Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients
The purpose of the present work is to solve the characterization problem, which consists of identi cation of necessary and su cient conditions on the scattering data ensuring that the reconstructed potential belongs to particular class.
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| Опубліковано в: : | Журнал математической физики, анализа, геометрии |
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| Дата: | 2006 |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients / R.F. Efendiev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 73-86. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859582366798839808 |
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| author | Efendiev, R.F. |
| author_facet | Efendiev, R.F. |
| citation_txt | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients / R.F. Efendiev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 73-86. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Журнал математической физики, анализа, геометрии |
| description | The purpose of the present work is to solve the characterization problem, which consists of identi cation of necessary and su cient conditions on the scattering data ensuring that the reconstructed potential belongs to particular class.
|
| first_indexed | 2025-11-27T06:55:00Z |
| format | Article |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 1, pp. 73�86
Complete Solution of an Inverse Problem for One Class
of the High Order Ordinary Di�erential Operators
with Periodic Coe�cients
R.F. Efendiev
Baku State University, Institute of Applied Mathematics
23 Z. Khalilov Str., Baku, AZ-1148, Azerbaijan
E-mail:rakibaz@yahoo.com
Received February 8, 2005
The purpose of the present work is to solve the characterization prob-
lem, which consists of identi�cation of necessary and su�cient conditions
on the scattering data ensuring that the reconstructed potential belongs to
particular class.
Key words: inverse problem, characterization problem, scattering data,
transformation operator.
Mathematics Subject Classi�cation 2000: 34B25, 34L05, 34L25, 47A40,
81U40.
1. Introduction
The purpose of the present work is solving the characterization problem, which
consists of identi�cation of necessary and su�cient conditions on the scattering
data ensuring that the reconstructed potential belongs to a particular class. In our
case Q2 is the class of all 2�-periodic complex-valued functions on the real axis R,
belonging to the space L2[0; 2�], and Q
2
+ is its subclass consisting of the functions
p
(x) =
1X
n=1
p
nexp (inx);
2m�2X
=0
1X
n=1
n
jp
nj <1: (1.1)
The object under consideration is the operator L, generated by the di�erential
expression
l(y) = (�1)my(2m) +
2m�2X
=0
p
(x)y
(
)(x) (1.2)
c
R.F. Efendiev, 2006
R.F. Efendiev
in the space L2 (�1;1), with the coe�cients p
(x) 2 Q2
+.
Note, that some of the characterizations for the Sturm�Liouville operators in
the real-valued potentials belonging to the L1
1 (R) (L1
�
(R) is the class of mea-
surable potentials satisfying the condition
R
R
dx(1 + jxj)� jp
(x)j <1), have been
given by A. Melin [1] and V.A. Marchenko [2]. More details review can be found
in the papers [3�5].
The inverse problem for the coe�cients (1.1) for the �rst time was formulated
and solved in paper [6], where it was shown that the equation l (y) = �2my has
the solution
'(x; �!� ) = ei�!�x +
2m�1X
j=1
1X
�=1
�X
n=1
V
(j)
n�
n+ �!� (1� !j)
e(i�!�+i�)x; � = 0; 2m� 1;
!j = exp (ij�=m) : (1.3)
and Wronskian of the system of solutions '(x; �!� ) being equal to (i�)m(2m�1)A,
where
A =
��������
1 1 :: 1
!1 !2 :: !2m�1
:: :: :: ::
!2m�1
1 !2m�1
2 :: !2m�1
2m�1
��������
is nonzero if � 6= 0.
The limit 'nj (x) � lim
�!��nj
(�+ �nj)' (x; �), �nj = � n
1�!j
, n 2 N , j =
1; 2m� 1, is also a solution of the equation l (y) = �2my but already linearly
depending on ' (x; �nj!j). Therefore, there exist the numbers ~Snj; n 2 N; j =
1; 2m� 1, for which the conditions
'nj(x) = ~Snj'(x; �nj!j) (1.4)
are ful�lled.
