BOUNDARY-VALUE PROBLEMS FOR SECOND ORDER DIFFERENCE EQUATIONS WITH A SPECTRAL PARAMETER IN THE BOUNDARY CONDITIONS

220

BOUNDARY-VALUE PROBLEMS FOR SECOND ORDER DIFFERENCE

EQUATIONS WITH A SPECTRAL PARAMETER IN THE BOUNDARY

CONDITIONS

Aytekin Eryilmaz1& Bilender P. Allahverdiev2

1

Department of Mathematics, Nevsehir University, Nevsehir, Turkey

2

Department of mathematics, Süleyman Demirel University, Isparta, Turkey Email: 1eryilmazaytekin@gmail.com;2bilender@fef.sdu.edu.tr

ABSTRACT

This paper is concern with the boundary-value problem in the Hilbert space 2

(

N

)

w

l

generated by an infinite Jacobi matrix with a spectral parameter in the boundary condition. We proved theorems on the completeness of the system of eigenvalues and eigenvectors of operator generated by boundary value problem.

Keywords: Second order difference equation, Jacobi matrix, Dilation, Dissipative operator, Completeness of the system of eigenvectors and associated vectors, Scattering matrix, Functional model, Characteristic function.

Subject Classification 2000: 47A20, 47A40, 47A45, 47B39, 47B44

1. INTRODUCTION

Spectral theory of operators plays a major role in mathematics and applied sciences. The studies of boundary value problems with a spectral parameter in the boundary conditions take a significant place in spectral theory of operators (see [1-5,7-9,16-18]). First general method in the spectral analysis of nonselfadjoint differential operators is the method of contour integration of resolvent which is studied by Naimark (see[13]). But this method is not effective in studying the spectral analysis of boundary value problem. This method is also called functional model method (see1-5,7,14,15). The rich and comprehensive theory has been developed since pioneering works of Brodskii, Livsiç and Nagy-Foiaş.The functional model techniques are based on the fundamental theorem of Nagy-Foia

s

. In 1960's independently from Nagy-Foia

s

, Lax and Phillips developed abstract scattering programme that is very important in scattering theory. This theory is based on the notion of incoming and outcoming subspaces for get information about analytical properties of scattering matrix by utilizing properties of orijinal unitary group.

By combining the results of Nagy-Foia

s

and Lax-Phillips, characteristic function is expressed with scattering matrix and the dilation of dissipative operator is set up. By means of different spectral representation of dilation, given operator can be written very simply and functional models are obtained. The eigevalues, eigenvectors and spectral projection of model operator is expressed obviously by characteristic function. The problem on completeness of the system of eigenvectors is solved by writing characteristic function as factorization.

In this paper, we consider to study nonselfadjoint boundary value problem, generated by infinite Jakobi matrix, with a spectral parameter in the boundary conditions. We applied this approach to the study of dissipative difference operator generated in the space

l

_w2

( )

N

. Our goal is to compute the characteristic function of the dissipative extensions. To accomplish this goal we constructed a selfadjoint dilation of dissipative operator and its incoming and outcoming spectral representations. Because this makes it possible to determine the scattering matrix of dilation according to the Lax and Phillips scheme [11]. Moreover, we constructed a functional model of dissipative operator by means of the incoming and outcoming spectral representations and defined its characteristic function. Finally, we proved a theorem on completeness of the system of eigenvectors and associated vectors of dissipative operators under consideration.

2.CONSTRUCTION OF DIFFERENCE OPERATOR IN THE HILBERT SPACE

221

.

.

.

.

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.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0

0

.

.

.

0

0

.

.

.

0

0

.

.

.

0

0

0

=

3 3 2

2 2 1

1 1 0

0 0









































a

b

a

a

b

a

a

b

a

a

b

J

For all sequences

y

=

 

y

_n

(

n



N

)

that composed of complex numbers

y

₀

,

y

₁

,...

of

ly

which has components form of

( ) (

ly

_n

n



N

)

is defined as follows:

 

:=

1

(

)

=

1

(

_{0 0 0 1}

)

0 0 0

0

b

y

a

y

w

Jy

w

ly



1,

),

(

1

=

)

(

1

:=

)

(

a

1

y

1



b

y



a

y

1

n



w

Jy

w

ly

n n n n n n

n n n n

where

_w

>

0

(

_n



N

)

n .

