Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics
The system of four equations of Marchenko type allowing to restore the bundle by the scattering matrix is derived for a fourth order di®erential bundle in L₂ (0;+∞) in the case of multiple ±i roots of the main characteristic polynomial. The uniqueness of solution of the inverse problem is proved whe...
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irk-123456789-1066352016-10-02T03:02:49Z Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics Orudzhev, E.G. The system of four equations of Marchenko type allowing to restore the bundle by the scattering matrix is derived for a fourth order di®erential bundle in L₂ (0;+∞) in the case of multiple ±i roots of the main characteristic polynomial. The uniqueness of solution of the inverse problem is proved when the bundle has a pure continuous spectrum. 2010 Article Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics / E.G. Orudzhev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 84-95. — Бібліогр.: 4 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106635 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The system of four equations of Marchenko type allowing to restore the bundle by the scattering matrix is derived for a fourth order di®erential bundle in L₂ (0;+∞) in the case of multiple ±i roots of the main characteristic polynomial. The uniqueness of solution of the inverse problem is proved when the bundle has a pure continuous spectrum. |
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Article |
| author |
Orudzhev, E.G. |
| spellingShingle |
Orudzhev, E.G. Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics Журнал математической физики, анализа, геометрии |
| author_facet |
Orudzhev, E.G. |
| author_sort |
Orudzhev, E.G. |
| title |
Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics |
| title_short |
Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics |
| title_full |
Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics |
| title_fullStr |
Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics |
| title_full_unstemmed |
Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics |
| title_sort |
uniqueness of solution of the inverse problem of scattering theory for a fourth order differential bundle with multiple characteristics |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/106635 |
| citation_txt |
Uniqueness of Solution of the Inverse Problem of Scattering Theory for a Fourth Order Differential Bundle with Multiple Characteristics / E.G. Orudzhev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 84-95. — Бібліогр.: 4 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT orudzheveg uniquenessofsolutionoftheinverseproblemofscatteringtheoryforafourthorderdifferentialbundlewithmultiplecharacteristics |
| first_indexed |
2025-07-07T18:47:55Z |
| last_indexed |
2025-07-07T18:47:55Z |
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1837015054115930112 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 1, pp. 84–95
Uniqueness of Solution of the Inverse Problem of
Scattering Theory for a Fourth Order Differential Bundle
with Multiple Characteristics
E.G. Orudzhev
Department of Applied Mathematics and Cybernetics Baku State University
23 Z. Khalilov Str., Baku, AZ1148, Azerbaijan
E-mail:elsharorucov63@mail.ru
Received March 10, 2009
The system of four equations of Marchenko type allowing to
restore the bundle by the scattering matrix is derived for a fourth order
differential bundle in L2 (0;+∞) in the case of multiple ±i roots of the main
characteristic polynomial. The uniqueness of solution of the inverse problem
is proved when the bundle has a pure continuous spectrum.
Key words: bundle, spectrum, spectral expansion, scattering matrix,
inverse problem.
Mathematics Subject Classification 2000: 34L05, 47E05.
The author devotes this paper to the memory of academician
of NAS of Azerbaijan, professor M.G. Gasymov
Let us consider a bundle Lλ in the space L2 (0;+∞) generated by a differential
equation
l
(
x,
d
dx
, λ
)
y =
(
d2
dx2
+ λ2
)2
y + r (x) y′ + (λp (x) + q (x)) y = 0 (1)
with the boundary conditions
y (0, λ) = y′ (0, λ) = 0, (2)
where the complex-valued functions p (x) , q (x) , r (x) satisfy the conditions
eεx |p (x)| ≤ c1, eεx
∣∣r (x) , r′ (x)
∣∣ ≤ c2, eεx |q (x)| ≤ c3, ε > 0,
and λ is a spectral parameter.
