, № 3. – P. 396 – 404.
24. Shepelsky D. G. The inverse problem of reconstruction of the medium’s conductivity in a class of
discontinuous and increasing functions // Adv. Soviet Math. – 1994. – 19. – P. 209 – 231.
25. Anderssen R. S. The effect of discontinuities in density and shear velocity on the asypmtotic overtone
sturcture of toritonal eigenfrequencies of the Earth // Geophys. J. Roy. Astron. Soc. – 1997. – 50. –
P. 303 – 309.
26. Lapwood F. R., Usami T. Free Oscillations of the Earth. – Cambridge: Cambridge Univ. Press, 1981.
27. He X., Volkmer H. Riesz bazes of solutions of Sturm – Liouville equations // Fourier Anal. Appl. – 2001.
– 7. – P. 297 – 307.
28. Amirov R. Kh., Topsakal N. A representation for solutions of Sturm – Liouville equations with Coulomb
potential inside finite interval // J. Cumhuriyet Univ. Natur. Sci. – 2007. – 28, № 2. – P. 11 – 38.
29. Amirov R. Kh., Topsakal N. On Sturm – Liouville operators with Coulomb potential which have di-
scontinuity conditions inside an interval // Integrю Transforms Spec. Funct. – 2008. – 19, № 12. –
P. 923 – 937.
30. Fulton C. T. Two-point boundary-value problems with eigenvalue parameter contained in the boundary
conditions // Proc. Roy. Soc. Edinburgh A. – 1977. – A77. – P. 293 – 308.
31. Fulton C. T. Singular eigenvalue problems with eigenvalue parameter contained in the boundary condi-
tions // Ibid. – 1980. – A87. – P. 1 – 34.
32. Guliyev N. J. Inverse eigenvalue problems for Sturm – Liouville equaitons with spectral parameter
linearly contained in one of the boundary conditions // Inverse Problems. – 2005. – 21. – P. 1315 – 1330.
33. Binding P. A., Browne P. J., Seddighi Sturm – Liouville problems with eigenparameter depent boundary
conditions // Proc. Roy. Soc. Edinburgh. – 1993. – 37. – P. 57 – 72.
34. Binding P. A., Browne P. J. Oscillation theory for indefinite Sturm – Liouville problems with
eigenparameter- dependent boundary conditions // Proc. R. Soc. Edinburgh A. – 1997. – A127. – P. 1123 –
1136.
35. Kapustin N. Yu., Mossiev E. I. Oscillation properties of solutions to a nonselfadjoint spectral problem
with spectral parameter in the boundary condition // Different. Equat. – 1999. – 35. – P. 1031 – 1034.
36. Pivovarchik V. N., Van der Mee C. The inverse generalized Regge problem // Inverse Problems. – 2001.
– 17. – P. 1831 – 1845.
37. Tikhonov A. N., Samarskii A. A. Equations of mathematical physics. – Oxford: Pergamon.
38. Van der Mee C., Pivovarchik V. N. A Sturm – Liouville inverse spectral problem with boundary conditions
depending on the spectral parameter // Funct. Anal. and Appl. – 2002. – 36, № 4. – P. 314 – 317.
39. Voitovich N. N., Katsenelbaum B. Z., Sivov N. Generalized method of eigen-vibration in the theory of
diffraction. – Moskow: Nauka, 1997.
40. Likov A. V., Mkhailov Yu. A. The theory of heat and mass transfer. – Moskow: Gosenergoizdat.
41. Walter J. Regular eigenvalue problems with eigenvalue parameter in the boundary condition // Math. Z.
– 1973. – 133. – P. 301 – 312.
42. Levitan B. M. Inverse Sturm – Liouville problems. – Moskow: Nauka, 1984. (English transl.: VNU Sci.
Press, Utrecht, 1987).
Received 22.02.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
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| id | umjimathkievua-article-2946 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:33:20Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fb/4100fb489ca876b794717157cb0968fb.pdf |
| spelling | umjimathkievua-article-29462020-03-18T19:41:02Z On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions Про імпульсні оператори Штурма-Ліувілля з потенціалом Кулона та спектральним параметром, що лінійно міститься в граничних умовах Amirov, R. Kh. Güldü, Y. Topsakal, N. Аміров, Р. Х. Гюль, Ю. Топсакал, Н. The Sturm–Liouville problem with linear discontinuities is investigated in the case where an eigenparameter appears not only in a differential equation but also in boundary conditions. Properties and the asymptotic behavior of spectral characteristics are studied for the Sturm–Liouville operators with Coulomb potential that have discontinuity conditions inside a finite interval. Moreover, the Weyl function for this problem is defined and uniqueness theorems are proved for a solution of the inverse problem with respect to this function. Досліджено задачу Штурма-Ліувілля з лінійними розривами у випадку, коли власний параметр міститься не лише у диференціальному рівнянні, але й у граничних умовах. Вивчено властивості та асимптотичну поведінку спектральної характеристики для операторів Штурма-Ліувілля з потенціалом Кулона, що мають умову розривності всередині скінченного інтервалу. Крім того, для розглядуваної задачі визначено функцію Вейля та доведено теореми єдиності для розв'язку оберненої задачі відповідно до цієї функції. Institute of Mathematics, NAS of Ukraine 2010-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2946 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 9 (2010); 1155–1172 Український математичний журнал; Том 62 № 9 (2010); 1155–1172 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2946/2642 https://umj.imath.kiev.ua/index.php/umj/article/view/2946/2643 Copyright (c) 2010 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 Coulomb potential and spectral parameter linearly contained in boundary conditions |
| title | On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions |
| title_alt | Про імпульсні оператори Штурма-Ліувілля з потенціалом Кулона та спектральним параметром, що лінійно міститься в граничних умовах |
| title_full | On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions |
| title_fullStr | On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions |
| title_full_unstemmed | On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions |
| title_short | On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions |
| title_sort | on impulsive sturm–liouville operators with coulomb potential and spectral parameter linearly contained in boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2946 |
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