It was established by M.G. Gasymov [6] that if
I.
1P
n=1
n
��� ~Sn��� <1,
II. 4m�1 am
1P
n=1
j ~Snj
n+1
= p < 1, where
am = max
1� j� l� 2m�1
1� n; r <1
j(1� ! j) (n+ r)j
j r (1� ! j) � n (1� ! l) ! j j
; ~Sn =
2m�1X
j=1
n2m�2j ~Snjj;
(1.5)
then there exist the uniquely de�ned functions p
(x),
= 0; 2m� 2 of (1.1), for
which the numbers f ~Sng are de�ned by formulae (1.3)�(1.4). Then the complete
74 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
An inverse problem for the high order periodic di�erential operators
solution of this problem atm = 1 was given in paper [3], where the authors proved
the following theorem:
Theorem 1. In order the given sequence of complex numbers fŜng to be a set
of spectral data of the operator L =
�
d
dx
�2
+ p0 (x) with the potential p0(x) 2 Q2
+
it is necessary and su�cient, that the following conditions are ful�lled:
1) fnŜng
1
n=1 2 l2;
2) the in�nite determinant D (z) �
Ænk + 2Ŝk
n+k
ei
n+k
2
z
1
n;k=1
exists (Ænk is Kro-
necker's symbol), is continuous, not equal to zero in the closed half-plane C+ =
fz : Im z � 0g and analytical inside of the open half-plane C+ = fz : Im z > 0g.
In the present work the complete inverse problem (characterization problem)
for the high order ordinary di�erential operators (1.2) with the coe�cients (1.1)
is solved.
Let us formulate now the basic result of the present work.
De�nition. The sequence f ~Snjg
1;2m�1
n=1;j=1, constructed by means of the formulae
(1.4), is called a set of spectral data of the operator (1.2) with the coe�cients (1.1).
Theorem 2. For a given sequence of complex numbers f ~Snjg
1;2m�1
n=1;j=1 to be a set
of spectral data of the operator L, generated by the di�erential expression (1.2)
and coe�cients (1.1), it is necessary and su�cient that the following conditions
are ful�lled:
fn ~Sng
1
n=1 2 l1; (1.6)
2) the in�nite determinant
D (z) � det
ÆrnE2m�1 �
i (1� !l) ~Snj
r!l (1� !j)� n (1� !l)
e
i
n
1�!j
z
e
�i
r!
l
1�!
l
z
2m�1
j;l=1
1
r;n=1
(1.7)
exists, (En is the unit n� n matrix), is continuous, not equal to zero in the close
half-plane C+ = fz : Im z � 0g, and analytical inside of the open half-plane
C+ = fz : Im z > 0g.
2. On the inverse problem of scattering theory on the semiaxis
On the base of the proof of the Theorem 2 we will study the equation l (y) =
�2my.
Denoting
x = it; � = �ik; y (x) = Y (t) ; (2.1)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 75
R.F. Efendiev
we obtain the equation
(�1)
m Y (2m) (t) +
2m�2X
=0
Q
(t) Y
(
) (t) = k2mY (t) ; (2.2)
in which
Q
(t) = (�1)m(�i)
1X
n=1
p
ne
�nt;
2m�2X
=0
1X
n=1
n
jp
nj <1: (2.3)
As a result we obtain the equation (2.1) whose coe�cient exponentially de-
crease as t!1.
Lemma 1. The kernel of the transformation operator of equation (2.2)
K (t; u), u � t, attached to +1, with the coe�cients (2.3) permits the represen-
tation
K(t; u) =
2m�1X
j=1
1X
n=1
1X
�=n
V
(j)
n�
i(1 � !j)
e
��t+ n
1�!j
(t�u)
;
in which the series
2m�1X
j=1
1X
n=1
1
n
1X
�=n
�2m�1 (�� n)
���V (j)
n�
���;
2m�1X
j=1
1X
�=1
�2m�1
���V (j)
��
���
are convergent.
P r o o f. It is shown in [7] that equation (2.2) with the coe�cients (2.3) has
the solution
f (t; k!� ) = eik!� t +
2m�1X
j=1
1X
�=1
�X
n=1
V
(j)
n�
in+ k!� (1� !j)
e(ik!���)t; � = 0; 2m � 1;
(2.4)
and the numbers V
(j)
n� are de�ned from the following recurrent formulae"�
��
n
(1� !j)
�2m
�
�
n
(1� !j)
�2m
#
V (j)
n�
= (�1)m+1
2m�1X
=0
��1X
s=n
�
i
�
s�
n
(1� !j)
��
P
;s�nV
(j)
ns
(2.5)
76 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
An inverse problem for the high order periodic di�erential operators
at
� = 2; 3; : : : ; n = 1; 2; : : : ; �� 1; j = 1; 2; : : : ; 2m� 1;
i
p
� +
2m�1X
j=1
�X
n=1
dj
(n; �)V
(j)
n�
+
2m�2X
�=
+1
2m�1X
j=1
X
r+s=�
sX
n=1
dj
(n; s; �)p�rV
(j)
ns
= 0;
(2.6)
where
1
n+ k(1� !j)
�
(i�+ k)2m � k2m � (i� + knj)
2m + k2m
nj
�
=
2m�2X
=0
dj
(n; �) k
; j = 1; 2m� 1;
(is+ k)� � (is+ knj)
�
in+ k (1� !j)
=
v�1X
=0
dj
(n; s; v) k
;
and the series (2.4) permits 2m times term by term di�erentiation. Then according
to conditions (2.3), we have
f (t; k) = eikt +
1Z
t
K (t; u) eikudu; (2.7)
where
K (t; u) =
2m�1X
j=1
1X
n=1
1X
�=n
V
(j)
n�
i (1� !j)
e
��t+ n
1�!j
(t�u)
: (2.8)
The lemma is proved.
Then it is possible to get equality [8]
fnj(t) = Snjf(t; knj!j); (2.9)
where
fnj (t) = lim
k!knj
[in+ k (1� !j)] f (t; k); knj = �
in
1� !j
; j = 1; 2m� 1; n 2 N:
Rewriting equality (2.9) in the form
1X
�=n
V (j)
n� e
��te
n
1�!j
t
= Snje
n!j
1�!j
t
+
2m�1X
l=1
1X
r=1
1X
�=r
i (1� !j)V
(l)
nr Snj
n!j (1� !l)� r (1� !j)
e
�
��+
n!
j
1�!j
�
t
(2.10)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 77
R.F. Efendiev
and denoting by
~F (t+ u) =
2m�1X
j=1
1X
n=1
Snj
i (1� !j)
e
n
1�!j
(t!j�u)
; t � u; (2.11)
we obtain the Marchenko type equation
K(t; u) = ~F (t+ u) +
1Z
t
K(t; s) ~F (s+ u)ds (2.12)
from equation (2.10). So, it is proved the following
Lemma 2. If the coe�cients Q
(t) of equation (2.2) have form (2.3), then
at every t � 0 the kernel of the transformation operator (2.8) satis�es to the
equation of the Marchenko type (2.12) in which the transition function ~F (t) has
form (2.11), and the numbers Snj are de�ned by equality (2.9), from which it is
obtained, that Snj = V
(j)
nn .
The coe�cients Q
(t) are reconstructed by the kernel of the transformation
operator by means of the recurrent formulae (2.5)�(2.6). Hence, the basic equa-
tion (2.12) and form of the transition function (2.11) make natural the formula-
tion of the inverse problem for reconstruction of coe�cients of equation (2.1) by
numbers Snj. In this formulation, which employs the transformation operator,
an important moment is a proof of unique solubility of the basic equation (2.12).
Lemma 3. The homogeneous equation
g (s)�
1Z
0
~F (u+ s) g (u) du = 0 (2.13)
corresponding to the coe�cients Q
(t) 2 Q2
+ has only a trivial solution.
P r o o f. Let g 2 L2(R
+) be a solution of equation (2.13) and f be a solution
of the equation
f (s) +
sZ
0
K (t; s) f (t) dt = g (s): (2.14)
Substituting g into (2.14) and taking into account equation (2.12), we get
f(s) +
sZ
0
K(t; s)f(t)dt+
1Z
0
[f(u) +
uZ
0
K(t; u)f(t)dt] ~F (u+ s)du
78 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
An inverse problem for the high order periodic di�erential operators
= f(s) +
1Z
s
f(t)[ ~F (t+ s) +
1Z
t
K(t; u) ~F (u+ s)du] = 0:
As at t � s the estimation������ ~F (t+ s) +
1Z
0
K (t; u) ~F (u+ s) du
������ � Ce�s
is ful�lled, hence it follows that f = 0; g = 0 and lemma is proved.
Lemma 4. At every �xed value a; (Im a � 0) the homogeneous equation
g(s) �
1Z
0
~F (u+ s� 2ia)g(u)du = 0 (2.15)
has only a trivial solution in the space L2 (R
+).
P r o o f. We substitute x by x+ a, where Im a � 0 in equation (1.2). Then
we obtain the same equation with the coe�cient Qa
(x) = Q
(x+ a) satisfying
condition (1.1). Let us remark, that the functions ' (x+ a; �!j) are solutions of
the equation
(�1)m y(2m) (x) +
2m�2X
=0
Qa
(x) y
(
) (x) = �2my (x)
that at x!1 have the form
' (x+ a; �!j) = ei�!jaei�!jx + o (1) :
Therefore the functions
'a (x; �!j) = e�i�!ja' (x+ a; �!j)
are also solutions of type (1.3). Then let us denote by Snj(a) the spectral data of
the operator L with the potential Qa
(x)
L � (�1)m
d2m
dx2m
+
2m�2X
=0
Qa
(x)
d
dx
:
According to (1.4), we have
Snj (a)'
a (x; �nj!j) = lim
�!�nj
[n+ � (1� !j)]'
a (x; �)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 79
R.F. Efendiev
= lim
�!�nj
[n+ �(1� !j)]'(x + a; �) e�i�a
= e
i
n
1�!j
a
Snj' (x+ a; �nj!j) = e
i
n
1�!j
a
e
�i
n!j
1�!j
a
Snj'
a (x; �nj!j)
= einaSnj'
a (x; �nj!j) :
Hence
Snj(a) = einaSnj: (2.16)
Now arguing as above, we obtain the basic equation of form (2.12) with the
transition function
~Fa(t+u) =
2m�1X
j=1
1X
n=1
Snj(a)
i(1 � !j)
e
n
1�!j
(t!j�u)
= ~F (t� ia+ u� ia) = ~F (t+u� 2ia):
From this lemma follows
Theorem 3. The coe�cients Q
(t) of equation (2.1), satisfying to condition
(2.2), are uniquely de�ned by the numbers Snj.
3. Proof of Theorem 2
Necessity. From the relation (2.9) and form of the function fnj (t) we ob-
tained that
Snj = V (j)
nn
:
Therefore
2m�1X
j=1
1X
n=1
n2m�1 jSnjj �
2m�1X
j=1
1X
n=1
n2m�1
���V (j)
nn
��� <1;
i.e., n2m�1 jSnjj 2 l1. The necessity of condition (1) is proved.
