Spectral Properties of a Transport Operator in a Slab with Integral Boundary Conditions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

222, 192]207 Ž1998.

AY985931

Spectral Properties of a Transport Operator in a Slab with Integral Boundary ConditionsU Zhang Xianwen Department of Mathematics, Xinyang Teacher’s College, Henan Pro¨ ince, Xinyang 464000, People’s Republic of China Submitted by Joyce R. McLaughlin Received July 16, 1996

In this paper, the spectrum of a transport operator A in a slab with integral boundary conditions of Maxwell type is discussed in its natural space L1 Ž G .. It is shown that the essential spectrum of the transport operator A occupies the left half plane Re l F yS and there exist only a finite number of eigenvalues of the transport operator A in each vertical band contained in the right half plane Re l ) yS. Q 1998 Academic Press

1. INTRODUCTION AND MAIN RESULTS Consider the following linearized transport equation in a uniform slab with integral boundary conditions of Maxwell type: ­ ­ k 1 f Ž x, m , t . s ym f Ž x, m , t . y S f Ž x, m , t . q f Ž x, mX , t . d mX , ­t ­x 2 y1

H

Ž x, m . g G s w ya, a x = w y1, 1 x , m f Ž ya, m , t . s

0

Hy1a

< m < f Ž a, m , t . s

q

1

t ) 0,

< mX < f Ž ya, mX , t . d mX ,

X ym f

H0 a

Ž a, mX , t . d mX ,

Ž 1. m g Iqs Ž 0, 1 x , Ž 2 .

m g Iys y1, 0 . ,

Ž 3.

where S, k and a " are positive constants such that the scattering coefficients aq and ay on the boundary belong to the interval Ž0, 1x. For the physical significance of S and k, see w6x. * This work was supported by the Educational Commission of Henan Province and the Natural Science Foundation of China. 192 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

193

PROPERTIES OF A TRANSPORT OPERATOR

Define linear operators B and K as B: L1 Ž G . ª L1 Ž G . ;

Ž B f . Ž x, m . s ym

­ ­x

f Ž x, m . y S f Ž x, m . ,

D Ž B. s  f g L1 Ž G .< Ži. f Ž x, m . is absolutely continuous about x on wya, ax for almost all m g wy1, 1x; Žii. < m < f Ž"a, m . is integrable on wy1, 1x and satisfies the boundary conditions Ž2. and Ž3.4 , K : L1 Ž G . ª L1 Ž G . ;

Ž K f . Ž x, m . s

k

1

X

X

H f Ž x, m . d m , 2 y1

D Ž K . s L1 Ž G . . Obviously, K is a bounded linear operator such that 5K 5 s k . Let A s B q K and DŽA. s DŽB.. Then A is the so-called linear transport operator with integral boundary conditions of Maxwell type. In this paper, we call B the streaming operator. If a "s 0, then the boundary conditions Ž2. and Ž3. become the free boundary conditions as stated in w6x. In this case, we denote the streaming operator B and the transport operator A by B 0 and A 0 , respectively. In the case of the free boundary conditions, Lehner and Wing w6, 7x have proved that: Ži. The spectrum of A 0 for Re l ) yS consists of a finite but positive number of eigenvalues, each of which has finite algebraic multiplicity and is of index 1. Žii. The continuous spectrum of A 0 occupies the half plane Re l F yS. These results are demonstrated in L2 space. In 1971, Suhadolc w10x generalized partly these results to L1 space, which is physically more appropriate to the transport problem than L2 space. At the same time, many authors begin to discuss the spectral properties of the transport operator with reflection boundary conditions Žsee, e.g., w1]3, 8, 9x. and tried to generalize Lehner]Wing’s results. Most of them restricted their discussions to L2 space. However, the study of the spectrum of the transport operator with integral boundary conditions of Maxwell type in L1 space is still lacking: the aim of this paper is to fill a gap in this field. The main results of this paper can be summarized as follows. THEOREM 1. Ži. The spectrum of A for Re l ) yS consists of at most countably many isolated eigen¨ alues, each of which has finite algebraic multiplicity.

194

ZHANG XIANWEN

Žii. The essential spectrum of A occupies the left half plane Re l F S; i.e., sessŽA. s  l g C
aq ay

HI

.

mX m

exp "

ž

2 aŽ l q S . Y

m

.

2 aŽ l q S .

mX

/

d mY ,

Ž m , mX . g I "= I " , h" Ž l ; m . s

a

X

d mX

Hya dx HI

"

= exp "

ž

q

a

aq ay < m< .

HI

2 aŽ l q Ý.

mY

dxX

Hya HI

a"

< m< .

.

exp "

ž

Ž a . xX . Ž l q S . mX

Ž a " xX . Ž l q S . mX

/

/

g Ž xX , mX . d mY

g Ž xX , mX . d mX ,

m g I". Define linear operators Ml" :

Ž Ml" f . Ž m . s H

I"

Ml" , Hl" , R " l ,

and Pl as

L Ž I " , < m < d m . ª L1 Ž I " , < m < d m . , 1

M " Ž l ; m , mX . f Ž mX . d mX ,

Hl" : L1 Ž G . ª L1 Ž I " , < m < d m . ;

f g L1 Ž I " , < m < d m . ,

Ž Hl" g . Ž m . s h " Ž l ; m . , g g L1 Ž G . ,

1 1 < < R" l : L Ž I" , m dm . ª L Ž G " . ;

Ž R "l f . Ž x, m . s exp

ž

.

Ž a . x . Ž l q S. m

/

f Ž m. ,

f g L1 Ž I " , < m < d m . ,

195

PROPERTIES OF A TRANSPORT OPERATOR

where G "s S = I " and S s wya, ax, Pl : L1 Ž G . ª L1 Ž G . ; x

1

ya

m

Ž Pl g . Ž x, m . s H

exp y

ž

Ž x y xX . Ž l q S . m

/

g Ž xX , m . dxX , for m ) 0,

and

Ž Pl g . Ž x, m . s H

a

x

1

exp y

< m<

ž

Ž x y xX . Ž l q S . m

/

g Ž xX , m . dxX , for m - 0.

LEMMA 1 w14x. If Re l ) yS, then Ml" , Hl" , R " l , and Pl are all bounded linear operators such that 5Ml" 5 - 1,

5Hl" 5 - 1,

5R " 5 l F

1 Re l q S

, 5Pl 5 F

1 Re l q S

.

Furthermore, r ŽB. >  l g C
Ž l I y B.

y1

LEMMA 2.

s Pl q

q Rq l Ž I y Ml .

y1

0 y Ry l Ž I y Ml .

0

y1

Hq l . Ž 4. Hy l

If Re l ) yS, then Ml" and Hl" are compact operators.

Proof. In fact, Ml" and Hl" are bounded finite rank operators by Lemma 1 and their definitions. Let

gs

`

ds

H1 exp Ž y2 aŽ l q S . s . s

2

,

for Re l ) yS.

Then < g < - 1. LEMMA 3.

If Re l ) yS, then

Ži. ŽMl". 2 s aq ay g 2 Ml" , Ml" Hl" s aq ay g 2 Hl" . Žii. Ž I y Ml".y1 s I q Ž1 y aq ay g 2 .y1 Ml" .

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ZHANG XIANWEN

Proof. Ži. For any f g L1 Ž Iq, < m < d m ., we have 2 Y Y Y q q q q Ž Mq l . f Ž m . s Ml Ž Ml f . Ž m . s H Ml Ž l ; m , m . Ž Ml f . Ž m . d m

Iq

s

1

0

mY

H0 Hy1a 1

q ay

0

m

exp

ž

q ay

=exp

ž

Z

m

y

2 aŽ l q S .

mY

/

d mZ

mX

H0 Hy1a

=

ž

2 aŽ l q S .

mY

2 aŽ l q S . XXXX

m

y

2 aŽ l q S .

mX

d mZX

/ /

=f Ž mX . d mX d mY 1

s aq ay =

ž ž žH ž H0 exp

0

y1

1

q ay

2 aŽ l q S .

mX m

d mY

mZ

/ / / /

0

2 aŽ l q S .

mY

2 aŽ l q S .

exp

H0 a

=

y

žH ž y1

exp

d mZ

ZX

m

y

2 aŽ l q Ý.

mX

d mZX

/ /

= f Ž mX . d mX s aq ay g 2 Ž Mq l f . Ž m. . . 2 s aq ay g 2 Mq This proves that ŽMq l l . By a similar procedure we can show the other three equations. Žii. This is a direct consequence of Ži.. Define linear operators L and J as L: L1 Ž S . ª L1 Ž S . ; J: L1 Ž G . ª L1 Ž S . ;

ŽL w . Ž x . s kw Ž x . , 1

w g L1 Ž S . ,

Ž J f . Ž x . s 12 H f Ž x, mX . d mX , y1

f g L1 Ž G . .

197

PROPERTIES OF A TRANSPORT OPERATOR

It follows from w6x that JPl L is a bounded integral operator for Re l ) yS and can be represented by

Ž JPl L w . Ž x . s H E1 Ž l ; x, xX . w Ž xX . dxX ,

w g L1 Ž S . ,

S

where E1 Ž l ; x, xX . s

k

`

X

H exp Ž y Ž l q S . < x y x < s . 2 1

ds s

Ž x, xX . g S = S.

,

Hence, ŽJPl L. 2 is a bounded integral operator for Re l ) yS and can be represented by 2 Ž JPl L. w Ž x . s H E2 Ž l ; x, xX . w Ž xX . dxX ,

w g L1 Ž S . ,

S

where E2 Ž l ; x, xX . s LEMMA 4.

Y

Y

X

Y

HSE Ž l ; x, x . E Ž l ; x , x . dx . 1

1

For Re l ) yS, E2 Ž l; x, xX . g L`Ž S = S ..

Proof. If Re l ) yS, then it follows from w6x that E1 Ž l ; x, xX . s

k 2

yln Ž Ž l q S . < x y xX < .

q

`

Ý ms1

Ž y1.

m

m = m!

m Ž l q S . < x y xX < m y C ,

Ž 5.

where C is the Euler constant. Since lim t ª 0 t 1r3 ln t s 0, we can rewrite formula Ž5. as X

E1 Ž l ; x, x . s

C Ž l ; x, xX . < x y xX < 1r3

,

C Ž l ; x, xX . g L` Ž S = S . .

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ZHANG XIANWEN

In the following, we employ the method in w13x to estimate < E2 Ž l; x, xX .<. Let Cl s ess supŽ x, xX .g S=S < C Ž l; x, xX .<. We then have < E2 Ž l ; x, xX . < F Cl2 < x y xY
Ž x, xX . g S = S.

HS

If x s xX , then < E2 Ž l ; x, xX . < F Cl2

a

Y y2 r3

Hya< x y x < 1r3

s 3 Ž a y x.

dxY

q Ž a q x.

1r3

Cl2 F 6 Ž 2 a .

1r3

Cl2 .

If x / xX , we can assume that x ) xX . Then for x y xX - a, < E2 Ž l ; x, xX . < F Cl2

ayx

y1 r3

Hyayx< t < 2a

y1 r3

F Cl2

Hy2 a< t <

F Cl2

H0

t

q Cl2

H0

2 a y1r3 <

< t q x y xX
t q x y xX
2 a y1r3 <

t

F Cl2 Ž x y xX . Ž

F Cl2 Ž x y xX .

t y x q xX
1r3

H02 ar xyx

=

< t q x y xX
X

.

ty1r3 Ž t q 1 .

1r3

1 y1r3

H0 t

q 2

2

H1 Ž t y 1.

F 3Cl2 Ž 2 a .

F 6 Ž 2 a.

1r3

1r3

Cl2 .

dt q t

y1 r3

q ty1r3 < t y 1
H12 ar xyx t

1r2 1r3 y1r3

H0

q

y1 r3

Ž

dt q

dt q

q Ž 2 a y x q xX .

. y2r3

1

H1r22

1r3

H22 ar xyx Ž

X

.

dt

Ž1 y t .

Ž t y 1.

y Ž x y xX .

1r3

q2y1r3 Ž x y xX .

1r3

1r3

y1 r3

y2r3

dt

dt

199

PROPERTIES OF A TRANSPORT OPERATOR

For x y xX G a, we have < E2 Ž l ; x, xX . < F Cl2 Ž 2 a . 1r3 F Cl2 Ž 2 a .

1r3

2

ty1r3 Ž t q 1 .

H0

1 y1r3

dt q

H0 t

q

2

3 2

q

3 2

dt q

t

H1 Ž t y 1.

1r3

2

1r2 1r3 y1r3

2

q ty1r3 < t y 1
H1 Ž t y 1.

q

H0

s Cl2 Ž 2 a .

y1r3

y1 r3

y1r3

1

H1r22

dt

1r3

Ž1 y t .

y1 r3

dt

dt

q 6 = 2y1r3 q

3 2

F 9 Ž 2 a.

1r3

Cl2 .

Hence, E2 Ž l; x, xX . g L`Ž S = S .. PROPOSITION 1. y B.y1 L.

If Re l ) yS, then Wl2 is compact, where Wl s JŽ l I

Proof. By Lemmas 1 and 3, we obtain

Wl s JPl L q J

Ž 1 y aq ay g 2 .

y1

q Rq l Hl

0

Ž 1 y aq ay g 2 .

0

y1

y Ry l Hl

L.

Denote the second term of the right-hand side of the above equation by Ql; it follows from Lemma 2 that Ql is compact. But 2

Wl2 s Ž JPl L . q

3

Ý Ql , i , is1

where each Ql, i contains a factor Ql Ž i s 1, 2, 3.. Hence Ý3is1Ql , i is compact. On the other hand, it follows from the proof of Lemma 4 and w10x that ŽJPl L. 2 is compact. So, Wl2 is compact. LEMMA 5 w14x. If Re l ) yS, then l g Ps ŽA. if and only if 1 g Ps ŽWl ., where Ps ŽA. is the point spectrum of the operator A. PROPOSITION 2.

sessŽB. s  l g C
Proof. We have by Lemma 1 that l g r ŽB. for Re l ) yS, and formula Ž4. is valid. On the other hand, it follows from Lehner and Wing

200

ZHANG XIANWEN

w6x that Ž l I y B 0 .y1 s Pl for Re l ) yS. Hence, we can rewrite formula Ž4. as

Ž l I y B. s

y1

y Ž l I y B0 .

q Rq l Ž I y Ml .

y1

y1

0 y Ry l Ž I y Ml .

0

y1

Hq l y . Hl

Ž 6.

The right-hand side of Ž6. is a finite rank operator and hence a compact operator, we obtain by Kato w4, problem 5.38; p. 244x that sessŽB. s sessŽB 0 .. On the other hand, we can show by the same method used in w1, 6x that sessŽB 0 . s  l g C
3 y aq ay g 2 1 y aq ay g

=

2

k Re l q S

,

Re l ) yS.

But lim lªq`g s 0 and lim l ªq` krŽRe l q S . s 0; hence lim l ªq` 5KŽ l I y B.y1 5 s 0. From the above discussion and Voigt w12x, we obtain that  l g C
Ž l I y A.

y1

s Ž l I y B.

y1

`

Ý

K Ž l I y B.

y1

n

.

Ž 7.

ns0

On the other hand, it follows from w10x that l g r ŽA 0 . for Re l ) yS q k and

Ž lI y A0 .

y1

s Ž l I y B0 .

y1

`

Ý ns0

K Ž l I y B0 .

y1

n

.

Ž 8.

201

PROPERTIES OF A TRANSPORT OPERATOR

From Ž4., Ž7., and Ž8., we obtain for l large enough

Ž l I y A.

y1

y Ž lI y A0 .

s Ž l I y B.

y1

q Ž l I y B.

y1

y Ž l I y B0 . `

y1

y1

K Ž l I y B.

Ý

y1

n

ns1

y Ž l I y B0 .

`

y1

K Ž l I y B0 .

Ý

y1

n

ns1

s Fl q Ž l I y B 0 .

y1

q Fl

`

Ý  K Ž l I y B0 . y1 q KFl 4

n

ns1

y Ž l I y B0 .

y1

`

Ý

K Ž l I y B0 .

y1

n

,

ns1

where Fl s

q Rq l Ž I y Ml .

y1

Hq l

y Ry l Ž I y Ml .

y1

Hy l

is a finite rank operator. So, we have

Ž l I y A.

y1

y Ž lI y A0 .

s Fl q

y1

`

Ý ½ Ž l I y B0 . y1Fl , n q Fl K Ž lI y B0 . y1

ns1

n

q Fl Fl , n , Ž 9 .

5

where Fl, n is a finite rank operator Ž n s 1, 2, 3, . . . .. Obviously, each term of the right-hand side of Ž9. is a finite rank operator and the operator series Ž9. is convergent in the uniform operator topology for l large enough, so Ž l I y A.y1 y Ž l I y A 0 .y1 is compact for l large enough. It follows from w4, Problem 5.38, p. 244x that sessŽA. s sessŽA 0 .. By virtue of the method used in w5x, we can show that sessŽA 0 . s  l g C
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ZHANG XIANWEN

countably many isolated eigenvalues. The following two lemmas give the connections of Pas ŽA 0 .1 and PasŽA 0 . 2 . LEMMA 6. Pas ŽA 0 .1 s Pas ŽA 0 . 2 . Proof. The inclusion Pas ŽA 0 . 2 ; Pas ŽA 0 .1 follows immediately from L2 Ž G . ; L1 Ž G .. On the other hand, if l g Pas ŽA 0 .1 , then there exists an f g L1 Ž G . such that Ž l I y A 0 . f s 0. Hence ŽŽ l I y B 0 .y1 K. 3 f s f, i.e., f s Ž l I y B 0 .y1 LWl2, 0 J f, where Wl , 0 s JŽ l I y B 0 .y1 L. But k

Ž LWl2, 0 J f . Ž x, m . s 2 H E2 Ž l ; x, xX . f Ž xX , mX . dxX d mX . G

It follows from Lemma 4 that < Ž LWl2, 0 J f . Ž x, m . < F

k 2

X

X

X

HG< E Ž l ; x, x . < < f Ž x , m . < dx 2

X

d mX F C 5 f 5 1 ,

where C is a positive constant and 5 ? 5 1 is the norm of L1 Ž G .. Hence, < f Ž x, m . < F Ž Re l I y B 0 . y1
1

x

Hya m exp

F C 5 f 51

x

ž

y 1

Ž x y xX . Ž Re l q S .

Hya m exp

m

ž

< LWl2, 0 J f . Ž xX , m . < dxX

/Ž

Ž Re l q Ý . Ž xX y x . m

/

dxX F

C Re l q S

5 f 51.

If m - 0, we get by a similar method that < f Ž x, m . < F

C Re l q S

5 f 51.

So, f g L`Ž G . ; L2 Ž G ., which proves l g Pas ŽA 0 . 2 and hence PasŽA 0 .1 ; Pas ŽA 0 . 2 . LEMMA 7.

If l g Pas ŽA 0 .1 s PasŽA 0 . 2 , then

N Ž l I y A 0 . Ž in L1 Ž G . . s N Ž l I y A 0 . Ž in L2 Ž G . . ; L` Ž G . , Ž 10 . NŽŽ lI y A0 .

2

. Ž in Li Ž G . . s N Ž l I y A 0 . Ž in Li Ž G . . ,

i s 1, 2, Ž 11 .

where N Ž l I y A 0 . Ž in Li Ž G .. is the null space of the operator l I y A 0 in space Li Ž G . Ž i s 1, 2..

203

PROPERTIES OF A TRANSPORT OPERATOR

Proof. In fact, formula Ž10. is valid by the proof of Lemma 6, and formula Ž11. for the case of i s 2 is valid by w7x. We only need to show formula Ž11. for the case of i s 1. Let f g N ŽŽ l I y A 0 . 2 .Žin L1 Ž G .., that is, f g L1 Ž G . and Ž l I y A 0 . 2 f s 0. It follows from formula Ž10. that g s Ž l I y A 0 . f g L`Ž G . and f s Ž l I y B0 .

y1

K f s K Ž l I y B0 . K Ž l I y B0 .

y1

g q Ž l I y B0 .

y1

y1

K f,

g q K Ž l I y B0 .

K f s K Ž l I y B0 .

y1 2

y1

Ž 12 .

K f,

Ž 13 .

g q LWl2, 0 J f .

Ž 14 .

From the proof of Lemma 6, we know that LWl2, 0 J f g L`Ž G . and KŽ l I y B 0 .y1 L`Ž G . ; L`Ž G .. So, KŽ l I y B 0 .y1 K f g L`Ž G . by Ž14.. Similarly, we can show that K f and f belong to L`Ž G . by Ž13. and Ž12.. It follows from w7x that Ž l I y A 0 . f s 0. In order to prove Theorem 2, we also need the following lemma, the proof of which can be found in w1, 9x. LEMMA 8. If b 2 ) b 1 ) yS, then there exists a t 0 ) 0 such that 5Wl , 0 5 F 12 for Re l g w b 1 , b 2 x and
k 2 Ž 1 y aq ay g

2

.

H0

qa .

1

1

aq ay g

1

H0

m 1

m

exp y

ž

exp y

ž

Ž a " x . Ž l q S. m

Ž a . x . Ž l q S. m

/

/

dm

dm ,

x g S, Re l ) yS,

h" Ž l, x . s

1

H0 exp

ž

y

Ž a . x . Ž l q Ý. m

/

dm ,

x g S, Re l ) yS.

PROPOSITION 3. Suppose Re l ) yS and Im l / 0. Then l is an eigen¨ alue of the transport operator A if and only if 1 q ² fq Ž l , ? . , hq Ž l , ? . : ² fy Ž l , ? . , hq Ž l , ? . :

² fq Ž l , ? . , hy Ž l , ? . : 1 q ² fy Ž l , ? . , hy Ž l , ? . :

s 0,

where f " Ž l, x . is the unique solution of the following integral equation in L1 Ž S .: f " Ž l, x . s

X

HSE Ž l ; x, x . f 1

"

Ž l , xX . dxX y G" Ž l , x . .

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ZHANG XIANWEN

Proof. For a meromorphic function w Ž z . in domain D and a linear operator T in a Banach space X, define two integer-valued functions ¨ Ž z, w . and ¨˜Ž z, T. as follows: ¨ Ž z, w . s

¡k, ¢0,

if z is a zero of w Ž z . of order k, if z is a pole of w Ž z . of order k, otherwise;

~yk,

¡0,

¨˜Ž z, T . s

~

if z g r Ž T . , if z g s Ž T . _ sess Ž T . and Pz is the eigenprojection associated with z, if z g sess Ž T . .

dim Pz ,

¢q`

It follows from Lemmas 1 and 3 that Wl s JPl L q Ql , where Ql s J

Ž 1 y aq ay g 2 .

y1

q Rq l Hl

0

Ž 1 y aq ay g 2 .

0

y1

y Ry l Hl

L.

By some complicated calculation, we obtain for w g L1 Ž S .,

Ž Ql w . Ž x . s ² w Ž ? . , hq Ž l , ? . : Gq Ž l , x . q ² w Ž ? . , hy Ž l , ? . : Gy Ž l , x . , where ² w Ž?., h " Ž l, ? .: s HS w Ž x .h " Ž l, x . dx. From w4, p. 245x, we now that the W y A determinant v Ž z . associated with JPl L and Ql is 1q² Ž JPl LyzI . y1 GqŽ l , ? . , hq Ž l , ? . :

² Ž JPl LyzI . y1 GqŽ l , ? . , hyŽ l , ? . :

² Ž JPl LyzI . y1 GyŽ l , ? . , hqŽ l , ? . :

1q² Ž JPl LyzI . y1 GyŽ l , ? . , hyŽ l , ? . :

.

Thus, we have for Re l ) yS w4, Theorem 6.2; p. 247x. ¨˜Ž 1, Wl . s ¨˜Ž 1, JPl L . q ¨ Ž 1, v . .

Ž 15 .

If Re l ) yS and Im l / 0, then it follows from Lemma 6 and w6x that 1 g r ŽJPl L. s r ŽJŽ l I y B 0 .y1 L.. Hence ¨˜Ž1, JPl L. s 0 and formula Ž15. becomes ¨˜Ž 1, Wl . s ¨ Ž 1, v . ,

Re l ) yS,

and Im l / 0.

Ž 16 .

On the other hand, we know by Proposition 1 and w11x that 1 g s ŽWl . if and only if 1 g Ps ŽWl . for Re l ) yS. Thus we have by formula Ž16. that 1 g Ps ŽWl . if and only if ¨ Ž1, v . ) 1, i.e., 1 g Ps ŽWl . if and only if

205

PROPERTIES OF A TRANSPORT OPERATOR

v Ž1. s 0. In consideration of Lemma 5, we obtain that l g Ps ŽA. if and only if v Ž 1 . s 0.

Ž 17 .

Let f " Ž l, x . s ŽJPl L y I .y1 G" Ž l, x .. Then G" Ž l, x . s ŽJPl L y I . f " Ž l, x ., i.e., f " Ž l, x . is the unique solution of the following integral equation in L1 Ž S ., f " Ž l, x . s

X

HSE Ž l ; x, x . f 1

"

Ž l , xX . dxX y G" Ž l , x . ,

and formula Ž17. becomes 1 q ² fq Ž l , ? . , hq Ž l , ? . : ² fy Ž l , ? . , h q Ž l , ? . :

² fq Ž l , ? . , hy Ž l , ? . : 1 q ² fy Ž l , ? . , hy Ž l , ? . :

s 0.

Proof of Theorem 2. If there exist infinitely many eigenvalues of A in the band  l g C < y S - b 1 F Re l F b 2 4 , then there exists a sequence  l n4`ns3 ;  l g C < b 1 F Re l F b 2 4 l Ps ŽA. such that l i / l j Ž i / j .. It follows from Theorem 1 that any subsequence of  l n4 cannot be convergent, so, we may assume that bn s Re l n ª b 0 g w b 1 , b 2 x and
² fq Ž l n , ? . , hy Ž l n , ? . : 1 q ² fy Ž l n , ? . , hy Ž l n , ? . :

s 0, Ž 18 .

where f " Ž l n , ? . is the unique solution of the following integral equation in L1 Ž S . for any n ) N1: f " Ž ln , x . s

X

HSE Ž l ; x, x . f 1

n

"

Ž l n , xX . dxX y G" Ž l n , x . .

Ž 19 .

On the other hand, it follows from Lemma 8 that there exists a N ) N1 such that 5Wln , 0 5 F 12 for n ) N; hence, we have for n ) N, 5 f " Ž l n , ? . 5 F 5Wl

n, 0

f " Ž l n , ? . 5 q 5 G" Ž l n , ? . 5

F 12 5 f " Ž l n , ? . 5 q 5 G" Ž b 1 , ? . 5 , 5 f " Ž l n , x . 5 F 2 5 G" Ž b 1 , ? . 5 ,

n)N

Ž 20 .

206

ZHANG XIANWEN

From Ž19., we obtain for n ) N, 5 f " Ž ln , ?. 5 F

X

HS< E Ž l ; x, x . < < f n

2

"

Ž l n , xX . < dxX

q < E1 Ž l n ; x, xX . < < G" Ž l n , xX . < dxX q < G" Ž l n , x . <

HS

F

X

HS< E Ž b ; x, x . < < f 2

1

"

Ž l n , xX . < dxX q < G" Ž b 1 , x . <

q < E1 Ž b 1 ; x, xX . < < G" Ž b 1 , xX . < dxX s F Ž b 1 , x . .

HS

From Lemma 4, we know that E2 Ž b 1; x, xX . g L`Ž S = S .. In consideration of formula Ž20., we get X

HS< E Ž b ; x, x . < < f 2

1

"

Ž l n , xX . < dxX F C 5 G" Ž b 1 , ? . 5 s const.

The other two terms of F Ž b 1 , x . are obviously integrable on S; hence F Ž b 1 , x . g L1 Ž S . and < f " Ž l n , x .< F F Ž b 1 , x . Ž n ) N .. But
n ) N, x g S,

Ž 21 .

< f " Ž l n , x . h. Ž l n , x . < F F Ž b 1 , x . ,

n ) N, x g S.

Ž 22 .

By the Riemann]Lebesgue’s lemma, we have `

H exp Žy Ž a . x . Ž b nª` 1 lim

n

q S . s . exp Ž y Ž a . x . stn i .

ds s

s 0,

x g S,

that is, lim h " Ž l n , x . s 0,

nª`

x g S.

Ž 23 .

Applying formulas Ž21., Ž22., and Ž23. and Lebesgue’s convergence theorem, we obtain lim ² f " Ž l n , ? . , h " Ž l n , ? . : s 0,

nª`

lim ² f " Ž l n , ? . , h. Ž l n , ? . : s 0;

nª`

PROPERTIES OF A TRANSPORT OPERATOR

207

hence we have lim

nª`

1 q ² fq Ž l n , ? . , hq Ž l n , ? . : ² fy Ž l n , ? . , hq Ž l n , ? . :

² fq Ž l n , ? . , hy Ž l n , ? . : 1 q ² fy Ž l n , ? . , hy Ž l n , ? . :

s 1,

which contradicts formula Ž18. and the proof is complete.

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