On the structure of discrete spectrum of the non-selfadjoint system of differential equations in the first order

ON THE STRUCTURE OF DISCRETE SPECTRUM OF THE NON-SELFADJOINT SYSTEM OF DIFFERENTIAL EQUATIONS IN THE FIRST ORDER

OMER AKIN AND ELGIZ BAIRAMOV

1.

Introduction

This paper is concerned with the problem given below

(1.1)

(1.2)

.dUI(X,A) () ( ') ' ( )

z dx + ql X

x,

x, A

.dU2(X,A) () ( ) ( )

- z

dx + q2 X

x, A

AU2 x, >. ,

OSx<oo

where A is a complex parameter and h is a non-zero complex number.

By applying the transformation of

1 1

Yl(X, A) = 2:[Ul(X, A) + U2(X, A)], Y2(X, A)

2i [U2(X, A) - Ul(X, A)]

to the system (1.1) we can see that it has become the following (1.3)

Where

dY(X,A)

B dx + n(x)y(x, A)

AY(X, >')

f3(x) ) -a(x)

Received March 15, 1994.

AMS Classification: 47AI0.

Key words: Spectrum, spectral theory, differential operator, non-selfadjoint system.

402 OmerAkin and Elgiz Bairamov

This equation (1.3) is called the canonical Dirac system of which mass

IS

zero.

We have already known that some problems which are very close to this problem has been treated by' some authors, such as in the works [8, 9], the'structUre of the discrete spectrum of non-self adjoint SchrOdinger operator is examined. The eigen values and spectral singu- larities of the dissipative SchrOdinger operators with rapidly decreasing potential were investigated in [4}. In the work [7} has been proved that the eigenvalues and the spectral singularities of the problem (1.1 )-(1.2) are finite under the conditions

(1.4) Iqi(X)1

Ce

> 0, i = 1,2,C constant.

In the present work, we have proved that the eigenvalues and the spectral singularities of the problem, given by (1.1)-(1.2), are finite under the following conditions

(1.5)

Our proof is different from the proof used in [7]. The reason for that is the proof of [7] is not applicable in the case of (1.5). It can easily be seen that the conditions (1.5) are weaker than (1.4).

Furthermore when the functions qi,

= 1, 2, are decreasing as polynomials the structure of spectral sets of the operator L are also examined in this work.

2. On the solutions of the system (1.1) Let us define the operator L on the space of

with the help of (1.1) and (1.2) as follows: For

L

(0, 00; C

which

are differentiable,

D(L)

{u :

L

(O,OOjC

absolutely continuous, l(u)

L 2 (O,

C 2), U2(0) - hUl (0) =O}

Then if

D(L) we define Lu

l( u)

Suppose that

q2

C(O,

and satisfy the inequality (2.1)

(2.4)

where C > 0,

> O.

Let qi(X) == 0, i = 1,2, the equation (1.1) has the solution given below

j(x,A) = (A.eXP(-:-iAX)) B· exp(zAx) A, B are constants.

Let A be a real number and

A) is a bounded solution of (1.1) with the following conditions

(2.2) Ul(X, A) = Ae-i>'x + 0(1), U2(X, A) = Bei>'x + 0(1), x

In [2], [7] have been shown that the solution (2.2) has the following form:

(2.3)

Ul(X, A) = ^Ae-i>'x + A 100 Hll(x, s)e-i>'Sds + B 100 ^H 12 (x, s)ei>'Sds U2(X, A) = Bei>'x + ^A 100 ^H 21 (X, s)e-iASds + ^B 100 H22 (x, s)eiASds where Hi,j(X,S), i,j = 1,2, are the solutions of the following two systems of Volterra integral equationsj

Hll(x, s) = ^-i 100

^{21 (t, t +} s - x)ql(t)dt, 0 ~ ^x ~ ^S

H 21 (X,S) = -~ ^{q2(X +2 S) + i} r=¥ ^{Hll(t,x +} s - t)q2(t)dt,

2z Jx

(2.5)

404 OmerAkinand Elgiz Bairamov

In the works [2] and [7] have been shown that these systems (2.4) and (2.5), have the bounded solutions which are unique. IT the func- tions qi(X), i

1,2, satisfy the estimate (2.1) then for the functions Hij(X,S), 0::; x::; s, i,j

1,2, the estimate given below holds good (2.6)

The matrix function

H(x,s):= ("11(:1;,S) H 12 (X,s»)

H 21 ^{(x,s) H}

(X,S)

has the role of the kernel of the operator transformation in the quantum

UH·

Let consider the vector functions

Where>. is a real number and Hi,j(X, x + t), x,

0, i,j = 1,2, were given as the solutions of (2.4) and (2.5). We can easily see that the vector functions

and the

are the proper cases of the system of (2.3) Le. these are the solutions of (1.1). Since the Wronskian

is independent of

we can take the limit of

as

+00, we

that lim

>.),

,X)] = 1 Therefore these solutions are the fundamental solutions of the system (1.1).

It is clear from the definition of e(l)(x,'x) and the relation (2.6) that the function

>.) has an analytic continuation on ,X to the lower half-plane. The similar result is also true for the function

to the upper half-plane.

(2.7)

(2) (2)

D+('x)

e

(0, >.) - he

(O,'x)

D_('x)

e~l)(O,'x) - hep)(0, ,X)

DEFINITION

2.1. The roots of D+(A)

° on the upper plane, ImA > 0, and the roots of D_(A)

° on the lower plane, ImA < 0 are called the singular numbers of the operator L. Multiplicity of. the root is called the multiplicity of that singular number of the operator L.

LEMMA

2.1. The singular numbers of the operator L are identi- cal when its non-real eigenvalues and the multiplicity of that singular numbers are also identical with the corresponding eigenvalues.

Proof.

We only prove the first part of the lemma, because the sec- ond part is easily obtained from the definition 2.1. Suppose that Ak, ImAk > 0, be a singular number of L, then D+(Ak)

0, the function e(2)(x, Ak) is a member of the space L 2 ^(0,

^C

and it is a solution of the problem given by the formulae (1.1) and (1.2). There- fore e(2)(x, Ak) is the eigen function of the operator L corresponding to the eigenvalue Ak. On the opposite side if Aa, ImA

> 0, is an arbitrary eigenvalue of L, then the problem (1.1)-(1.2) has a solution with the properties u(x, Aa) ;t 0, u

L

(0,

C

Under these con- ditions

Aa) and U2(0, Aa) both are non-zero numbers. Otherwise the uniqueness of the solution of the Cauchy problem related to (1.1) and (1.2) gives that u(x, Aa) == 0.

The Wronskian of u(x, Aa) and e(2)(x, Aa) is not depending on x then

W[

^{(2)] _}

^l'

^{{ (}

x, Aa ⁾

^{( 2 » )}

(x, Aa - U2(X, Aa)e ^{(2»)} _} ₁ (x, Aa - 0

x ...00

holds good and because of

must be satisfied. That is to say D+(Aa)

° is obtained. This shows us that Aa is a singular number of the operator L. We can show the result for D_(A) with the similar way. This completes the proof.

By using [6] we can easily see that the zeros fo D+(A) and D_(A)

on the real line are spectral singularities of the operator L. This result

406 Cmer Akin and EIgiz Bairamov

and lemma

show

that the structure of discrete spectrum of the operator L is the same as the structure of the roots of D+{A) = ° ^on

the closed upper plane and the roots of D _ (A)

° on the closed lower plane. For the sake of shortness and having the same structure of the roots of D+{A) = ° ^{and D_{A)}

° examine the structure of D+{A) on the upper'semi-plane.

3. The structure of the discrete spectrum of L

DEFINITION

3.1. H the complex valued functions qi{X), i =

continuous on the interval (O, 00) and (3.1)

Im.\

0, D+{A)

O}

ImA > 0, D+{A) = ^O}

, is satisfied for

> O,Ti- =

n, i = 1,2, n

Z+ for some constant C

then the functions qj{x) are called in the class of Mn,E'

Specially if the inequality (3.1) is satisfied for all n in Z+ U { +oo} then we call qj{x) in the class of Moo where Ck

R+ constants and inde- pendent of x. '

We give the following notations for the use of sequel.

s+

^{A:

C'+ ._ {\ •

VO . - A .

si ^:= p: ^{ImA = 0, D+{ A) = O}}

si

p : ^D+{A)

0, the multiplicity of A is infinite}

st ^:= p : 3{An), ^ImA

> 0, D+{An) ⁼ 0, An

A,

oo}

=

{The limit points of eigenvalues of the operator L on the upper half-plane}

LEMMA

3.1. If qi{X) in class Mk,E then the integral

(3.2)

is convergent and the function

is analytic for ImA > O. Furtheremore all the derivatives up to k of D+(A) are continuous for ImA

O. The derivatives, mentioned above, has the property given below

(3.3) Finally (3.4)

Proof. We know that if qi(X) E Mk,E' i = 1,2, then with the help of [2] and the systems of (2.4)-(2.5)

(3.5) IHij(x,s)1 $ ek(l + x +

i,j

1,2,

holds good. We can easily see that with the help of (3.5) the integrals (3.2) are convergent. Since the functions e~2)(0, ^{A) and} e~2)(0, ^{A) are} analytic for ImA > 0, D+( A) is analytic on the open upper-plane. Since the equality

D+(A) = e~2)(0, A) - he~2)(0, A)

[':to

r

= 1 + 10 ^H

^(0, t)e' dt - h 10 ^H

(O, t)e' dt

holds (3.3) and (3.4) can be obtained easily. This completes the proof.

Now we are going to prove some theorems for the structure of the discrete spectrum. for the operator L. It is clear that st ⁿ st ⁼

0, st ^u st ^{= S+} and for all points in st are the eigenvalues with the finite multiplicity of the operator L.

THEOREM

3.1. Ifqi(X)

Mo,E then the set st is closed, its Lebesgue measure is zero, st c st ^{and st} c st are satisfied.

Proof. Let us consider

1

D+(A) = ¹ +

[H

(0, t) - hH

(O, t)]e' dt.

408 Omer Akin and EIgiz Bairamov

We can obtain the following result by using this relation;

(3.6)

= 1 + 0(1) as I-XI- +00, Im-X

The asymtotic equality (3.6) gives us that the zeros of D+(-X) on the closed upper half-plane are in a bounded region. We know that D+(-X) is continuous up to the real axis, therefore si ^{is closed,} st c si, si c

Si can be obtained easily. By the uniqueness theorems of analytic functions the Lebesgue measure of Si is zero. (See [10]). This com- pletes the proof.

THEOREM 3.2. IT

qi(X)

tben

1) Tbe set of eigenvalues of tbe operator L satisfies·

2::Im-Xv < ob·

v

Here we also regard tbe multiple eigenvalues as tbe sum of multiplicity numbers.

2) Tbe following condition bolds.

(3.7) L ¹¹

^ll

^ll

^{l > -00}

v

wbere 11

l is the length of the complementary interval Iv of the set si ^{and the sum of (3.7) has been} taken on the bounded complementary intervals.

Proof. By the asympt()tic equality (3.6) the non-real zeros {Av} of the function D+(-X) on the open upper half-plane are in a bounded region and the limits of {-Xv} lie on .. the real line, i.e.

ImAv

0 as

00,

By the Lemma 3.1. the function D+(-X) is a bounded analytic function in Im-X > 0, then the following factorization holds ([11]),

Im-X >0

where c is constant, Icl

B()") is a Blaschke product and defined by

B()")

ITv).. - )..v

ITv(l- 2Im)..v), Im).. > O.

)..-)..v )..-)..v

In the B1aschke product )..v are all the zeros of D+()..) on the open upper half-plane and in this product the multiple zeros are repeated of the multiplicity numbers.

Now, we can take the number R > 0

big

all the zeros of D+()..) on the open upper-half plane are stayed on the following semi-disc P;

P

lm).. > 0, 1)..1

R}

Since for all arbitrary ).. belonging to P limv...

).vl

Im).. =f 0 we have

. Im)..v

hmv_oo I).. _ )..vl = o.

We also know that the for all ).. in

Blasehke product is absolutely couvergent, therefore we have the following result;

L ---:::-- < ^2Im)..v

v /).. -)..v/

By the last inequality and by the statement 0

I).. - f v I 5 2R, which holds for all ).. in

we obtain that

L ^{Im)..v <}

v

is satisfied.

By the hypothesis of the theorem

()..) exists and continuous on the real line.

We can prove

by using the uniqueness theorem of Beurling

We state the theorem for the readers convenience;

Suppose that the function g( z) is analytic in the unit disc, contin- uous upto its boundary and it satisfies the Holder's condition on the boundary.

the function

is zero on the set X C

of which Lebesgue measure zero, and the condition

L

holds then g( z) == 0, Hence the proof is completed.

410 Omer Akin and Elgiz Bairamov

THEOREM

3.3. Let qi(X) be in Moo. Then the si is closed, its Lebesgue measure zero and st ^c st satisfied. Furtheremore

(3.8) 100 1n T(s)d8(st,s) >

is satisfied, where T(s)

inf

Btf' , ⁹⁽ st ^{,s) the} linear measure of the neigbbom-hood of st. {We know that Bk is given by (3.2)J.

Proof. Since qi(X)

Moo then D+(A) is infinitely differentiable on the real axis. Moreover the function D+(A) and its all derivatives are continuous upto the real axis. This implies that st is closed, its Lebesgue measure zero and st c Sr ^{To prove} (3.8) we use the following uniqueness theorem of Pavlov [9]: Let g(z) is analytic in the unit disc, continuous with all derivatives upto its boundary and

If g( e

= 0 on the set F C [0,2,r] of which Lebesgue measure zero and the condition Jo ^oo lnT(s)d9(F,S)

holds then g(z) == O.

Hence the proof is completed.

THEOREM

3.4. H the conditions

, (3.9) Iqi(X)1

C exp( -exa),i

1,2,

e > 0, 0 <

< 1/2

is satisfied then st is closed, its Lebesgue measure is zero and the condition

:L: ^II

<

bolds where Ilv I is the length of the complementazy intervals Iv of the set si.

Proof. If qi has the property (3.9) we know that from the 'systems

of (2.4), (2.5)

and then (3.10)

is satisfied, where Al and d

are constants depending on 6 and a. By using

(1 + ~) ^{le/Ot <}

⁽¹ +

<

kle <

and (3.10) we obtain that

(3.11) Bk S Adleklkk(l-Ot)/OI

is satisfied, where A and d are constants depending on 6 and a. From (3.3) we have

(3.12)

IDCk)(A)1 S CdkklkkCI-Ot)/OI

(3.12) gives us that D+(A)

G

and si ^rt. EOt/ CI - OI), where

shows the Gevrey class of the order aj(l -

and EOI/(l-OI) shows the class of uniqueness set (see [3]). Then using Carleson's the- orem [1 J we obtain that si is closed, its Lebesgue measure zero and the following inequality is satisfied

L

<

v

Thus the proof of theorem 3.4 is completed.

We showed that D+(A)

and si ^rt. EOI/(l-Ot). For the structure

si see also Hruscev's theorem given in [3J.

THEOREM

3.5. Hthe functions q.(x), i = 1,2, satisfy the inequal- ities of

(3.13) Iqi(X)1 s Cexp(-6vx),

= 1,2, 6> 0 then the set si ^{is empty.}

Proof. By using the hypothesis (3.13) and the relation (3.11) we can

obtain the following result

412 Omer Akin and Elgiz Bairamov

(3.15)

Comparing the relation (3.13) and the hypothesis

theorem 3.3 we can see that under (3.13) the theorem 3.3 holds good. Using kk :5 k!ek we arrive the result as follows

. cdks kk2k

(3.14) T(s)S:irf k! S:irf(dkskek,kk):5Cexp(-d-le-ls-l) Getting help from (3.8) and (3.14) we

that

1

^dO( '1' ^{s) <}

is satisfied, where 1

is the length of the longest bounded complemen- tary intervals of st. The condition (3.15) is satisfied if and only if the linear measure 6(St, s) = (),rerall·s, This explains·that st' ^{must be}

an

empty set. Thus we have completed the proof of the theorem.

According to the theorem 3.3 si ^c st it is easy to see that ~der the condition (3.13) the set st ^{is empty.}

COROLLARY

3.1. When tbe inequality (3.13) is satisfied tben tbe eigen values and spectral singularities of tbe operator L is finite. Tills is also true for the multipliCity of tbem.

When tbe condition (3.13) bolds tbe spectral expansion formula re- lated witb tbe eigen functions of tbe operator L will be examined in a different work.

ACKNOWLEDGEMENT.

The authors are grateful to the referee for this kind interest and valuable suggestions for the improvements of this paper.

References

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Ankara University, Faculty of Sciences Department of Mathematics

Ankara-TURKEY