To proof the necessity of condition (2) let us demonstrate �rst of all that from
the trivial solubility of the basic equation (2.12) at t = 0 in the class of functions,
satisfying to the inequality kg (u)k � Ce�
u
2 , u � 0, follows the trivial solubility
in l2
�
1;1; R2m�1
�
of the in�nite system of equations
gjn �
2m�1X
l=1
1X
r=1
i(1� !j)Srl
n!j(1� !l)� r(1� !j)
glr = 0: (3.1)
Really, if fgjng 2 l2, j = 1; 2m� 1; is a solution of this system, then the
function
g (u) = (g1 (u) ; g2 (u) ; : : : ; g2m�1 (u)) =
2m�1X
j=1
1X
n=1
Snjgjne
�
n
1�!j
u
; (3.2)
80 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
An inverse problem for the high order periodic di�erential operators
de�ned for all u � 0, satis�es to the inequality
jg (u)j � Ce
�
u
1�!j ; u � 0;
and is a solution of equation (2.13)
g(u) �
1Z
0
g(s) ~F (u+ s)ds =
2m�1X
j=1
1X
n=1
Snjgjne
�
n
1�!j
u
�
1Z
0
(
2m�1X
j=1
1X
n=1
Snjgjne
�
n
1�!j
s
)(
2m�1X
l=1
1X
r=1
Srl
i(1� !j)
e
r
1�!
l
(s!l�u))ds
=
2m�1X
j=1
1X
n=1
Snjgjne
�
n
1�!j
u
�
2m�1X
j=1
1X
n=1
2m�1X
l=1
1X
r=1
SnjSrl
i(1 � !l)
gjne
�
r
1�!
l
u
1Z
0
e
�
n
1�!j
s
e
r!
l
1�!
l
s
ds
=
2m�1X
j=1
1X
n=1
Snjgjne
�
n
1�!j
u
�
2m�1X
j=1
1X
n=1
2m�1X
l=1
1X
r=1
i(1� !j)SnjSrl
n!j(1� !l)� r(1� !j)
glre
�
n
1�!j
u
=
2m�1X
j=1
1X
n=1
Snje
�
n
1�!j
u
[gjn �
2m�1X
l=1
1X
r=1
i(1� !j)Srl
n!j(1� !l)� r(1� !j)
glr] = 0:
Since, g(u) = 0, then Snjgjn = 0 for all n � 1, j = 1; 2m� 1, and
gjn = 0, j = 1; 2m� 1, n � 1, according to (3.1). Let us introduce in the space
l2
�
1;1;R2m�1
�
the operator F (t), given by the matrix
Frn (t) =
F jl
rn
2m�1
j;l=1
=
i (1� !l)Snj
r!l (1� !j)� n (1� !l)
e
�
n
1�!j
t
e
r!
l
1�!
l
t
2m�1
j;l=1
; (3.3)
and let '2k�1 = f(Æ�1) Ækrg
2m�1;1
�;r=1 , '2k = f(Æ�2) Ækrg
2m�1;1
�;r=1 ((Æij) is
a column vector) be the orthonormal system in this space. Then we obtain from
n2m�1 jSnjj 2 l1
1P
j;k=1
���(F'j ; 'k)l2(1;1;R2m�1)
��� < 1, i.e., F (t) is a kernel ope-
rator [9]. Therefore, there exists the determinant �(t) = det (E � F (t)) of the
operator E � F (t), connected, as easy to see, with the determinant D(z) from
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 81
R.F. Efendiev
condition 2) of the Theorem 2, with relation �(�iz) = det (E � F (�iz)) �
D (z).
The determinant of system (3.1) isD (0) and determinant of the similar system
corresponding to the coe�cient Qz
= Q
(x+ z), Im z � 0, is
D(z) � det
ÆrnE2m�1 �
i(1 � !l)Snj(z)
r!l(1� !j)� n(1� !l)
2m�1
j;l=1
1
r;n=1
= det
ÆrnE2m�1 �
i(1� !l)Snj
r!l(1� !j)� n(1� !l)
einz
2m�1
j;l=1
1
r;n=1
= det
ÆrnE2m�1 �
i(1� !l)Snj
r!l(1� !j)� n(1� !l)
e
i
n
1�!j
z
e
�i
r!
l
1�!
l
z
2m�1
j;l=1
1
r;n=1
:
Therefore, in order to prove the necessity of condition 2) of Theorem 2 one
should check that �(0) = D (0) 6= 0. System (3.1) can be written in
l2
�
1;1;R2m�1
�
as the equation
g � F (0) g = 0:
As F (0) is the kernel operator, we can apply to this equation the Fredholm
theory, according to which its trivial solvability is equivalent to condition that
det (E � F (0)) is not equal to zero [10]. The necessity of condition 2) is proved.