c© E.G. Orudzhev, 2010
Uniqueness of Solution of the Inverse Problem of Scattering Theory
In papers [1, 2] equation (1) is studied and the transformation operators
transforming the solutions of the equation
(
d2
dx2 + λ2
)2
y = 0 to the solutions of
equation (1) are constructed. In particular, it is obtained in [2] that equation
(1) has a fundamental system of solutions F±
j (x, λ) , j = 0, 1, satisfying the
conditions
lim
x→∞
[
F±
j (x, λ)− xje±iλx
]
= 0, ±Imλ > 0, (3)
and there exist the kernels K±
j (x, t) such that
F±
j (x, λ) = xje±iλx +
∞∫
x
K±
j (x, t) e±iλtdt. (4)
These kernels satisfy the equations
l
(
x,
∂
∂x
,±i
∂
∂t
)
K±
j (x, t) = 0,
and there holds
lim
x+t→∞
∂α+βK±
j (x, t)
∂xα∂tβ
= 0, α + β ≤ 3,
∞∫
x
∣∣∣K±
j (x, t)
∣∣∣
2
dt < ∞. (5)
Furthermore, for the kernels K±
j (x, t) the following conditions are fulfilled on the
characteristics t = x :
d
dx
K±
j (x, t) = ± 1
4i
∞∫
x
pj (ξ) dξ−1
4
∞∫
x
rj (ξ) dξ,
(
∂2K±
j
∂t2
− ∂2K±
j
∂x2
)∣∣∣∣∣
t=x
= ±1
4
pj (x)− 1
4
rj (x) +
1
2
∞∫
x
qj (ξ) dξ
±1
4
∞∫
x
[ip (ξ)− r (ξ)]K±
j (ξ, ξ) dξ, (6)
where pj (u) = ujp (u) , rj (u) = ujr (u) , qj (u) = uj [q (u)− r′ (u)] .
In the given paper for a bundle Lλ we derive a system of four equations
of Marchenko type allowing to restore the bundle by the scattering matrix.
The uniqueness of the solution of the inverse problem is proved when the bundle
has a pure continuous spectrum.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 85
E.G. Orudzhev
The eigenfunctions ϕ±i (x, λ), i = 1, 2, of the continuous spectrum −∞ < λ < ∞
of the bundle Lλ are determined in the following way:
ϕ±i (x, λ) = F±
0 (x, λ) S±i1 (λ) + F±
1 (x, λ) S±i2 (λ)− F±
i−1 (x, λ) , i = 1, 2. (7)
Here S±ik (λ), i, k = 1, 2, are found from the conditions ϕ±i (0, λ) = ϕ±
′
i (0, λ) = 0,
i = 1, 2, and for them as |λ| → ∞ there hold
S+
11 (λ) = 1− 2K+
1 (0, 0) + O
(
1
λ
)
,
S+
12 (λ) = −2iλ + 2
(
K+
0 (0, 0)−K+
0 (0, 0)
)
+ O
(
1
λ
)
,
S+
21 (λ) = O
(
1
λ
)
, S+
22 (λ) = 1 + O
(
1
λ
)
,
S−11 (λ) =
1
1−K− (0, 0)
+ O
(
1
λ
)
, S−21 (λ) = O
(
1
λ
)
, (8)
S−12 (λ) =
iλ
1−K− (0, 0)
− K+ (0, 0)
1−K− (0, 0)
+ O
(
1
λ
)
,
S−22 (λ) =
2iλ
1−K− (0, 0)
− K+ (0, 0)−K− (0, 0)
1−K− (0, 0)
+ O
(
1
λ
)
.
Thus, assuming
1− 2K+
1 (0, 0) = a+
11, −2iλ + 2
(
K+
0 (0, 0)−K+
0 (0, 0)
)
= a+
12, a+
21 = 0, a+
22 = 1
(9)
and denoting S−ik (λ) = a−ik + O
(
1
λ
)
, we find that the functions
F±
kj (t) =
1
2π
∞∫
−∞
{
S±kj (λ)− a±kj (λ)
}
eiλtdλ (10)
belong to the space L2 (−∞; +∞).