Definition 1. For all

_n



N

,

Green's formula is defined as fallows





 

=0

( )

( )

=

,

(

).

n

j

j

w ly z

j j



w y l z

j j j



y z

_n

n



N

(2.1)

Then let us construct Hilbert space 2

(

N

)

w

l

( :=

w

 

w

_n

n



N

)

consisting of all complex sequences

 

(

)

=

_y

_n



N

y

_n provided _n=0

w

_n

y

_n 2

<



, with the inner product

=0

( , ) =

_{n n} n n

y z

w y z





in order to pass from the matrix

J

to operators. Let us denote with

D

the set of

_y

=

 

_y

(

_n



N

)

n sequences in

2

( )

w

l

N

providing

)

(

2

_N

w

l

ly



. Define

L

on

D

by the equality

Ly

=

ly

. For all

y

,

z



D

, we obtain existing and being finite of

the limit

 

 

_n

n

y

z

z

y

,

=

lim

,

 

 from (2.1). Therefore, if we pass to the limit as

n





in (2.1), it is obtained

 

,

.

=

)

,

(

)

,

(

Ly

z



y

Lz



y

z

_ (2.2)

Let us take into account the set of linear

D

₀' consisting of finite vector having only finite many nonzero components

in 2

(

N

)

w

l

. We denote the restriction of

L

operator in

D

₀' by

L

₀'

.

It is obtained from (2.2) that

L

'₀ operator is symmetric. The clousure of

'

L

0 operator is denoted by

L

0

.

The domain of

L

0 operator is

D

0 and it consists of

the vector of

y



D

satisfying the condition

 

y

,

z

_

=

0,



z



D

.

The operator

L

₀ is a closed symmetric operator with defect index

(0,0)

and

(1.1)

. Furthermore,

L

=

L

₀(see [1,6,8,18]). The operators

L

₀ and

L

are called respectively the minimal and maximal operators generated by infinite Jakobi matrix J. The operator

L

₀ is a selfadjoint operator for defect index

(0,0).

Namely

L

₀

=

L

₀

=

L

.

Suppose the solution of equation of a second order difference equation on the half-line

1,2,...)

=

(

=

1 1

1

y

b

y

a

y

w

y

n

a

_{n n}



_{n n}



_{n n}



_{n n} (2.3)

222

0 1

0 0

0 0 1

0

1

=

)

(

0,

=

)

(

,

=

)

(

1,

=

)

(

a

Q

Q

a

b

w

P

P













(2.4)

be

P

(



)

=



P

_n

(



)



and

Q

(



)

=



Q

_n

(



)



.

Here the function

P

_n

(



)

is called the first kind polynomial of degree

n

in



and the function

Q

_n

(



)

is called the second kind polynomial of degree

n



1

in



(see [1]). For

1,



n

P

(



)

is a solution of

(

Jy

)

_n

=



w

_n

y

_n is

P

_n

(



)

. But

Q

(



)

is not a solution of

(

JQ

)

_n

=



w

_n

Q

_n

because of

(

)

=

=

0

1

=

1

=

0

=

0

,

0 0 0 1 0 0 0

0

Q

a

a

b

Q

a

Q

b

JQ









For

n



N

and under boundary condition

0,

=

1



y

the equation

(

Jy

)

_n

=



w

_n

y

_n is equivalent to (2.3). For two arbitrary sequences

y

=

 

y

_n and

 

z

n

z

=

, The Wronskian of the solutions

y

and

z

of the equation (2.3) is defined as follows:

1 1

( , ) :=

(

) = [ , ] , (

).

n n n n n n n

W y z

a y z

_



y

_

z

y z

n



N

The Wronskian of the two solutions of (2.3) does not depend on

n

,

and two solutions of this equations are linearly indepent if only if

W

_n

(

y

,

z

)



0

. From the condition (2.4), we get

W

₀

(

P

,

Q

)

=

1

. Consequently, from Wronskian constacy, the solutions of

P

(



)

and

Q

(



)

form a fundamental system of solutions (2.3).

Suppose that the minimal symmetric operator

L

₀ has defect index (1,1) so that the Weyl limit circle case holds for the expression

ly

(see [1-4], [6-8]). As the defect index of

L

₀ is (1,1) for all





C

the solutions of

P

(



)

and

)

(



Q

belong to 2

(

N

).

w

l

The solutions of

u

=

 

u

_n and

v

=

 

v

_n of the equality (2.3) be

u

=

P

(0)

and

(0)

=

Q

v

satisfying the initial condition of

0 1 0 0 0 1 0

1

=

0,

=

,

=

1,

=

a

v

v

a

b

u

u



while



=

0.

