Necessary and sufficient conditions
for the inverse problem of one class ordinary
differential operators with complex periodic coefficients
Efendiev R.F.
Institute of Applied Mathematics BSU,
Z.Khalilova 23, 370148, Baku, Azerbaijan.
rakibaz@yahoo.com
Abstract :
The basic purpose of the present paper is the full solutions of the inverse
problem (i.e. a finding of necessary and sufficient conditions) for the
operator with complex periodic coefficients.
Mathematics Subject classification:
34B25, 34L05, 34L25, 47A40, 81U40
1. Statement of the problem and the formulation of the basic results.
Let's consider a class Q 2 of all 2 periodic complex valid functions on the
real axis R, belonging to space L2 [0,2 ] , and its subclass Q 2 consisting of the
functions of a type

p ( x)   pn exp inx  .
n 1
2 m2 
n



0 n 1
pn   .
(1.1)
The basic purpose of the present paper is the full solutions of the inverse
problem (i.e. a finding of necessary and sufficient conditions) for the operator
L, generated by differential expression
l ( y)  (1) m y ( 2m) 
2 m2
p ( x) y 


( )
( x)
(1.2)
0
in the space L2 (, ) , with potentials of a type (1.1).
The inverse problem, for potentials of a type (1.1) for the first time is put
and solved in work [1] where is shown, that the equation l ( y)  2 m y has the
solutions
 ( x,  )  e i  x 
2 m 1 

Vn(j )
e (i  i ) x ,   0,2m  1 ,  j  exp( ij  / m) (1.3)

n


(
1


)
j 1  1 n 1

j
and Wronckian of the systems of solutions  ( x,   ) being equal to (i ) m( 2m1) A ,
where
A=
1
1
1
2
..
..
12 m1  22 m1
..
1
..  2 m1
..
..
..  22mm11
it is not zero at   0 , and a limit
,
 nj ( x)  lim (   nj ) ( x,  ) . n  N , j  1,2m  1
   nj
also is the solution of the equation l ( y)  2 m y , but linearly dependent on
~
 ( x,  nj j ) , because there are such numbers S nj , n  N , j  1,2m  1 , that satisfy
the relation
~
 nj ( x)  S nj ( x, nj j ) . nj  
n
.
1 j
(1.4)
Further, M.G.Gasymov [1] has established, that if

I.
~
n S
n 1
II. 4
m 1

n

аm 
n 1
where
am 
~
Sn
n 1
 p 1
1   n  r 
r 1     n 1   
~
j
max
1 j  l  2 m 1
1  n, r  
j
l
, Sn 
j
2 m 1
n
j 1
2 m2
~
| S nj | ,
(1.5)
then exist unique determined functions p ( x),   0,2m  2 of the form (1.1), for
~
which numbers {S n } are under formulas (1.3) - (1.4).
The full solution of this problem at m=1 is given in work [5](the present paper
doesn’t include the concerning reference, the detail reference reader can find
in [5,7]) where it is proved the following statement:
In order to the set of sequence of complex numbers {S n } was a set of the
2
d
spectral data of the operator L     p0 ( x) , with potential p0 ( x)  Q2 , it is
 dx 
necessary and sufficient the simultaneously satisfying of the conditions:
1) {nS n }n 1  l1 ;
2) an infinite determinant
D( z )   nk
2S k i n 2 k z