Using paper [3, Th. 4], we can write a spectral expansion of the bundle Lλ.
There holds the following theorem:
Theorem 1. Let the bundle have neither eigenvalues nor spectral properties
and let a smooth up to sixth order function f (x) finite in the vicinity of zero
and infinity be given. Then it holds a uniformly converging for all x ∈ [0;∞)
spectral expansion
f (x) =
1
2πi
∞∫
−∞
λ3 [E1 (λ)ϕ1 (x, λ) + E2 (λ) ϕ2 (x, λ)]dλ, (11)
86 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Uniqueness of Solution of the Inverse Problem of Scattering Theory
where E1 (λ) =
∞∫
0
ϕ∗1 (ξ, λ)f (ξ) dξ, E2 (λ) =
∞∫
0
ϕ∗2 (ξ, λ)f (ξ) dξ and ϕ∗1 (x, λ),
ϕ∗2 (x, λ) are the solutions of the adjoint equation l∗
(
x, d
dx , λ
)
z = 0 satisfying
the conditions ϕ∗i (0, λ) = ϕ∗i
′ (0, λ) = 0, i = 1, 2.
From (7) we take the function ϕ+
i (x, λ) and multiply it by 1
2πeiλt, then we
integrate it with respect to λ on (−∞;∞) under assumption that t > x. Thus,
in the sense of convergence in a class of distributions we get
∞∫
−∞
eiλxeiλtS+
11 (λ)dλ +
∞∫
−∞
S+
11 (λ)
∞∫
x
K+
0 (x, ξ) eiλξdξ
eiλtdλ
+
∞∫
−∞
xeiλxeiλtS+
11 (λ)dλ +
∞∫
−∞
S+
12 (λ)
∞∫
x
K+
1 (x, ξ) eiλξdξ
eiλtdλ
−
∞∫
−∞
e−iλxeiλtdλ−
∞∫
−∞
∞∫
x
K+
0 (x, ξ)e−iλξdξ
eiλtdλ = 0,
∞∫
−∞
eiλxeiλtS+
12 (λ)dλ +
∞∫
−∞
S+
11 (λ)
∞∫
x
K+
0 (x, ξ) eiλξdξ
eiλtdλ
+
∞∫
−∞
xeiλxeiλtS+
22 (λ)dλ +
∞∫
−∞
S+
22 (λ)
∞∫
x
K+
1 (x, ξ) eiλξdξ
eiλtdλ
−
∞∫
−∞
e−iλxeiλtdλ−
∞∫
−∞
∞∫
x
K+
1 (x, ξ)e−iλξdξ
eiλtdλ = 0. (12)
Analogously, by multiplying the function ϕ−i (x, λ) by 1
2πe−iλt, we get the similar
relations from (7). In sequel, we consider the asymptotic formulas (8) and arrive
at the following theorem.
Theorem 2. If the bundle Lλ has neither eigenvalues nor spectral properties,
then for each x ≥ 0 the kernels K±
i (x, t), (x < t < ∞), of the transformation
operator satisfy the following system of the main equations of Marchenko type [4]:
F±
11 (x + t) +
∞∫
x
K±
0 (x, ξ)F±
11 (t + ξ) dξ + xF±
12 (x + t)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 87
E.G. Orudzhev
+
∞∫
x
K±
1 (x, ξ)F±
12 (t + ξ) dξ −K±
0 (x, t) = 0,
F±
21 (x + t) +
∞∫
x
K±
0 (x, ξ)F±
21 (t + ξ) dξ + xF±
22 (x + t)
+
∞∫
x
K±
1 (x, ξ)F±
22 (t + ξ) dξ −K±
1 (x, t) = 0. (13)
Let us study the properties of the transition function F±
kj (x) . In the main
equations (13) we assume t = x and t = 2x, and find
F±
11 (2x) + xF±
12 (2x) +
∞∫
x
K±
0 (x, ξ)F±
11 (x + ξ) dξ
+
∞∫
x
K±
1 (x, ξ)F±
12 (x + ξ) dξ = K±
0 (x, x),
F±
11 (3x) + xF±
12 (3x) +
∞∫
x
K±
0 (x, ξ)F±
11 (2x + ξ) dξ
+
∞∫
x
K±
1 (x, ξ)F±
12 (2x + ξ) dξ = K±
0 (x, 2x).