In addition it is

u

,

v



D

and

(

Ju

)

_n

=

0,

(

n



IN

),

(

Jv

)

_n

=

0,

n



1.

Lemma 1. [2]For arbitrary vectors

y

=

 

y

_n



D

and

z

=

 

z

_n



D

it is

   

,

,

[ , ]

_n

 

,

[ , ]

_n

 

n n n

y z = y u

z v



y v

z u , (n

  

N

) .

Theorem 1. [2]The domain

D

₀ of the operator

L

₀ consists precisely of those vectors

y



D

satisfying the following boundary conditions

   

y

,

u

_

=

y

,

v

_

=

0.

Consider the boundary value problem governed by

1,

,

,

=

)

(

ly

_n



y

_n

y



D

n



(2.5)

subject to the boundary conditions

0

>

0,

=

1

0

hy

Imh

y



_ (2.6)

 

,

₂

 

,

=

(

₁

 

,

₂

 

,

)

1

y

v





y

u



y

v





y

u



' '











(2.7)

for the following difference expression

 

:=

1

(

)

=

1

(

0 0 0 1

)

0 0 0

0

b

y

a

y

w

Jy

w

ly



1

),

(

1

=

)

(

1

:=

)

(

a

1

y

1



b

y



a

y

1

n



w

Jy

w

ly

n n n n n n

223

where



is spectral parameter and

   

₁

,

₂

,

₁'

,

₂'



R

and



is defined by

0.

>

=

:=

_{1 2 1 2}

2 2

1

1 ' '

' '





















For the sake of brevity, let us suppose that the followings

 

,

 

,

,

:=

)

(

_{1  2 }



y

y

v



y

u

M





 

,

 

,

,

:=

)

(

1  2 



y

y

v



y

u

M

'



'



'

,

:=

)

(

₁

0

1

y

y



N

,

:=

)

(

0

0

2

y

y

N

 

,

,

:=

)

(

1 



v

y

y

N

 

,

,

:=

)

(

2 



u

y

y

N

).

(

)

(

:=

)

(

₂0 ₁0

0

y

N

y

hN

y

M



Lemma 2. [9]For arbitrary

y

,

z



D

suppose that

M

_

(

z

)

=

M

_

(

z

),

M

_'

(

z

)

=

M

_'

(

z

)

and

)

(

=

)

(

,

)

(

=

)

(

₁0 ₂0 ₂0

0

1

z

N

z

N

z

N

z

N

then it is

i)

 

_







_

  





(

)

(

)

(

)

(

)

1

=

,

z

M

y

M

z

M

y

M

z

y

' '



(2.8)

ii)

 

y

,

z

_1

=

N

₁0

(

y

).

N

₂0

(

z

)



N

₁0

(

z

).

N

₂0

(

y

).

(2.9)

Supposing

f

(1)



l

_w2

( ),

N

f

(2)



C

,

we denote linear space

H

=

l N

_w2

( )



C

with two component of elements

of

=

₍₂₎

.

(1)















f

f

f

Supposing

2 2

1 1

:=















, if



>

0

and

(1) (1)

(1) (1) (1) (1) (2) (2)

=

f

,

=

g

,

= (

_n

),

= (

_n

)(

),

f

g

H f

f

g

g

n

N

f

g



_

_



_

_





















then the formula

 1 ₍₂₎ (2) (1)

0 =

1

=

,

g

f

g

w

f

g

f

_{n n n n}



















 





(2.10)

defines an inner product in Hilbert space

H

. In terms of this inner product, linear space

H

is a Hilbert space. Thus it is Hilbert space which is suitable for boundary value problem that has been defined. Let us define operator of

H

H

A

_h

:



with equalities suitable for boundary value problem







































(

)

=

(

,

:

=

=

)

(

(1) ₀ (2) (1)

(2) (1)

f

M

f

M

D

f

H

f

f

f

A

D

_h (2.11)

224

 

 

.

:=

)

(

=

₍₁₎

(1)



















f

M

f

l

f

l

f

A

_h

:

(2.12)

Lemma 3.[9]In Hilbert space

H

=

l

_w2

( )

N



C

for

A

_h operator defined with equalities (2.11) and (2.12) the equality

(1) (1) (1) (1) 1

,

,

=

,

,

h h

A f g

f A g

f

g

f

g

 

 







 













 





_

_



_

_



 





 



(2.13)



(

)

(

)

(

)

(

)



1

(1) (1) (1) (1)

g

M

f

M

g

M

f

M

_{ }





_



_





is provided.