Su�ciency. Let us multiply equation (2.12) by e
r!
l
1�!
l
u
and integrate it over
u 2 [t;1). We obtain
k (t) = F (t) e (t) + k (t)F (t) ; (3.4)
in which the operator F (t) is de�ned by the matrix kFrn (t)k
1
r;n=1of the form (3.3),
e (t) = kenj (t)k
1;2m�1
n;j=1
=
e
n!j
1�!j
t
1;2m�1
n;j=1
;
k (t) = kklr (t)k
2m�1;1
l;r=1 =
1Z
t
K (t; u) e
r!
l
1�!
l
u
du
2m�1;1
l;r=1
:
As F (t) is the trace class for t � 0 and the condition �(t) = det (E � F (t)) 6= 0
holds, there exists the bounded in l2 inverse operator R (t) = (E � F (t))�1. Since
F (t) e (t) 2 l2, then from (3.4) we get
k(t) = R(t)F (t)e(t): (3.5)
82 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
An inverse problem for the high order periodic di�erential operators
Now denoting hf; gi =
1P
n=1
fngn, we �nd from (2.12) that
K (t; u) = he (t) ; A (u)i+ hk (t) ; A (u)i = he (t) ; A (u)i+ hR (t)F (t) e (t) ; A (u)i
= he (t) +R (t)F (t) e (t) ; A (u)i = hR (t) e (t) ; A (u)i ; (3.6)
where A (u) is de�ned by the matrix
A(u) = kajn(u)k
2m�1;1
j;n=1
=
Snj
i(1� !j)
e
�
n
1�!j
u
2m�1;1
j;n=1
:
Now assume that the conditions of the theorem are ful�lled. According to the
stated considerations, de�ne the function K (t; u) at 0 � t � u by equality (3.6).
Then at u � t we have
K (t; u)�
1Z
t
K (t; s)F (s+ u) ds
= hR (t) e (t) ; A (u)i �
1Z
t
hR (t) e (t) ; A (s) he (s) ; A (u)iids
= hR(t)e(t); A(u)i �
*
R(t)e(t);
* 1Z
t
A(s)e(s)ds;A(u)
++
= hR(t)e(t); A(u)i
� hR (t) e (t) ; hA (u) ; F (t)ii = hR (t) e (t) ; A (u)i � hR (t) e (t) ; A (u)F � (t)i
= hR (t) e (t) ; A (u)�A (u)F � (t)i = he (t) ; A (u)i = F (t+ u) ;
where the symbol ½* � means transition to the matrix, adjoint to the F (t) with
respect to the bilinear form h: ; :i. So, we have
Lemma 5. For any t � 0 the kernel K (t; u) of the transformation operator
satis�es to the basic equation
K (t; u) = ~F (t+ u) +
1Z
t
K (t; s) ~F (s+ u) ds:
From Lemma 3 it follows unique solubility of the basic equation. By the direct
substitution it is easy to calculate that the solution of the basic equation is
K (t; u) =
2m�1X
j=1
1X
n=1
1X
�=n
V
(j)
n�
i (1� !j)
e
��t+ n
1�!j
(t�u)
;
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 83
R.F. Efendiev
where the numbers V
(j)
n� are de�ned from the recurrent relations
V (j)
nn
= Snj ;
V
(j)
n�+n
= (1� !j)V
(j)
nn
2m�1X
l=1
�X
r=1
V
(l)
r�
r(1� !j)� n(1� !l)!j
:
Passing to the proof of the basic statement that the coe�cients Q
(t) have
form (2.2), let us �rst establish the estimations for the matrix elements Rrn (t) of
the operator R (t) ���Rjl
rn
(t)
��� � ÆrnÆjl + CSn ; (3.7)���� @2m��@t2m��
Rjl
rn
(t)
���� � CSn; � = 1; 2m� 1; (3.8)
where C = maxfCk > 0; k = 1; 2m� 1g is a constant, and Sn =
2m�1P
j=1
n2m�1 jSnjj.
Indeed, it follows from the identity R (t) = E +R (t)F (t) that
���Rlj
rn(t)
��� � ÆrnÆlj +
2m�1X
�=1
1X
p=1
���Rl�
rp(t)
��� ��F �j
pn(t)
��
� ÆrnÆlj + 2
2m�1X
�=1
(
1X
p=1
���Rl�
rp(t)
���2) 12 ( 1X
p=1
��F �j
pn(t)
��2) 12
� ÆrnÆlj + C1am((R(t)R�(t))pp
1X
p=1
1
(n+ p)2
)
1
2Sn � ÆrnÆlj + C1 kR(t)k
l2!l2
Sn:
On the other hand, as it has been noted above, the operator-function R (t) =
(E � F (t))�1 exists and is bounded in the l2(because F (t) is the kernel operator
at t � 0 and �(t) = det (E � F (t)) 6= 0) that proves �rst inequality (3.7).
In order to proof the second estimation (3.8) we �rst obtain
���� ddtRlj
rn
(t)
���� �
2m�1X
�=1
1X
p;q=1
���Rl�
rp
(t)
���
���� ddtF �k
pq
(t)
����
���Rkj
qn
(t)
���
�
2m�1X
�=1
1X
p;q=1
(ÆrpÆl� + C2Sp)Sq (ÆqnÆkj + C3Sn)
� (1 + C4
1X
p=1
Sp)
2Sn � CSn:
84 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
An inverse problem for the high order periodic di�erential operators
Then from the equality d
l
dtl
R (t) =
lP
n=1
Cn
l
�
d
l�n
dtl�n
R (t)
� �
d
n
dtn
F (t)
�
R (t),
l = 1; 2m� 1, by the help of mathematical induction the inequality (3.8) is proved.
In [11] the following relations were proved (for correspondence to our case, assume
that q2m�2�
(x) = Q
(x)):
(�1)m
@2m
@x2m
K (x; t) +
2m�2X
=0
q2m�2�
(x)
@
@x
K (x; t)�
@2m
@t2m
K (x; t) = 0;
q0 (x) = 2m
d
dx
K (x; x) ;
qk+1 (x) =
kX
�=0
q� (x)
kX
s=�
Ck�s
2m�3�s
�
@s��
@xs��
K (x; t)
����
t=x
�(k�s)
+
k+2X
k=0
Ck+2��
2m�1��
�
@�
@x�
K (x; t)
����
t=x
�(k+2��)
� (�1)k
@k+2
@tk+2
K (x; t)
����
t=x
;
k = 0; 1; : : : ; 2m� 3:
Now it is easy to show, that
q0 (x) =
2m�1X
j=1
1X
n=1
n � Snj
i (1� !j)
e�nt +�0 (t);
where �0 (t) =
1P
n;p;q;�=1
R
ej
npF
j�
pq eq�e�q =< R (t)F (t) e (t) ; A (t) > is 2i� periodic
function and has bounded derivative until (2m-1) order. It follows from this
fact that the Fourier coe�cients of the function �0 (�ix), x 2 R, are such that
1P
n=1
��n2m�1�n
��2 <1. But then
1P
n=1
n2m�2 j�nj <1. So, the Fourier coe�cients
P2m�2;n of the function Q2m�2 (x) = q0 (x) satisfy condition (2.2). Similarly,
for all other coe�cients Q
(x),
= 0; 2m� 3, it is established that the Fourier
coe�cients p
n of the function Q
(x) = q2m�2�
(x),
= 0; 2m � 3, satisfy
condition (2.2). And it means that the Fourier coe�cients of the function P
(x),
= 0; 2m � 2 satisfy condition (1.1).