Hence
F±
11 (η) +
η
2
F±
12 (η) +
∞∫
η
2
K±
0
(η
2
, ξ
)
F±
11
(η
2
+ ξ
)
dξ
+
∞∫
η
2
K±
1
(η
2
, ξ
)
F±
12
(η
2
+ ξ
)
dξ = K±
0
(η
2
,
η
2
)
,
F±
11 (η) +
η
3
F±
12 (η) +
∞∫
η
3
K±
0
(η
3
, ξ
)
F±
11
(
2η
3
+ ξ
)
dξ
+
∞∫
η
3
K±
1
(η
3
, ξ
)
F±
12
(
2η
3
+ ξ
)
dξ = K±
0
(
η
3
,
2η
3
)
.
88 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Uniqueness of Solution of the Inverse Problem of Scattering Theory
From formula (52) of paper [2] for the estimation of the kernels of transformation
operators ∣∣∣K±
j (x, t)
∣∣∣ ≤ 1
4
σj
(
x + t
2
)
eτ(x),
where
σj
(
x + t
2
)
=
∞∫
x+t
2
sj+1
{|p (s)|+ |r (s)|+ s
[∣∣r′ (s)∣∣ + |q (s)|]} ds, j = 0, 1,
τ (x) =
∞∫
x
s2
{|p (s)|+ |r (s)|+ s
(∣∣r′ (s)∣∣ + |q (s)|)} ds,
and for their derivatives, for example, for the first derivatives, where the estima-
tions
{∣∣∣∣∣
∂K±
j (x, t)
∂x
∣∣∣∣∣ ,
∣∣∣∣∣
∂K±
j (x, t)
∂t
∣∣∣∣∣
}
≤
∞∫
x+t
2
ξ
{∣∣p′j (ξ)
∣∣ +
∣∣r′ (ξ)∣∣ + |qj (ξ)|} dξ
+
1
2
∞∫
x+t
2
{|pj (ξ)|+ |rj (ξ)|} dξ +
1
4
σj
(
x + t
2
)
σ0 (x) eτ(x),
hold, it follows that
∣∣∣F±
11 (η) +
η
2
F±
12 (η)
∣∣∣ ≤
∣∣∣K±
0
(η
2
,
η
2
)∣∣∣+
∞∫
η
2
∣∣∣K±
0
(η
2
, ξ
)∣∣∣
2
dξ
∞∫
η
2
∣∣∣F±
11
(η
2
+ ξ
)∣∣∣
2
dξ
1
2
+
∞∫
η
2
∣∣∣K±
1
(η
2
, ξ
)∣∣∣
2
dξ
∞∫
η
2
∣∣∣F±
12
(η
2
+ ξ
)∣∣∣
2
dξ
1
2
≤ C1e
−δη, δ > 0 (14)
and also ∣∣∣F±
11 (η) +
η
3
F±
12 (η)
∣∣∣ ≤ Ce−δη,
where C1, C and δ are constant numbers, δ < ε
2 .
In the similar way we show
∣∣∣F±
21 (η) +
η
2
F±
22 (η)
∣∣∣ ≤ C2e
−δη, η > 0. (15)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 89
E.G. Orudzhev
Using these estimations, from (13) we find that F±
kj (x) has the same number of
the derivatives as K±
i (x, t). Furthermore, the derivatives of the function expo-
nentially decrease.