Theorem 2.[9]

A

_h operator is dissipative in

H

.

For all





C

,

the solutions of (2.5) be



(



)

and



(



)

for the following conditions:

1,

=

)

(

=

))

(

(

₁

0

1













N

,

=

=

))

(

(

0

0

2

y

h

N





,

=

))

(

(

_{2 2}

1







_

_

_

_

N

1

( ( )) =

1 1

N



 

 



(2.14)

From (2.9) for



₁

(



)

having Wronskian is





1





1

1

(

)

:=

(

),

(

)



=

(

),

(

)



























))

(

(

))

(

(

))

(

(

))

(

(

=



N

₁0





N

₂0







N

₁0





N

₂0





))

(

(

))

(

(

=

N

₂0







hN

₁0





)).

(

(

=

M

₀





From (2.8) for



_

(



)

having Wronskian is



















(



)

:=



(



),



(



)

=



(



),



(



)



(

(

)

(

(

))

(

(

))

(

(

))



1

=



















 













M

M

M

M

Therefore, in terms of the definition of



, it is

))

(

(

)))

(

(

))(

(

(

)))

(

(

[(

1

=

)

(



_



1 1







2 2







1 1







2 2





 

 















N

N

N

N

))]

(

(

))

(

(

))(

(

(

))

(

(

_{2 2 1 1 2 2}

1

1

























_



_

_

 

_





N

N

N

N











(

(

))

(

(

))

(

(

))

(

(

))



1

=



₁



₂



₂



_{1 1}





₂





₂





₁













 



 



N

N

N

N

 











1

(

(

))

1 1 2

(

(

))

2 2



1

=































 

N

N



(

(

))

(

(

))



))

(

(

))

(

(

=



₁

N

₁









₂

N

₂











₁

N

₁













₂

N



₂





)).

(

(

)

(

(

225

Lemma 4.[9]Boundary values problem

   

2.5



2.7

has eigenvalues iff it consists of zeroes of



(



)

.





(



)

=



₁

(



)

=





(



)



.

Definition 2.If the system of vectors of

y

₀

,

y

₁

,

y

₂

,...,

y

_n corresponding to the eigenvalue



₀ are

 

y

₀

=

₀

y

₀

,

l



 

y

_{0 0}

M

 

y

₀

=

0,

M

_





_

 

₀

=

0,

0

y

M

 

y

_s



₀

y

_s



y

_s1

=

0,

l



 

_{0 }

 

_

 

_1

=

0,



y

s



M

y

s



M

y

s

M



 

0 s

= 0, = 1, 2,..., .

M

y

s

n

(2.15)

Then the system of vectors of

y

₀

,

y

₁

,

y

₂

,...,

y

_n corresponding to the eigenvalue



₀ is called a chain of eigenvectors and associated vectors of boundary value problem

(2.5)



(2.7)

.

Lemma 5.[9]The eigenvalue of boundary value problem

(2.5)



(2.7)

coincides with the eigenvalue of dissipative h

A

operator. Additionally each chain of eigenvectors and associated vectors

y

₀

,

y

₁

,

y

₂

,...,

y

_n corresponding to

the eigenvalue



₀ corresponds to the chain eigenvectors and associated vectors

y

y

y

y

_n









,...,

,

,

_{1 2}

0

corresponding to the same eigenvalue



₀ of dissipative

A

_h operator. In this case, the equality

 

k

n

y

M

y

y

k k

k

=



,

=

0,1,2,...,















is valid.

3.MAIN RESULT

In this section we mention about selfadjoint dilation, scattering theory dilation and functional model of the maximal dissipative operator

A

_h

.

One of the completeness proofs of nonselfadjoint oprators is the method of dilation and characteristic function. To do this, we first build a selfadjoint dilation of the maximal dissipative operator

A

_h

.

Definition 3. Let add the " incoming" and " outgoing" subspaces

D

_

=

L

2







,0



and

D

_

=

L

2

 

0,



to

C

N

₎



(

=

2

w

l

H

. The orthogonal sum H

=

D

_



H



D

_{ is called mainHilbertspace of the dilation}

_.