e
nk

n , k 1
exists (here and further we use the definition  rn is Kronecker delta, En
identity matrix of order n  n ), it is continuous, does not zero in closed half
plane C  = {z : Im z  0} and analytical inside of open half plane
C  {z : Im z  0}.
In the present work, using a technique of works [2], [5] the full inverse
problem is solved for the operator (1.2) with potentials of the type (1.1).
Now let's formulate the basic result of the present work.
~
Definition. The sequence {S nj }n,12,mj11 constructed with the help of formulas (1.4),
we call as a set of the spectral data of the operator L generated by differential
expression (1.2), with potential (1.1).
~
Thoerem1. In order to the given sequence of complex numbers {S nj }n,12,mj11
would be the set of the spectral data of operator L generated by differential
expression (1.2) and potential (1.1), it is necessary and sufficient that the
conditions satisfied in the same time;
~
1) {nS n }n1  l1 ;
(1.6)
2) The infinite determinant
D( z )  det  rn E 2 m1 
~
i (1   l ) S nj
i
r l (1   j )  n(1   l )
e
n
z
1 j
i
e
r l
z
1 l
2 m 1 
(1.7)
j ,l 1 r , n 1
exists, is continuous, is not zero in closed half plane C  = {z : Im z  0} and
analytical inside of open half plane C  {z : Im z  0}.
2. About the inverse problem of the theory of scattering on half-line.
Taking x  it ,   ik , y ( x)  Y (t ) in the equation l ( y)  2 m y , we receive
(1) m Y ( 2 m) (t ) 
2 m2
Q (t )Y 


( )
(t )  k 2 mY (t ),
(2.1)
0
where

Q (t )  (1) m (i )   pn e  nt ,
n 1
2 m2 
n



0 n 1
pn   .
(2.2)
As a result we have the equation (2.1) where potential exponentially decreases
t .
3. About the operator of transformation.
In [1] it is proved, that (2.1) has solutions type of
f ( x, k  )  e
ik x
2 m 1 

Vn(j )
 
e (ik  ) x .   0,2m  1
j 1  1 n 1 in  k  (1   j )
(3.1)
where numbers Vn(j ) are determined from the following recurrent formulas:
2m
2m

2 m 1 1

 n   ( j)
n
 
 Vn  (1) m1 
 
 (1   )  
(1   j ) 

 0 s  n
j 





n 
( j)
i s 
 P , s  nVns (3.2)

(1   j ) 
 
at   2,3,...; n  1,2,...,  1; j  1,2,...,2m  1,
i  p 
2 m 1 
2 m  2 2 m 1
 d j (n, )Vn(j )  
   d  (n, s, ) p V
  1 j 1 r  s 
j 1 n 1
s
n 1
j
r
( j)
ns
0,
(3.3)
where


2 m2
1
(i  k ) 2 m  k 2 m  (i  k nj ) 2 m  k nj2 m    d j (n, )k  ; j  1,2m  1,
n  k (1   j )
 0
(is  k )  (is  k nj )
in  k (1   j )
v 1
  d j (n, s, v)k  ,
 0
and a series (3.1) are 2m time term by term differentiable.
Consequence: Let’s the condition (2.2) is satisfied. Then

f (t , k )  e ikt   K (t , u )e iku du ,
(3.4)
t
where


n
( t u ) 
 t 
Vn(j )
1 j

.
K (t , u )  
e 
j 1 n 1   n i (1   j )
2 m 1 

(3.5)
4. The basic equations of the inverse problem.
Using a technique of work the [1,2] it is possible to receive equality such as
(1.4) for solutions (3.1)
f nj (t )  S nj f (t , k nj j ) ,
(4.1)
where
f nj (t )  lim [in  k (1   j )] f (t , k ) , k nj  
k k nj
in
.
1 j
Equality (4.1) we shall write as

Vn(j ) e t e
 n
n
t
1 j
n j
 S nj e
1 j
t

2 m 1 


n

l 1 r 1
r
i(1   j )Vnr(l ) S nj
j
(1   l )  r (1   j )
(  
e
n j
1 j
)t
(4.2)
n

u
1
1 j
Let's multiply both parts of (4.2) by
and denote
e
i(1   j )
2 m 1 
~
F (t  u )  
j 1 n 1
S nj
i(1   j )
e
n
( t j u )
1 j
,t  u ,
(4.3)
then from the equation (4.2) we shall receive Marchenko's type equations