For each fixed x = x0 ∈ [0;∞) the main system of equations (13) gives four
equations for determining K±
0 (x, t) and K±
1 (x, t) , and it follows from (14), (15)
that the kernels of these equations generate the Hilbert-Schmidt type operators.
Thus, the equations
F±
11 (x0 + t) + x0F
±
12 (x0 + t) +
∞∫
x0
K±
0 (x0, ξ) F±
11 (ξ + t)dξ
+
∞∫
x0
K±
1 (x0, ξ) F±
12 (ξ + t)dξ −K±
0 (x0, t) = 0,
F±
21 (x0 + t) + x0F
±
22 (x0 + t) +
∞∫
x0
K±
0 (x0, ξ) F±
21 (ξ + t)dξ (16)
+
∞∫
x0
K±
1 (x0, ξ) F±
22 (ξ + t)dξ −K±
1 (x0, t) = 0
are of the Fredholm type.
It is directly verified that if the coefficients of equation (1) are real, then
K±
i (x0, t) = K∓
i (x0, t) , i = 0, 1.
Definition. We call the matrix function S± (λ) =
{
S±ij (λ)
}2
i,j=1
a scattering
matrix of the bundle Lλ, and its elements S±ij (λ) we call the scattering data of
the bundle Lλ.
Notice that the scattering data are completely determined by the asymptotics
of eigenfunctions ϕ±i (x, λ) of a continuous spectrum as x →∞.
Now we put a problem on the restoration of the bundle Lλ by a scattering
matrix.
Theorem 3. For each fixed x = x0 ∈ [0;∞) the main system of integral
equations (13) in the class of twice differentiable functions from L2 (x0,∞) has
a unique solution K±
0 (x0, t) K±
1 (x0, t).
90 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Uniqueness of Solution of the Inverse Problem of Scattering Theory
P r o o f. By the Fredholm property of the system (16) it suffices to show
that the system of homogeneous equations
∞∫
x0
[
f0 (ξ) F±
11 (ξ + t) + f1 (ξ) F±
12 (ξ + t)
]
dξ − f0 (t) = 0,
∞∫
x0
[
f0 (ξ) F±
21 (ξ + t) + f1 (ξ) F±
22 (ξ + t)
]
dξ − f1 (t) = 0
(17)
has only a zero solution from L2 (x0,∞).
Consider the contrary: the system (17) has nonzero solutions f0 (t) and f1 (t),
possessing twice differentiable derivative from L2 (x0;∞) . Assume
Fj (λ) =
1
2π
∞∫
x0
fj (ξ)eiλξdξ, (18)
fj (ξ) =
∞∫
−∞
Fj (λ)e−iλξdξ, fj (ξ) =
∞∫
−∞
Fj (λ)eiλξdλ. (19)
Multiply (17) by the derivatives of the twice differential functions g0 (t) , g1 (t)
from L2 (x0;∞) and integrate it with respect to t ∈ [x0;∞) . Then
∞∫
−∞
[
S+
11 (λ) F0 (λ) + S+
12 (λ) F1 (λ)
]
G0 (λ)dλ
−
∞∫
−∞
[
a+
11 (λ) F0 (λ) + a+
12 (λ) F1 (λ)
]
G0 (λ)dλ−
∞∫
−∞
F0 (λ)G0 (λ) dλ = 0,
∞∫
−∞
[
S+
21 (λ) F0 (λ) + S+
22 (λ) F1 (λ)
]
G1 (λ)dλ
−
∞∫
−∞
[
a+
21 (λ) F0 (λ) + a+
22 (λ) F1 (λ)
]
G1 (λ)dλ−
∞∫
−∞
F1 (λ)G1 (λ) dλ = 0,
where Gj (λ) =
∞∫
x0
gj (t)eiλtdt, and a+
ij determines the asymptotics of S+
ij (λ).