Let, �nally,
n
~Snj
o
be the spectral data set of the operator
�
L� k2mE
�
with
the constructed coe�cients P
(x). For completing of the proof it remains to show
that fSnjg coincide with the initial set
n
~Snj
o
. This follows from the equality
~Snj = V
(j)
nn = Snj. The theorem is proved.
The author is grateful to Prof. M.G. Gasymov and Prof. I.M. Guseynov for
useful discussions.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 85
R.F. Efendiev
References
[1] A. Melin, Operator methods for inverse scattering on the real line. � Comm. Part.
Di�. Eq. 10 (1985), 677�766.
[2] V.A. Marchenko, Sturm�Liouville operators and applications. Birkhauser, Basel,
1986.
[3] L.A. Pastur and V.A. Tkachenko, An inverse problem for one class of one-
dimensional Shchrodinger's operators with complex periodic potentials. � Funkts.
Anal. Prilozen. 54 (1990), 1252�1269. (Russian)
[4] P. Deift and D. Trubowitz, Inverse scattering on the line. � Comm. Pure Appl.
Math. 32 (1979), 121�251.
[5] T. Aktosun and M. Klaus, Inverse theory: problem on the line. Ch. 2.2.4. Scattering
(E.R. Pike and P.C. Sabatier, Eds.). Acad. Press, London, 2001.
[6] M.G. Gasymov, Spectral analysis of one-class nonselfadjoint ordinary di�erential op-
erators with periodic coe�cients. � Spectr. Theory Operators. Publ. House "ELM",
Baku 4 (1982), 56�97. (Russian)
[7] M.G. Gasymov, Uniqueness of the solution of an inverse problem of scattering
theory for one class of even order ordinary di�erential operators. � Dokl. Akad.
Nauk USSR 266 (1982), 1033�1036. (Russian)
[8] R.F. Efendiev, An inverse problem for one class of ordinary di�erential operators
with the periodic coe�cients. � Mat. �z., analiz, geom. 11 (2004), 114�121. (Rus-
sian)
[9] I.T. Gohberg and M.G. Crein, Introduction to the theory of linear nonselfadjoint
operators. Nauka, Moscow, 1965. (Russian)
[10] V.I. Smirnov, Course of higher mathematics. V. 4. GITTL, Moscow, Leningrad,
1951. (Russian)
[11] I.G. Khachatrian, On existence of a transformation operator for high order di�eren-
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86 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
|
| id | nasplib_isofts_kiev_ua-123456789-106582 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-11-27T06:55:00Z |
| publishDate | 2006 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Efendiev, R.F. 2016-09-30T19:23:35Z 2016-09-30T19:23:35Z 2006 Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients / R.F. Efendiev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 73-86. — Бібліогр.: 11 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106582 The purpose of the present work is to solve the characterization problem, which consists of identi cation of necessary and su cient conditions on the scattering data ensuring that the reconstructed potential belongs to particular class. The author is grateful to Prof. M.G. Gasymov and Prof. I.M. Guseynov for useful discussions. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients Article published earlier |
| spellingShingle | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients Efendiev, R.F. |
| title | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients |
| title_full | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients |
| title_fullStr | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients |
| title_full_unstemmed | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients |
| title_short | Complete Solution of an Inverse Problem for One Class of the High Order Ordinary Differential Operators with Periodic Coefficients |
| title_sort | complete solution of an inverse problem for one class of the high order ordinary differential operators with periodic coefficients |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106582 |
| work_keys_str_mv | AT efendievrf completesolutionofaninverseproblemforoneclassofthehighorderordinarydifferentialoperatorswithperiodiccoefficients |