~
~
K (t , u )  F (t  u )   K (t , s) F ( s  u )ds .
(4.4)
t
Lemma 1: If the coefficients Q (t ) of the equation (2.1) are (2.2) type, then at
all t  0 the operator of transformation (3.5) satisfies Marchenko's type
~
equation (4.4) in which the function of transition F (t ) has type (4.3), and
numbers S nj are determined by equality (4.1). From this it is obtained that
S nj  Vnn( j ) .
4.1 Resolvability of the basic equation and uniqueness of the solution of the
inverse problem.
By nucleus of the operator of transformation the coefficients Q (t ) are
reconstructed with the help of recurrent formulas (3.2) - (3.3). As
consequences, the basic equation (4.4) and type (4.3) of the functions of
transition of the inverse problem put natural statement about reconstructed of
coefficients of the equation (2.1) on numbers S nj . In this statement is an
important moment the proof of unequivocal resolvability of the basic equation
(4.4).
Lemma 2. The homogeneous equation

~
g ( s)   F (u  s) g (u )du  0
(4.5)
0
corresponding to potentials Q (t ) Q2 has only trivial solution.
The proof: Let g  L2 ( R  ) be solution of the equation (4.5) and let f be
solution of
s
f ( s)   K (t , s) f (t )dt  g ( s)
(4.6)
0
Substituting g in (4.6) and taking into account the equation (4.4) we receive
s

u
0
0
0
~
f ( s)   K (t , s) f (t )dt   [ f (u )   K (t , u ) f (t )dt ]F (u  s)du 


s
t
~
~
 f ( s)   f (t )[ F (t  s)   K (t , u ) F (u  s)du ]  0 .
As at all t  s the estimation

~
~
F (t  s )   K (t , u ) F (u  s )du  Ce  s
0
is satisfied, it follows, that f  0, g  0 and lemma it is proved. From this
lemma follows
Theorem 2: Coefficients Q (t ) of the equation (2.1) satisfying
condition (2.2) is unequivocally determined by numbers S nj .
In the equation (1.2) we shall replace x by x  a , where Im a  0 . Then we
shall receive equation of the same type with potential Qa ( x)  Q ( x  a)
satisfying the condition (1.1). Note, that the functions  ( x  a,  j ) are
solutions of the equation
(1) m y ( 2 m) ( x) 
2 m2
Q ( x) y 


a
( )
( x )  2 m y ( x )
0
which, at x   has a form
 ( x  a,  j )  e
i j a
e
i j x
 o(1) .
Therefore functions
 a ( x,  j ) = e
 i j a
 ( x  a,  j )
are solutions of type (1.3). We shall denote further through S nj (a ) the
spectral data of operator L with potential Qa (x)
L  (1) m
d 2m 2m2 a
d

Q
(
x
)
.


dx 2 m  0
dx 
According to (1.4) we have
S nj (a )  a ( x, nj j ) = lim [n   (1   j )] a ( x,  ) = lim [n   (1   j )] ( x  a,  ) e  ia
  n j
 n j
i
=e
n
a
1 j
S nj  ( x  a, nj j ) e
i
n
a
1 j
i
e
n j
1 j
a
= S nj  a ( x, nj j ) e ina = S nj  a ( x, nj j ) ,
as consequence
S nj (a ) = e ina S nj
(4.7)
Now considering as above we receive basic equations of type (4.4) with the
transitive function
n
2 m 1 
S nj (a) 1 j (t j u ) ~
~
~
Fa (t  u )  
e
 F (t  ia  u  ia )  F (t  u  2ia )
j 1 n 1 i (1   j )
and validity of the following lemma.
Lemma 3: At each fixed value a, (Im a  0) the homogeneous equation

~
g ( s)   F (u  s  2ia ) g (u )du  0
0
has only trivial solution in the space L2 ( R  ).
5. The proof of the basic theorem 2.
(4.8)
Necessity: From the relation (4.1) and a type of function f nj (t ) we shall
receive S nj  Vnnj .
Therefore
2 m 1 
 n
2 m 1
j 1 n 1
S nj 
2 m 1 
 n
2 m 1
j 1 n 1
Vnnj   ,
i.e. n 2 m 1 S nj  l1 . Necessity of the condition (1) of the theorem is proved. For
the proof of necessity of a condition (2) at first all we shall show, that from
trivial resolvability of the basic equation (4.4) at t  0 in a class of functions

u
2
satisfying to the inequality g (u)  Ce , u  0 , follows trivial resolvability of
the infinite systems of the equations in l 2 (1, , R 2m1 )
g jn 
i (1   j ) S rl
2 m 1 
 n
l 1 r 1
j
(1   l )  r (1   j )
g lr  0 .
(5.1)
Really, if g jn  l 2 , j  1,2m  1, and the solution of this system exists, then the
function
g (u )   g1 (u ), g 2 (u ),... g 2 m 1 (u )  

2 m 1 
 S
j 1 n 1
nj
g jn e
n
u
1 j
(5.2)
is determined for all u  0 and satisfies inequality
g (u)  Ce

u
1 j
; u  0.
also is the solution of the equation (4.5), because of
n

n


u

s 2 m 1 
2 m 1 
2 m 1 
~
1
1
g (u )   g ( s) F (u  s)ds   S nj g jn e j   (  S nj g jn e j )( 
j 1 n 1
0

2 m 1 
 S nj g jne
n

u
1 j

j 1 n 1

 S
j 1 n 1

2 m 1 
nj
g jn e
 S nj e
j 1 n 1

2 m 1  2 m 1 
S nj S rl
  i(1  
j 1 n 1 l 1 r 1

2 m 1 
0
n
u
1 j
n
u
1 j

j 1 n 1 l 1 r 1
[ g jn 
2 m 1 
 n
l 1 r 1
r

u
1 l
l)
e
l 1
n
r l

s
s
1 j
1 l
j

(1   l )  r (1   j )
i(1   j ) S rl
j (1   l )  r (1   j )
e
ds 
0
i(1   j ) S nj S rl
2 m 1  2 m 1 
  n
g jn e
j 1 n 1
g lr e
g lr ]  0 .
n
u
1 j

r
( s l u )
S rl
e 1l
)ds 
r 1 i (1   j )
Therefore, g (u )  0 , S nj g jn  0 at all, n  1, j  1,2m  1, hence
g jn  0, j  1,2m  1, n  0 by (5.1). Now we shall introduce in space l 2 (1, , R 2 m1 )
the operator F (t ) defined by matrix
jl 2 m 1
rn j ,l 1
Frn (t )  F

i(1   l ) S nj

e
r l (1   j )  n(1   l )
n
t
1 j
e
r l
t
1 l
2 m 1
(5.3)
j ,l 1
and let  2k 1  (  1 ) rk 2 m,r 11, ,  2k    2  rk 2m,r 11, { (  1 ) -column vector}
ortonormal system in this space, as n 2 m 1 S nj  l1 , we have

 F
j , k 1
j
,  k l
2 (1, , R
2 m 1
)
  , and the operator F (t ) is nuclear [3]. Therefore there
exists a determinant (t )  det( E  F (t )) of the operator E  F (t ) connected as it
is easy to see, with a determinant from the condition 2) of the Theorem2 by
the relations (iz )  det( E  F (iz ))  D( z ) . Determinant of system (5.1) is
D (0) , and a determinant of the similar system considering to potential
Qz  Q ( x  z), Im z  0 is
D( z )  det  rn E 2 m 1 
 det  rn E 2 m 1 
 det  rn E 2 m 1 
2 m 1 
i (1   l ) S nj ( z )
r l (1   j )  n(1   l )
i (1   l ) S nj
r l (1   j )  n(1   l )
i (1   l ) S nj
r l (1   j )  n(1   l )

j ,l 1 r , n 1
2 m 1 

e inz
j ,l 1 r , n 1
i
e
n
z
1 j
i
e
r l
z
1 l
2 m 1 
.
j ,l 1 r , n 1
Therefore for the proof of necessity of the condition 2) of the theorem should
be checked up, that (0)  D(0)  0. The system (5.1) can be written as the
equation in l 2 (1, , R 2m1 ) ,
g  F ( 0) g  0.
Since the operator F (0) is nuclear, we can apply to this equation Fredholm
theory, according to which, its trivial resolvability is equivalent to that
det( E  F (0)) is not zero [6]. Necessity of a condition 2) is proved.
Sufficiency: Let’s multiply the equation (4.4) e
Then we receive
r l
u
1 l
k (t )  F (t )e(t )  k (t ) F (t )
and integrate on u  [t , ) .
(5.4)

in which the operator F (t ) is defined by the matrix Frn (t ) r ,n1 of a type (5.3),
n j
e(t )  enj (t )  e
1 j
 , 2 m 1
t

2 m 1,
; k (t )  k lr (t ) l ,r 1   K (t , u)e
t
n , j 1
r l
u
1 l
2 m 1,
;
du
l , r 1
As F (t ) is nuclear at t  0 and conditions (t )  det( E  F (t ))  0 holds, there
exists bounded in l 2 the inverse operator R(t )  ( E  F (t )) 1 . Considering
F (t )e(t )  l 2 , from (5.4) we find that
k (t )  R(t ) F (t )e(t )
(5.5)

Defining f , g   f n g n , from (4.4) receive
n 1
K (t , u)  e(t ), A(u)  k (t ), A(u)  e(t ), A(u)  R(t ) F (t )e(t ), A(u)  e(t )  R(t ) F (t )e(t ), A(u) 
 R(t )e(t ), A(u) ,
(5.6)
where A(u ) is defined by the matrix
A(u )  a jn (u )
2 m 1,

j , n 1

S nj
i(1   j )
e
2 m 1,
n
u
1 j
j , n 1
Let's assume, now, that conditions of the theorem are satisfied. According to
the stated reasons we define at 0  t  u the function K (t , u ) by equality (5.6).
Then at u  t we have


t
t
~
K (t , u)   K (t , s) F ( s  u )ds  R(t )e(t ), A(u )   R(t )e(t ), A( s) e( s), A(u ) ds 

 R(t )e(t ), A(u )  R(t )e(t ),
 A(s)e(s)ds, A(u)
 R(t )e(t ), A(u ) 
t
 R(t )e(t ), A(u ), F (t )  R(t )e(t ), A(u )  R(t )e(t ), A(u ) F * (t )  R(t )e(t ), A(u )  A(u ) F * (t ) 
~
 e(t ), A(u )  F (t  u )
(5.7)
where “*” designates transition to the matrix connected with F (t ) rather
bilinear form .,. .The following lemma is proved.
Lemma 4: At everyone t  0 the nucleus K (t , u ) of the operator of
transformation satisfies to the basic equation

~
K (t , u )  F (t  u )   K (t , s) F ( s  u )ds
t
~
where F (t  u ) it is defined from (4.3).
Resolvability of the basic equation follows from the lemma 2 unequivocal. By
substitution it is easy to calculate, that the solution of the basic equation is

n

( t u ) 
 t 
Vn(j )
1 j

,
K (t , u )  
e 
j 1 n 1   n i (1   j )
2 m 1 

where numbers Vn(j ) are determined from the recurrent relations
Vnn( j )  S nj ,
2 m 1 
Vr(l )
.
r 1 r (1   j )  n(1   l ) j
Vn(j ) n  (1   j )Vnn( j ) 
l 1
Passing to the proof of the basic statement that coefficients Q (t ) look like
(2.2), all over again we shall establish for the matrix elements Rrn (t ) of the
operator R (t ) the estimation
Rrnjl (t )   rn jl  CS n ;
(5.8)
 2 m jl
Rrn (t )  CS n ;  1,2m  1
t 2 m
(5.9)
where C  max{ Ck  0, k  1,2m  1} - a constant, and S n 
2 m 1
n
j 1
2 m 1
S nj ;
Really from equality R(t )  E  R(t ) F (t ) follows
R (t )   rn lj 
lj
rn
2 m 1 


1 p 1
2 m 1

 1
p 1
2

j
R (t ) F pn (t )  rn lj  2  ( R (t ) ) 2 ( F pn
(t ) ) 2  rn lj 
l
rp
j
l
rp
2
p 1

1
)S n   rn lj  C1 R(t )
2
p 1 (n  p)
 C1am (( R(t ) R * (t )) pp 
l2 l2
Sn .
On the other hand, as it has already been shown, operator - function
R(t )  ( E  F (t )) 1 exists and is bounded in l 2 (valid nuclear F (t ) at t  0 and
conditions (t )  det( E  F (t ))  0 ), that proves the first inequality (5.8).
The proof of the second estimation (5.9), we do similarly, proceeding from
identity
l n
l
dl
dn
n d
R
(
t
)

C
(
R
(
t
))(
F (t )) R(t ), l  1,2m  1, and using the first

l
dt l
dt l n
dt n
n 1
estimation (5.8). Then
2 m 1 
2 m 1 
d lj
d k
kj
Rrn (t )    Rrpl (t ) Fpq
(t ) Rqn
(t )    ( rp l  C2 S p )S q ( qn kj  C3 S n ) 
dt
dt
 1 p , q 1
 1 p , q 1

 (1  C 4  S p ) 2 S n  CS n .
p 1
Further, proceeding from equality
l n
l
dl
dn
n d
R
(
t
)

C
(
R
(
t
))(
F (t )) R(t ), l  1,2m  1,

l
dt l
dt l n
dt n
n 1
with the help of a mathematical induction method the inequality (5.9) is
proved. In [4] it is established validity of the following relations (for
conformity with our case, we shall put q 2 m2 ( x)  Q ( x) ).
(1) m
2 m2
 2m

 2m
K
(
x
,
t
)

q
(
x
)
K
(
x
,
t
)

K ( x, t )  0

2 m  2 
x 2 m
x 
t 2 m
 0
q 0 ( x )  2m
d
K ( x, x )
dx
k
k
 0
s 
q k 1 ( x)   q ( x) C
k s
n  3 s
  s 

 s  K ( x, t ) 
 x
tx 

(k s)

 

 C
  K ( x, t ) 
 x
k 0
tx 

k  0,1,...2m  3.
k 2
( x  2  )
k  2 
n 1
 (1) k
 k 2
K ( x, t )
t k  2
tx
Further, it is easy to show, that
q 0 ( x) 
n  S nj
2 m 1 
 i(1  
j 1 n 1
j)
e  nt   0 (t )
where
 0 (t ) 

 R
n , p , q , 1
ej
np
Fpqj eq eq  R(t ) F (t )e(t ), A(t ) 
is 2i periodic function and has the bounded derivative up to the order (2m
1). Then Fourier coefficients
n
n 1
2
2 m 1
n
 .

But then
n
2 m2
n 1
 n   . Thus
Fourier coefficients P2 m  2,n of the function Q2m2 ( x)  q0 ( x) satisfy to the
condition (2.2). Similarly, for all other coefficients Q ( x),   0,2m  3 it is
established, that the Fourier coefficients pn functions
Q ( x)  q 2 n2 ( x),   0,2m  3 satisfy a condition (2.2). Thus Fourier coefficients
of function P ( x),   0,2m  2 satisfied to the condition (1.1).
Let, at least S nj be the set of spectral data of the operator ( L  k 2 m E ) with
constructed coefficients P (x) . For the end the proof it needs to be shown, that
S nj coincides with an initial set S~nj . It turns out from equality S~nj  Vnn( j )  S nj .
The theorem is proved.
Acknowledgments
The author has benefited from discussions with M.G.Casymov and
I.M.Guseynov.
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