Obviously,
∞∫
−∞
a+
ijFs (λ)Gj (λ) dλ = 0.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 91
E.G. Orudzhev
Therefore
∞∫
−∞
[
S+
11 (λ) F0 (λ) + S+
12 (λ) F1 (λ)− F0 (λ)
]
G0 (λ)dλ = 0,
∞∫
−∞
[
S+
21 (λ) F0 (λ) + S+
22 (λ) F1 (λ)− F1 (λ)
]
G1 (λ)dλ = 0.
It follows from the relations
∞∫
−∞
f ′i (x) e−iλxdx = fi (x) e−iλx
∣∣∣∣∣
∞
−∞
+iλ
∞∫
−∞
fi (x) e−iλxdx = iλFi (λ)
that Fi (λ) = O
(
1
λ
)
. Then there exist the functions γ0 (ξ) and γ1 (ξ) from
L2 (x0;∞) such that
S+
11 (λ) F0 (λ) + S+
12 (λ)F1 (λ)− F0 (λ) =
∞∫
−x0
γ0 (ξ) eiλξdξ = H0 (λ),
S+
21 (λ) F0 (λ) + S+
22 (λ)F1 (λ)− F1 (λ) =
∞∫
−x0
γ1 (ξ) eiλξdξ = H1 (λ).
In a matrix notation
S+ (λ) F (λ)− F (λ) = H (λ) , (20)
where
S+ (λ) =
(
S+
11 (λ) S+
12 (λ)
S+
21 (λ) S+
22 (λ)
)
, F (λ) =
(
F0 (λ)
F1 (λ)
)
, H (λ) =
(
H0 (λ)
H1 (λ)
)
.
The definition of S+ (λ) yields S+ (λ) S+ (λ) = E. Therefore, from (20) we get
S+ (λ)F (λ)− F (λ) = H (λ)
that is equivalent to
F (λ)− S+ (λ) F (λ) = S+ (λ) H (λ). (21)
By (19) and (20), S+ (λ) H (λ) = −H (λ) .
Consider the case of x0 = 0. It is directly verified that H (λ) and H (λ)
are holomorphic functions in the upper and lower half-planes, respectively, and
92 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Uniqueness of Solution of the Inverse Problem of Scattering Theory
that they decrease. The functions F+
1 (0, λ), F−
1 (0, λ) and F−
1 (0, λ) H (λ) =
−F+
1 (0, λ) H (λ) are also holomorphic. Thus, the function
M (λ) =
{
F+
1 (0, λ) H (λ) , Imλ > 0,
F−
1 (0, λ) H (λ) , Imλ < 0,
is an entire function of λ on a real axis M (λ) → 0 as |λ| → ∞. From the well-
known Privalov’s theorem we conclude that M (λ) ≡ 0. Then H (λ) ≡ 0, and
therefore S+ (λ) F (λ)− F (λ) ≡ 0. From hence, after simple transformations we
get F (λ) ≡ 0. Then f0 (t) ≡ f1 (t) ≡ 0, but this is a contradiction. The theorem
is proved for the case of x0 = 0.
Before proving the solvability of the main equations for x0 > 0 we should
notice that the dependence of the transient functions F+
kj (x, t) for x, t ≥ x0 on
the values of coefficients of the functions r (x) , p (x) and q (x) for x ≥ x0 is
revealed from the main integral equations. Therefore, in L2(x0;∞) we consider
a new boundary value problem
(
d2
dx2
+ λ2
)2
y + r (x) y′ + (λp (x) + q (x)) y = 0,
y (x0) = y′ (x0) = 0.
This problem has its own scattering matrix S̃ (x0, λ), and in the main equation
the transition function is determined as follows:
Fkj (x, t) =
1
2π
∞∫
−∞
{
S̃kj (x0, λ)− akj (x0) e−2iλx0
}
eiλ(x+t)dλ, x, t ≥ x0.
Thus, in the main equations for to study their solvability we can use this formula
for Fkj (x, t). Then the reasonings cited for x0 = 0 should be completely repeated.