In the space

H

,

we consider the operator

L

_h on the set

D

 

L

_h

,

_{its elements consisting of vectors}

_w

₌

,

,

,







_

y



_ generated by the expression





























 





_











d

d

i

y

l

d

d

i

y

,

=

,

,

,

L

_(3.1)

satisfying the conditions:



_



W

₂1







,0



,



_



W

₂1

 

0,



,

y



H

,



 

 

,

=

₂

1















y

y

y

y

 1



D

,

y

 2

=

M

_

 

y

 1

,

 

0

,

=

₁

1

0



hy



a







y

y

₀



h

y

₁

=

a

₁





 

0

,

where

W

21

 

.,.

are Sobolev spaces and

:=

2

,

2

Imh



226

Then we state our main result for the spectral analysis of the dissipative operator

A

_h

.

Theorem 3. For all the values of

h

with

Imh

>

0,

except possibly for a single value

h

=

h

₀

,

the characteristic function

S

_h

 



of the maximal dissipative operator

A

_h is a Blaschke product. The spectrum of

A

_h is purely discrete and belongs to the open upper half-plane.The operator

A

_h



h



h

₀



has a countable number of isolated eigenvalues with f nite multiplicity and limit po nts at inf nity. The system of all eigenvectors and associated vectors of the operator

A

_h



h



h

₀



is complete in the space

H

.

To prove theorem we have:

Theorem 4. The operator

L

_h is selfadjoint in

H

and it is a selfadjoint dilation of the operator

A

_h

.

Proof. We first prove that

L

_h is symmetric in

H

. Namely



L

h

f

,

g

 

H



f

,

L

h

g



H

=

0.

Let

f

,

g



D

 

L

h

,

 







,

,



=

y

f

and

=

,

,



.







_

z



_

g

Then we have



 



_



























































,

=



_

,

,



_

,



_

,

,



_



_

,

,



_

,



_

,

,



_

,

g

f

g

y

z

y

z

f

_h

h

L

L

L

L

_H H

.

,

,

,

,

,

,

,

,

,

,

=

_

























































































       

























d

d

i

z

l

d

d

i

y

z

d

d

i

y

l

d

d

i

Now, passing to the components of space

H

, it yields

















d

d

d

i

z

y

l

d

d

d

i

H

































































      

,

,

,

=

0 0

















d

d

d

i

z

l

y

d

d

d

i

H



































































     



,

,

0

,

0













d

l

y

z

i

d

i

' H       









































0 0

,

=













d

y

l

z

i

d

i

' H '       











































0 0

,

By

 

2.8

,



2.13



and integrating by parts, we get



 





   





   



 





1 1

1 1 1

,

,

=

,

,

g

f

g

y

z

y

z

f

_h

h H

L

H

L

   

   

   

0

0

   

0

0

1

(1) (1) (1) (1)

        













_













' ' ' ' '

i

i

y

M

y

M

y

M

y

M

   









   





   





   





   1





1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 2 1 1 1

,

=

y

z

_



i

y



hy

_

z



hz

_



y



hy

_

z



h

z

_

227

   







               1



1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 2 1 1 1

,

=

y

z

_



i

y

z



h

y

z

_



hy

_

z



h

h

y

_

z





               



1



0 1 1 1 0 1 1 1 1 1 0 1 1 1 0

2

y

z

hy

z

h

y

z

h

h

y

z

i

   













   







 

   

 

   1



0 1 1 1 1 1 0 2 1 1 1

,

=

y

z

_



i

h



h

y

z

_



h



h

y

_

z



   







 

   

 

   1



0 1 1 1 1 1 0 2 1 1 1

,

=

y

z

_



i

h



h

y

z

_



h



h

y

_

z



 1  1  1  1

1 1

=

y

,

z

y

,

z

= 0.

 



















(3.2)

Thus,

L

_h is symmetric operator. To prove that

L

_h is selfadjoint, we need to show that

L

_h



L

_h

.

_{We consider the}

bilinear form



L

h

f

,

g



H on elements

 

 











z

D

_h

g

₌



_,

_,



L

_{, where}

 

h

D

y

f





L







_

,

,



_

=

,

 

,

1 2 





W

R





_

 

0

=

0.

Integrating by parts, we get







   









d

d

i

z

d

d

i

g

h

=

,

,

L

_{, where} 1

 

_,

2 





W

R



H

z







. Similarly, if

f

=

0,

y

,0





D

 

L

_h

,





then integrating by parts in



L

h

f

,

g



H, we obtain

   

 

 

.