The theorem is proved.
We introduce the denotations:
F (x0 + t) =
F+
11 (x0 + t) + x0F
+
12 (x0 + t)
F+
21 (x0 + t) + x0F
+
22 (x0 + t)
F−
11 (x0 + t) + x0F
−
12 (x0 + t)
F−
21 (x0 + t) + x0F
−
22 (x0 + t)
, K (x0, t) =
K−
0 (x0, t)
K−
1 (x0, t)
K+
0 (x0, t)
K+
1 (x0, t)
,
H (ξ + t) =
F+
11 (ξ + t) F+
12 (ξ + t) 0 0
F+
21 (ξ + t) F+
22 (ξ + t) 0 0
0 0 F−
11 (ξ + t) F−
12 (ξ + t)
0 0 F−
21 (ξ + t) F−
22 (ξ + t)
.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 93
E.G. Orudzhev
Then the system of equations (16) can be written in the form
K (x0, t) = F (x0 + t) +
∞∫
x
K (x0, ξ)H (ξ + t)dξ,
or K = F+HK, where H is a linear integral operator in the space L2
(
[x0;∞) ;E4
)
;
E4 is a complex four-dimensional space, Hf =
∞∫
x
H (ξ + t)f(t)dt. Here f is any
function from L2
(
[x0;∞) ; E4
)
, f = (f0, f1, f2, f3), f
(k)
i (x) ∈ L2
(
[x0;∞) ;E4
)
,
k = 1, 2, i = 0, 3. A scalar product in this space is determined as
(f, g) =
3∑
j=0
∞∫
x0
fj (x)gj (x)dx.
From the introduced norm of the space L2
(
[x0;∞) ;E4
)
it is easy to
get ‖H‖ < 1, and therefore the system of the integral equations is solved by
a successive approximation method.
Considering that conditions (6) allow to determine uniquely all the coefficients
of equation (1) with respect to K±
j (x, t) , j = 0, 1, we arrive at the main theorem.
Theorem 4. If the bundle Lλ has neither eigenvalues nor spectral properties,
then by the scattering matrix S± (λ) equation (1) is determined in a unique way.
R e m a r k. The above stated inverse problem is overdetermined. There are
definite ties between the scattering data. We can find relations between them if
for −∞ < λ < ∞ we can write the representations ϕ−i (x, λ) = Ai (λ) ϕ+
1 (x, λ) +
Bi (λ) ϕ+
2 (x, λ), and then, taking into account a linear independence of the system
xje±iλx, j = 0, 1, from the last equalities as x →∞ we find the following relations:
S−11 = −A1 (λ) , S−12 = B1 (λ) , S−11S
+
11 − S−12S
+
21 = 1, S−11S
+
12 − S−12S
+
22 = 0,
S−21 = −A2 (λ) , S−22 = B2 (λ) , S−21S
+
12 − S−22S
+
22 = 1, S−21S
+
11 − S−22S
+
21 = 0.
94 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Uniqueness of Solution of the Inverse Problem of Scattering Theory
References
[1] M.G. Gasymov and A.M. Maherramov, The Existence of Transformation Operators
for Higher Order Differential Equations that Depend Polynomially on a Parameter.
— DAN SSSR 235 (1977), No. 2, 259–263.
[2] M.G. Gasymov and A.M. Maherramov, Transformation Operators with Conditions
at Infinity for a Class of Even Order Differential Bundles. Preprint No. 7. Inst. Phys.
AS of Azerbaijan SSR, Baku, 1986, 29 p.
[3] E.G. Orudzhev, Spectral Analysis of a Class of Differential Bundles with Multiple
Characteristics. — Rep. NAS Azerbaijan 58 (2002), No. 5–6, 40–46.
[4] V.A. Marchenko, Sturm-Liouville Operators and their Applications. Naukova
Dumka, Kiev, 1977.
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