=

,

,

,

,

=

,

,

=

z

1

D

z

2

M

z

1

d

d

i

z

l

d

d

i

z

g

h       

_

_







































L

L

_(3.3)

Consequently, from

(

3.3), we have



L

h

f

,

g

 

H



f

,

L

h

g



H

,



f



D

 

L

h

,

where the operator

L

is defined by

 

3.1

.

Therefore, the sum of the integrated terms in the bilinear form



L

h

f

,

g



H must be equal to zero:

   







   



   

   









_





_{    }

1 1 1 (1) (1) (1) (1) 1

1

1

,

,

z

y

z

M

y

M

y

M

y

M

y

y

' '



   

0

_

0

_

   

0

_

0

=

0.









'





'



i

i

(3.4)

Then from

 

2.8

, we get

   



1

,

1



₁







   

0







0







   

0





0

=

0.

' '

i

i

z

y

(3.5)

From the boundary conditions for

L

_h

,

_{we have}

 

 





 



 

 



_

















_{    }  



=

0

0

,

0

=

1

0

0

0

1

1















'

i

h

a

y

i

a

y

Then

 

2.9

and

 

3.5

, we get

 

 



0

0



0

 

0



 

0

 

0



1

1

     













_

_





z

i

h

z

i

' '















   

0

0

   

0

0

=

i



_ '



_



i



_'



_ (3.6)

Comparing the coefficients of





 

0

in

 

3,6

we obtain

228 or

 

0

=

1

0



hz







z

(3.7)

Analogously, comparing the coefficients of



_

 

0

in

 

3,6

we get

 

0

=

1

0



h

z







z

(3.8)

Therefore conditions

 

3,7

and

 

3,8

imply

D

 

L

h



D

 

L

h

,



hence

L

h

=

L

h

.

The selfadjoint operator

L

_h generates on

H

a unitary group

U

_t

₌

_exp

 

i

L

_h

t

₍

_t



R

_

₌

 

_0,



_{). Let}

denote by

P

_:

H



H

_and

_P

_:

_H



H

1 the mapping acting according to the formulae











_

y

_

y

P

:



,

,



and ₁

:

0,

,0



.









y

y

P

Let

Z

_t

:=

PU

_t

P

_1,

t



0,

by using

U

_t

.

The family

 

Z

_t



t



0



of operators is a strongly continuous semigroup of comletely nonunitary contraction on

H

.

Let us denote by

B

_h the generator of this semigroup

:

B

_h

=

lim

 

1

.

0

























 



it

Z

y

y

y

_t

t The domain of

B

h

consists of all the vectors for which the limit exists. The operator

B

_h is dissipative.The operator

L

_h is called the selfadjoint dilation of

B

_h

(

see



10,12



)

. We show that

B

_h

=

A

_h, hence

L

_h is selfadjoint dilation of

B

_h

.

For this, it is sufficient to verify the equality



I



1

P

₁

y

=



A

I



1

y

,

y

H

,

Imh

<

0

.

P

_{h h}













_



_



L

_(3.9)

For this purpose

,

we set





1 ₁

=

=

,

,



.









_



_

_

_

_

z

g

y

P

I

h

L

_Then





₌

_,

1





I

g

P

y

h



L

_{and hence}























z

z

l



y



_

 





_

 

e

i



0

=

,

=

and



_

 



=



_

 

0

e

i . Since

g



D

 

L

_h

,

_then



,0



,

2









L



it follows that





 

0

=

0,

and consequently

z

satisfies the boundary condition

0.

=

1 0



hy



y

Therefore

z



D

 

A

_h

,



and since point



with

Im



<

0

cannot be an eigenvalue of

dissipative operator

,

it follows that



_

 

0

is obtained from the formula

 

0

=



0 1



,

1

 





y



hy



Thus





























_





1 0 1 1 1

1

,

0,

=

A

I

y

y

h

y

y

P

I

_h

h







L

for



y

and

Im



<

0.

On applying the mapping

P

,

we obtain

 

3.9

,

and









1

0 1

1 1

=

=

P

I

P

iP

U

e

dtP

I

A

i t

t h

h







   









L





,

<

0,

=

=

i

Z

_t

e

it

dt

B

_h



I

1

Im









_

so this clearly shows that

A

_h

=

B

_h

.

This completes the proof of theorem 4.

The unitary group

 

U

_t has an important property which makes it possible to apply to it the Lax-Phillips [11].i.e, it has incoming and outcoming subspaces

D

_

=



L

2







,0



,0,0



and

D

_

=



0,0,

L

2

 

0,





having the following properties: