On the Spectrum of a One-Velocity Transport Operator with Maxwell Boundary Condition

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

202, 920]939 Ž1996.

0354

On the Spectrum of a One-Velocity Transport Operator with Maxwell Boundary Condition* Zhang Xianwen and Liang Benzhong Department of Mathematics, Xinyang Teachers College, Henan Pro¨ ince, Xinyang, 464000, People’s Republic of China Submitted by William F. Ames Received July 31, 1995

The spectrum of a one-velocity transport operator with Maxwell boundary condition is discussed in L1 space. First, it is proved that the spectrum of a streaming operator associated with the transport operator consists of infinitely isolated eigenvalues, each of which is simple; furthermore, a formula for computation of these eigenvalues is obtained. Second, it is proved that the essential spectrum of the transport operator is empty and it is pointed out that the dominant eigenvalue of the transport operator exists. Finally, a necessary and sufficient condition for the existence of a complex eigenvalue of the transport operator in a right half plane is given. Q 1996 Academic Press, Inc.

1. INTRODUCTION In a homogeneous sphere medium with spherical symmetry and isotropical scattering, the one-particle distribution function f Ž r, m , t . satisfies the following transport equation,

­ ­t

f Ž r , m , t . s ym

­f ­r

y

1 y m2 ­f r

­m

y Sf q

cS 2

1

Hy1f Ž r , m9, t . d m9, Ž 1.

where r g S s w0, R x, m g V s wy1, 1x, t G 0, and R is the radius of the sphere. Both S and c are positive constants. It is well known that properties of a solution for Eq. Ž1. are closely relevant to the spectral distribution of the transport operator defined by *Supported by the National Natural Science Foundation of China and the Natural Science Foundation of Henan Province. 920 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

ONE-VELOCITY TRANSPORT OPERATOR

921

the right half side of Eq. Ž1.. J. Lehner w3x discussed the spectrum of the transport operator with free boundary condition in L2 space. It was proved that the spectrum of the transport operator with free boundary condition Ždenoted by A 0 . consists of at most countable eigenvalues, each of which has finite algebraic multiplicity. R. Van Norton w6x and S. Ukai w5x pointed out that there exist countable infinite real eigenvalues of A 0 diverging to minus infinite and that there exists no complex eigenvalue of A 0 in the right hand plane Re l ) yS. Because every eigenfunction of A 0 is an L`-function, it is not difficult to see that these results can be expanded to L p-space Ž1 F p - q`.. In practice, a transport equation with Maxwell boundary condition is to some extend more important than that with free boundary condition. So, the following integral boundary condition of Maxwell type < m
1

H0 am9f Ž R, m9, t . d m9

Ž y1 F m - 0, t G 0 . Ž 2 .

is considered in this paper, where a g Ž0, 1x is the scattering coefficient on the boundary of the sphere. Equation Ž2. becomes the free boundary condition if a s 0. Spectral results of a transport operator with Maxwell boundary condition are scarce. We should mention here that Zhang and Liang w8x have made a systematic study of the spectrum for the slab geometric transport operator with Maxwell boundary condition. In the first part of this paper, the spectrum of the so-called streaming operator is observed. It is shown that the spectrum of the streaming operator is an infinitely countable set, each element of which is an eigenvalue whose geometric multiplicity and algebraic multiplicity are equal to one. Further, a formula for computation of these eigenvalues is found. By means of this formula, it is pointed out that all eigenvalues except for one are complex. The second part of this paper is devoted to discussing the spectral properties of the transport operator with Maxwell boundary condition; the main result is a necessary and sufficient condition for the existence of a complex eigenvalue in the right half plane Re l ) yS. This paper gives an answer to an open question proposed by P. F. Zweifel et al. w9x in ‘‘The 7th International Conference on Transport Theory,’’ ŽTexas, 1981.. In consideration of the physical significance of the distribution function f Ž r, m , t ., we choose L1 Ž G, r 2 dr d m . Ža weighted L1 space with weight function r 2 . as the state space, where G s w0, R x = wy1, 1x. From the physical significance of f Ž r, m , t ., we know that the function < m < f Ž R, m , t .

922

ZHANG AND LIANG

is m-integrable on V, i.e., f Ž R, m , t . g L1 Ž V, < m < d m .. Let D s Ž x, y .< x 2 q y 2 F R 2 , y G 04 and introduce the transformation x s rm ,

y s r 1 y m 2 Ž r g S, m g V . .

'

Ž 3.

It is easy to show that transformation Ž3. is one-to-one from G onto D. By Ž3. an isometry J from the weighted L1 space L1 Ž G, r 2 dr d m . onto L1 Ž D, y dx dy . can be constructed as J : L1 Ž G, r 2 dr d m . ª L1 Ž D, y dx dy . f Ž x, y . s Ž Jf . Ž x, y . s f

ž

'x

2

q y2 ,

x

'x

q y2

2

/

,

f g L1 Ž G, r 2 dr d m . . Under the isometry J, the transport equation Ž1. and the Maxwell boundary condition Ž2. are transformed into

­ ­t

f Ž x, y, t . s y

­f ­x

rs y R

2

y S f Ž x, y, t . q

'x

2

q y2 ,

f y R 2 y y 2 , y, t s

ž '

cS

r

H f Ž x9, 'r 2 r yr

Ž x, y . g D,

R

ay

/ H R'R 0

y9

yy

2

2

R2

2

y x9 2 , t . dx9 Ž 4 .

tG0 f

ž 'R

2

y y9 2 , y9, t dy9.

/

Ž 5. Define linear operators Ba , K, A a as Ba : L1 Ž D, y dx dy . ª L1 Ž D, y dx dy . ;

Ž Ba f . Ž x, y . s y

­f ­x

y S f Ž x, y . .

D ŽB a . s  f < f and ­ fr­ x belong to L1 Ž D, y dx dy . such that f Ž" R 2 y y 2 , y . belong to L1 Ž S, Ž yrR 2 . dy . and satisfies the boundary condition Ž5.4 .

'

K: L1 Ž D, y dx dy . ª L1 Ž D, y dx dy . ;

Ž K f . Ž x, y . s

cS

r

H f Ž x9, 'r 2 r yr

2

y x9 2 . dx9

D Ž K . s L1 Ž D, y dx dy . A a s Ba q K,

D Ž A a . s D Ž Ba . .

ONE-VELOCITY TRANSPORT OPERATOR

923

Ba is the streaming operator, K is the collision operator, and A a is the transport operator, where a is the scattering coefficient on the boundary. In this paper, denote L1 Ž D, y dx dy . by X and its norm is 5f5s

f Ž x, y . y dx dy.

HD

Denote L1 Ž S, Ž yrR 2 . dy . by Y and its norm is 5 g5 s

y

HS g Ž y .

R2

dy.

Both X and Y are Banach spaces. If T is a linear operator from X to X, then s ŽT ., se s s ŽT ., Ps ŽT ., and r ŽT . represent the spectrum, the essential spectrum, the eigenvalue spectrum, and the spectral radius of T, respectively; N ŽT . and RŽT . represent the null space and the range of T, respectively.

2. SPECTRAL ANALYSIS OF THE STREAMING OPERATOR Ba Define two linear integral operators as follows for every complex number l, Ma , l : Y ª Y ;

Ž Ma , l f . Ž y . s H

a y9

R

0

R R yy

'

2

2

ey2 Ž lqS.'R

2

yy 9 2

f Ž y9 . dy9

Ha , l : X ª Y ;

Ž Ha , l g . Ž y . s

R

H0

a y9

Hy''RRyy9 yy9 R'R 2

dy9

2

2

2

2

yy

2

eyŽ lqS.Ž 'R

2

yy 9 2 yx 9 .

g Ž x9, y9 . dx9.

LEMMA 1. For any l g C, both M a , l and H a , l are bounded linear operators. Furthermore, if Re l ) yS, then 5M a , l 5 F a ,

5H a , l 5 F

a R2

,

r Ž Ma , l . - a .

924

ZHANG AND LIANG

Proof. For any f g Y 5M a , l f 5 s F

y

R

H0

R

2

ay

R

H0 R'R

sa

a y9

R

H0 R'R

2

2

yy

yy

2

R

dy

H0

2

y9 R

R y2 ŽRe l qS. R 2 yy9 2

'

H0 e

ey2 Ž lqS.'R

2

yy 9 2

ey2 ŽRe lqS.'R

2

y9 R2

f Ž y9 . dy9 dy 2

yy 9 2

f Ž y9 . dy9

f Ž y9 . dy9.

If Re l ) yS, then R

5M a , l f 5 - a

H0

y9

f Ž y9 . dy9 s a 5 f 5 .

R2

Hence 5M a , l 5 F a . If Re l F S, then 5M a , l f 5 F a ey2 ŽRe lqS. R

y9

R

H0

f Ž y9 . dy9 s a ey2ŽRe lqS. R 5 f 5 .

R2

Hence 5M a , l 5 F a ey2 ŽRe lqS. R . Similarly, for Re l ) yS, and f g Y, 5M a2 , l f 5 F a 2 = Fa

2

ž

R

y0

H0 R'R

2

R y2

2

žH

e

y y0

'R yy 9

2

2

ey2'R

ŽRe l qS .

2

yy 0 2 ŽRe l qS .

f Ž y9 .

0

1 y ey2 ŽRe lqS. R 2 Ž Re l q S . R

5 f 5.

That is, 5M a2 , l 5 F a 2

1 y ey2 ŽRe lqS. R 2 Ž Re l q S . R

.

y9 R2

dy9

dy0

/

/

ONE-VELOCITY TRANSPORT OPERATOR

925

Hence

(

r Ž M a , l . F 5M a2 , l 5 1r2 F a

1 y ey2 ŽRe lqS. R 2 Ž Re l q S . R

- a.

The assertion about H a , l can be proved by a similar way. LEMMA 2.

For any l g C, M a , l is compact.

Proof. In fact, M a , l is a one-dimensional linear operator. LEMMA 3.

For any l g C,

½

s Ž M a , l . _  04 s

a Ž 1 y ey2 Ž lqS. R 2Ž l q S . R

5

.

Furthermore,

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R is an eigen¨ alue of the operator M a , l , where

a Ž 1 y ey2 Ž lqS. R .

s a.

2Ž l q S . R

lsyS

Proof. M a , l is a compact operator by Lemma 2, hence, any nonzero spectral point of M a , l must be an eigenvalue w4x. Now, suppose z g s ŽM a , l . _  04 , then there exists an f g Y such that f / 0 and zf Ž y . s ŽM a , l f .Ž y .. That is, zf Ž y . s

aR

'R

2

y y2

R y2 Ž l qS. R 2 yy 9 2

'

H0 e

R2

Multiply both sides of Eq. Ž6. by ey2 Ž lqS. 'R it from 0 to R about y. We obtain z

R y2 Ž l qS. R 2 yy 2

H0 e

'

y R

2

f Ž y . dy s

y9

ž

yy 2 Ž

ay

R

H0 R'R

=

2

žH

2

yy

2

f Ž y9 . dy9.

yrR 2 ., then integrate

ey2Ž lqS.'R

R y2 Ž l qS. R 2 yy 9 2

0

e

Ž 6.

'

y9 R2

2

yy 2

dy

/ /

f Ž y9 . dy9 .

926

ZHANG AND LIANG

Hence, zsa

R

H0 R'R

y 2

y y2

ey2 Ž lqS.'R

2

yy 2

dy s

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

.

On the other hand, if z s a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R, it is easy to verify that f Ž y . s 1r R 2 y y 2 satisfies zf Ž x . s ŽM a , l f .Ž y ., i.e., z is an eigenvalue of M a , l .

'

THEOREM 1. The resol¨ ent set and spectrum of the streaming operator Ba satisfy Ži. r ŽBa . s  l g C
x

Hy'R yy 2

2

2

yy 2 .

f y R2 y y 2 , y

ž '

/

eyŽ lqS.Ž xyx 9. g Ž x9, y . dx9.

Since f Ž x, y . satisfies the boundary condition Ž5., it is easy to verify that f Žy R 2 y y 2 , y . satisfies

'

f Ž y . s Ž Ma , l f . Ž y . q Ž Ha , l g . Ž y . .

Ž 7.

For Re l ) yS, r ŽM a , l . - a F 1 by Lemma 1, hence 1 g r ŽM a , l .. For Re l F yS satisfying a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R / 1, 1 g r ŽM a , l . by Lemma 3. In this case, f Žy R 2 y y 2 , y . is the unique solution of Eq. Ž7.. This shows that Ž l I y Ba .y1 exists and is defined on X. Furthermore, the unique solution f Ž x, y . of equation Ž l I y Ba . f s g can be represented by w4x

'

f Ž x, y . s eyŽ lqS.Ž xq 'R q

x

Hy'R yy 2

2

2

yy 2 .

y1 Ž I y M a , l . H a , l g Ž x, y .

eyŽ lqS.Ž xyx 9. g Ž x9, y . dx9.

ONE-VELOCITY TRANSPORT OPERATOR

927

By some computation, we obtain 5 f 5 s Ž l I y Ba . y1 g F 2 R 3 Ž I y M a , l . y1 5H a , l 5 q

½

1 Re l q S

5

5 g 5.

This shows that Ž l I y Ba .y1 is a bounded linear operator. Hence,

½

r Ž Ba . > l g C
5

/1 .

2Ž l q S . R On the other hand, if Re l F yS and

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

s1

then it is easy to show that the function f Ž x, y . s eyŽ lqS.Ž xq 'R

2

aR

yy 2 .

'R

2

y y2

belongs to DŽBa . such that Ž l I y Ba . f s 0. That is, l g Ps ŽBa ., hence, we have

½

r Ž Ba . s l g C
5

/1 .

2Ž l q S . R Žii. From the proof of Ži., we know that

s Ž Ba .

½

s Ps Ž Ba . s l g C
a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

5

s1 .

Now, let l g Ps ŽBa ., then there exists an f g DŽBa . _  04 such that Ž l I y Ba . f s 0. Hence f Ž x, y . s eyŽ lqS.Ž xq 'R

2

yy 2 .

f y R2 y y 2 , y .

ž '

/

928

ZHANG AND LIANG

Since f Ž x, y . verifies the boundary condition Ž5., we obtain y R

2

f y R2 y y 2 , y s

ž '

ay

R

/ H R'R s

0

2

y9

yy

la y R R2 y y 2

'

2

R2

f

ž 'R

2

y y9 2 , y9 dy9

/

,

where l s H0R Ž y9rR 2 . f Ž R 2 y y9 2 , y9. dy9 / 0. This shows that the geometric multiplicity of l is equal to one, or expressed by

'

dim N Ž l I y Ba . s 1,

½

N Ž l I y Ba . s span eyŽ lqS.Ž xq 'R

2

aR

yy 2 .

'R

2

y y2

5

.

Ž 8.

Now, suppose f g N ŽŽ l I y Ba . 2 ., then Ž l I y Ba . f g N Ž l I y Ba .. From Ž8. we know that there exists a complex number l such that

Ž l I y Ba . f s eyŽ lqS.Ž xq 'R yy 2

2

la R

.

'R

2

y y2

.

So, we obtain f Ž x, y . s eyŽ lqS.Ž xq 'R q

x

Hy'R yy

2

yy 2 .

eyŽ lqS.Ž xyx 9.

ž 'R

yy

= f y R2 y y 2 , y q x q

'R

2

'R 2

la R 2

2

2

2

'R

/ eyŽ lqS.Ž x 9q 'R

2

yy 2 .

dx9

yy 2 .

ž '

f

ž '

2

2

s eyŽ lqS.Ž xq 'R

Let x s

f y R2 y y 2 , y

/ ž

y y2

la R

/ 'R

2

y y2

.

y y 2 , then

y y 2 , y s ey2Ž lqS.'R

/

2

yy 2

f y R 2 y y 2 , y q 2 lR a .

ž '

/

ONE-VELOCITY TRANSPORT OPERATOR

Substitute f Ž R 2 y y 2 , y . in Ž5. with ey2Ž lqS. 'R q 2 l a R x and we have

'

y R

2

R

ay

2

y9

yy 2 w

929 f Žy

'R

' / H R'R y y R e = f ž y'R y y9 , y9 / q 2 lR a

f y R2 y y 2 , y s

ž '

2

0

y2Ž l qS. R 2 yy 9 2

2

Multiply both sides of the above equation by eyŽ lqS. 'R integrate it from 0 to R about y. We obtain 2 la

R

H0

R

y9ey2 Ž lqS.'R

2

yy 9 2

y y 2 , y.

2

2

2

2

dy9. 2

yy 2

, then

dy9 s 0.

If l verifies

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

s1

then R

H0

y9ey2 Ž lqS.'R

2

yy 9 2

dy9 / 0.

So l s 0, that is, Ž l I y Ba . f s 0. Hence, we have proved that 2

Ž 9.

2

Ž 10 .

N Ž l I y Ba . s N Ž Ž l I y Ba . . . Finally, we prove that for l satisfying

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

s1

we have R Ž l I y Ba . s R Ž Ž l I y Ba . . .

930

ZHANG AND LIANG

Suppose g g RŽ l I y Ba ., then there exists a f g DŽBa . such that Ž l I y Ba . f s g. By complicated computation we know that the function f Ž x, y . s eyŽ lqS.Ž xq 'R x

q

Hy'R yy 2

q xq

ž

2

'R

2

Ra

yy 2 .

'R

y y2

2

eyŽ lqS.Ž xyx 9.f Ž x9, y . dx9

2

y y 2 eyŽ lqS.Ž xq 'R

/

2

lR a

yy 2 .

'R

2

y y2

belongs to DŽBa2 . and satisfies Ž l I y Ba . 2 f s g, where ls

yŽ l q S .

'R yy dyH H 1 y a q 2Ž l q S . R 0 y 'R yy 2

R

2

2

2

eyŽ lqS.Ž 'R

2

yy 2 yx .

y R2

f Ž x, y . dx

Ž if l / yS . and lsy

1

'R yy dyH H R 0 y 'R yy 2

R

y

2

2

2

R2

f Ž x, y . dx

Ž if l s yS, in this case a must be 1 . . So g g RŽŽ l I y Ba . 2 ., and this proves that RŽ l I y Ba . ; RŽŽ l I y Ba . 2 .. But RŽŽ l I y Ba . 2 . ; RŽ l I y Ba . is obvious, hence Ž10. is verified. From Ž8. Ž9., Ž10., and w4x, we know that each l g Ps ŽBa . is an isolated simple eigenvalue. LEMMA 4.

If a g Ž0, 1x, then

Ži. The equation e z s 1 q Ž1ra . z has infinitely many conjugate complex roots. Further, there exists a unique root of the equation in each of the following two kinds of bands: Re z ) ln

ž

4 kp q p

/

, 2a p 2 kp - Im z - 2 kp q Ž k s 1, 2, 3, . . . . 2

ONE-VELOCITY TRANSPORT OPERATOR

931

and Re z ) ln

4 kp y p 2a

,

2 kp y

p 2

- Im z - 2 kp Ž k s y1, y2, y3, . . . . .

Žii. If a s 1, then the equation e z s 1 q Ž1ra . z has a unique real root z s 0; and if 0 - a - 1, then e z s 1 q Ž1ra . z has two real roots z1 s 0 and z 2 s l 0 , where l 0 is a positi¨ e constant larger than yln a . Proof. Ži. We only need to show that there exists a unique root of the equation e z s 1 q Ž1ra . z in each of the bands 2 kp - Im z - 2 kp q pr2 and Re z ) lnŽŽ4 kp q p .r2 a . Ž k s 1, 2, 3, . . . .. Let z s b q it , then z is a root of e z s 1 q Ž1ra . z if and only if Ž b , t . is a solution of the equation

¡e

~

¢e

b

b

1

cos t s 1 q sin t s

1

a

a

b

Ž 11 .

t.

By methods of calculus, it is not difficult to show that there exists a unique solution of Ž11. satisfying b ) lnŽŽ4 kp q p .r2 a . and 2 kp - t 2 kp q pr2 for each k s 1, 2, 3, . . . . Žii. Since the function f Ž x . s e x y 1 y x is strictly increasing on w0, q`. and strictly decreasing on Žy`, 0x, further, f Ž0. s 0. Hence, f Ž x . s 0 if and only if x s 0. That is, e z s 1 q Ž1ra , z has a unique root z s 0 for a s 1. Similarly, we can manage the case 0 - a - 1. From Theorem 1 and Lemma 4, we get THEOREM 2. Ži. The streaming operator Ba has infinitely many conjugate complex eigen¨ alues. Furthermore, there exists a unique complex eigen¨ alue in each of the following two kinds of bands Re l - yS y

1 2R

ln

Ry1 kp - Im l - Ry1 kp q

ž

ž

4 kp q p

p 4

2a

/Ž

/

,

k s 1, 2, 3, . . . .

932

ZHANG AND LIANG

and Re l - yS y Ry1 kp y

ž

Žii. Žii. l 0r2 R .. Živ.

p 4

/

1

ln

2R

4 kp y p 2a

,

- Im l - Ry1 kp Ž k s y1, y2, y3, . . . . .

If a s 1, then Ba has only one real eigen¨ alue l s yS. If 0 - a - 1, then Ba has only one real eigen¨ alue l s yŽ S q

se s s ŽBa . s F. 3. SPECTRAL PROPERTIES OF THE TRANSPORT OPERATOR A a

Define two linear operators as Pl : X ª X ;

Ž Pl f . Ž x, y . s H

x

'

y R 2yy 2

Rl: Y ª X ;

eyŽ lqS.Ž xyx 9. f Ž x9, y . dx9

Ž R l f . Ž x, y . s eyŽ lqS.Ž xq 'R yy . f Ž y . . 2

2

It is not difficult to show that Pl and R l are bounded linear operators for any complex l. Moreover, if Re l ) yS, then 5Pl 5 F

1 Re l q S

5R l 5 F

,

R2 Re l q S

.

From the above discussion, we get

Ž l I y Ba .

y1

s Rl Ž I y Ma , l .

y1

H a , l q Pl ,

l g r Ž Ba . . Ž 12 .

LEMMA 5. For any complex number l, both KR l H a , l and ŽKPl . 2 are weakly compact operators on X. Proof. By some computation, we know that KR l H a , l is the integral operator R

'R yy9 y 'R yy9

Ž KR l H a , l f . Ž x, y . s H dy9H 0

2

2

2

2

K a , l Ž x, y ; x9, y9 . f Ž x9, y9 . dx9,

ONE-VELOCITY TRANSPORT OPERATOR

933

where K a , l Ž x, y ; x9, y9 . cS

s

r

yŽ l qS.Ž

He 2 r yr

'R yy qz q 'R yy 9 2

2

2

2

2

a y9

qzyx 9.

R'R y r 2 q z 2 2

rs

'x

2

dz,

q y2 .

Defining G: D ª X; GŽ x9, y9. s K a , lŽ x, y; x9, y9., then for Re l G yS we have

max

Ž x 9, y9.gD

ž'

G Ž x9, y9 . s cS

2 x qy

Ž x 9, y9.gD

Hy''x xqyqy 2

2

max

2

2

2

2

K a , l Ž x, y ; x9, y9 . F

eyŽR e lqS.Ž 'R

2

yr 2 qz 2 q

=

F

R

H0

Hy''RRyyyy 2

y dy

R

2

2

2

ž'

2

2

y x2 y y2

2

2

2

2

/

dz dx

aR

2

2

Hy''RRyyyy

y dy

qzyx 9.

Hy''x xqyqy R'R

y dy 2

2

R'R 2 y r 2 q z 2

2 x2 q y2 2

2

a y9

cS

HyR dxH0'R yx 'R

F a cS

'R yy9

R

H0

2

2

y r2

/

dz dx

.

For Re l - yS, we get by a similar method that

max

Ž x 9, y9.gD

G Ž x9, y9 . F a cS ey4 ŽRe lqS. R

R

HyR dxH0'R yx 'R 2

y dy

2

2

y x2 y y2

.

So, we have for any l max G Ž x9, y9 . F C Ž l . , where C Ž l. is a positive constant.

Ž 13 .

934

ZHANG AND LIANG

On the other hand, let E ; D be measurable, then

HEy G Ž x9, y9.

Since, 1r

dx dy cS

ž

F

HE 2'x

s

HEcS e

'R

2

2

Hy''x xqyqy 2

q y2

2

2

2

e 4
y

4
'R

2

y x2 y y2

dz

'R 2 y r 2

/

y dx dy

dx dy.

y x 2 y y 2 g X, we have

HE y G Ž x9, y9.

lim max mEª0

dx dy s 0.

Ž 14 .

From Ž13., Ž14., and w1x, we know that KR l H a , l is weakly compact for each complex number l. Now let, r

Ž Vf . Ž r . s H f Ž z, 'r 2 y z 2 . dz

V: X ª L1 Ž S, r dr . ;

yr

Q: L Ž S, r dr . ª X ; 1

Ž Q f . Ž x, y . s

1 2 x q y2

'

2

f

ž 'x

2

q y 2 , f g L1 Ž S, r dr . .

/

Then V and Q are bounded linear operators such that 5V 5 s 5Q 5 s 1. Obviously, K s cSQV, and cSVPl Q: L1 Ž S, r dr . ª L1 Ž S, r dr . is a bounded linear operator for each l and can be represented by

Ž cSVPlQ f . Ž r . s H Ll Ž r , r 9 . f Ž r 9 . dr9, S

f g L1 Ž S, r dr . ,

where Ll Ž r , r 9 . s

cS 2

rqr 9 yŽ l qS. s

H< ryr 9< e

ds s

.

It is similar to the above procedure that we can show the weak compactness of cSVPl Q. Hence, ŽKPl . 2 is weakly compact for each l.

ONE-VELOCITY TRANSPORT OPERATOR

935

LEMMA 6. Ži.

If a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R / 1, then

Ž I y Ma , l . Žii.

y1

sIq

2Ž l q S . 2 Ž l q S . R y a Ž 1 y ey2Ž lqS. R .

Ma , l .

For any l g C, Ma , l Ha , l s

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

Ha , l .

Proof. Ži. In fact M a2 , l s

a Ž 1 y ey2 Ž lqS. R . 2Ž l q S . R

Ma , l ,

hence Ži. is valid. Žii. We can obtain this result by some direct computation. THEOREM 3. If a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R / 1, then the operator wKŽ l I y Ba .y1 x 4 is compact. Proof. If a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R / 1, then we obtain by Ž12. and Lemma 6 that K Ž l I y Ba .

y1 2

2

s l 2 Ž KR l H a , l . q l Ž KR l H a , l . Ž KPl . 2

q l Ž KPl . Ž KR l H a , l . q Ž KR l . , where ls

2Ž l q S . R 2 Ž l q S . R y a Ž 1 y cy2 Ž lqS. R .

.

From Lemma 5, we know that wKŽ l I y Ba .y1 x 2 is weakly compact, hence wKŽ l I y Ba .y1 x4 is compact w1x. THEOREM 4. se s s ŽA a . s F. That is, each of the spectral points of A a is an isolated eigen¨ alue with finite algebraic multiplicity.

936

ZHANG AND LIANG

Proof. By Lemma 1, Lemma 6, and the estimations of 5R l 5 and 5Pl 5, we get for real l K Ž l I y Ba .

y1

2 Ra

F

y2Ž l qS. R

2Ž l q S . R y a Ž 1 y e

.

1

q

lqS

5K 5 .

Hence, we have 5KŽ l I y Ba .y1 5 - 1 for l large enough. From Theorem 3 and w7x, we know that se s s ŽA a . s se s s ŽBa . s F. Remark. In fact, we can show that the transport operator A a generates an irreducibly positive C0-semigroup in the Banach lattice X. As a consequence, we can prove the existence of the dominant eigenvalue of the transport operator A a . Since the proof is very tedious, we leave it out here. In the following, the existence of a complex eigenvalue l of A a satisfying Re l ) yS is observed. LEMMA 7.

If a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R / 1, then R

Ž cSVR l Ž I y M a , l . y1 H a , l Q f . Ž r . s H0

D a , l Ž r , r 9 . f Ž r 9 . dr9, f g L1 Ž S, r dr . ,

where D a , l Ž r , r 9. s

a cS Ž l q S . 2 Ž l q S . y a Ž 1 y ey2Ž lqS. R .

=

ž

=

r

žH

dz

'

yŽ l qS. R 2 yr 2 qz 2

Hyre r9

yr 9

'R 2 y r 2 q z 2 eyŽ lqS.Ž z 9q 'R

2

yr 9 2 qz 9 2 .

/

dz9 ,

/ r , r 9gS.

Denote the streaming operator and the transport operator by B 0 and A 0 for a s 0. From w3, 6x, we know that s ŽB 0 . s F and Ž l I y B 0 .y1 s Pl . Moreo¨ er, the following lemma is ¨ alid. LEMMA 8. Ži. Žii. is real; Žiii.

l g Ps ŽA 0 . if and only if 1 g Ps Ž cSVPl Q.; Each spectral point of A 0 distributed in the half plane Re l ) yS s ŽA 0 . s Ps ŽA 0 . and se s s ŽA 0 . s F.

ONE-VELOCITY TRANSPORT OPERATOR

937

By a similar method used in w3x, we can pro¨ e the following lemma. LEMMA 9. If a Ž1 y ey2 Ž lqS. R .r2Ž l q S . R / 1, then l g Ps ŽA a . if and only if 1 g Ps Ž cSVŽ l I y Ba .y1 Q.. From Lemma 7, we know that cSVR lŽ I y M a , l .y1 H a , l Q is a degenerated one-dimensional operator and can be expressed by cSVR l Ž I y M a , l .

y1

H a , l Q f s ² f , e :h ,

f g L1 Ž S, r dr . , Ž 15 .

where eŽ r . s

a cS Ž l q S . 2 Ž l q S . R y a Ž 1 y ey2 Ž lqS. R . r

=

1

'

yŽ l qS.Ž z 9q R 2 yr 2 qz 9 2 .

Hyr r e

hŽ r. s

r

dz

'

yŽ l qS.Ž zq R 2 yr 2 qz 2 .

Hyre

dz9,

'R 2 y r 2 q z 2

and eŽ r . g L`Ž S, dr ., h Ž r . g L1 Ž S, r dr ., and ² f , e: s THEOREM 5.

R

H0

f Ž r . e Ž r . dr.

Suppose Re l ) yS and Im l / 0, then l is an eigen-

¨ alue of the transport operator A a if and only if

a cS Ž l q S .

R

y2 Ž l qS. R

H0 m Ž r . f Ž r . dr s a Ž 1 y e l

l

. y 2 Ž l q S . R,

where ml Ž r . s

r

'

yŽ l qS.Ž xq R 2 yr 2 qx 2 .

Hyre

dx,

and flŽ r . is the unique solution of the integral equation fl Ž r . s

cS 2

R

ž

rqr 9 yŽ l qS. s

H0 H< ryr 9< e

ds s

/

fl Ž r 9 . dr9 y

Rqr yŽ l qS. s

HRyr e

ds s

Proof. We get by Ž12. cSV Ž l I y Ba .

y1

Q s cSVPl Q q cSVR l Ž I y M a , l .

y1

H a , l Q.

.

938

ZHANG AND LIANG

Hence, cSVŽ l I y Ba .y1 Q is the perturbation of cSVPl Q by the onedimensional operator cSVR lŽ I y M a , l .y1 H a , l Q. From Ž15. and w2x, we obtain the W y A determinant associated with cSVPl Q and cSVR lŽ I y M a , l .y1 H a , l Q as

v Ž j . s 1 q ² Ž cSVPl Q y j I .

y1

h , e: ,

j g r Ž cSVPl Q . .

Ž 16 .

Let w Ž j . be a meromorphic function on complex plane and define an integer-valued function n Ž j ; w . as

¡k ¢0

n Ž j ; w . s~yk

if j is a zero of w of order k, if j is a pole of w of order k, if j is neither a zero nor a pole of w Ž j . .

Suppose T is a linear operator from X to X and define an integer-valued function n Ž j ; T . as

¡0 ¢q`

n Ž j ; T . s~dim P

if j g r Ž T . , if j g s Ž T . _ se s s Ž T . , if j g se s s Ž T . ,

where P is the eigen-projection corresponding to the eigenvalue j . Then from the above discussion and the W y A formula w2x, we obtain

n Ž 1; cSV Ž l I y Ba .

y1

Q . s n Ž 1; cSVPl Q . q n Ž 1, v . .

From Lemma 8, we know that 1 g r Ž cSVPl Q. for Re l ) yS and Im l / 0, i.e., n Ž1; cSVPl Q. s 0. Hence,

n Ž 1; cSV Ž l I y Ba .

y1

Q . s n Ž 1, v . .

From this formula and Lemma 9, we know that 1 g Ps ŽA a . if and only if Ž v 1. s 0. From Ž16., this is equivalent to

² Ž cSVPlQ y I . y1 h , e: s y1. Denote Ž cSVPl Q y I .y1h Ž r . by flŽ r ., then the above formula is equivalent to

a cS Ž l q S .

R

y2 Ž l qS. R

H0 m Ž r . f Ž r . dr s a Ž 1 y e l

l

. y 2 Ž l q S . R,

ONE-VELOCITY TRANSPORT OPERATOR

939

where ml Ž r . s

r

'

yŽ l qS.Ž xq R 2 yr 2 qx 2 .

Hyre

dx

and flŽ r . satisfies flŽ r . s Ž cSVPl Q. flŽ r . y h Ž r .. Setting s s z q 'R 2 y r 2 q z 2 in the expression of h Ž r ., and in consideration of the proof of Lemma 5, we know that flŽ r . is the unique solution of the integral equation fl Ž r . s

cS 2

R

ž

rqr 9 yŽ l qS. s

H0 H< ryr 9< e

ds s

/

fl Ž r 9 . dr9 y

Rqr yŽ l qS. s

HRyr e

ds s

.

This completes the proof.

REFERENCES 1. J. Diestel and J. J. Uhl, Jr., ‘‘Vector Measure,’’ Amer. Math. Soc., Providence, RI, 1977. 2. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ 2nd ed., Springer-Verlag, New YorkrBerlin, 1984. 3. J. Lehner, An unsymmetric operator arising in the theory of neutron diffusion, Comm. Pure Appl. Math. 9 Ž1956., 487]497. 4. A. E. Taylor and D. C. Lay, ‘‘Introduction to Functional Analysis,’’ 2nd ed., Wiley, New York, 1980. 5. S. Ukai, Eigenvalues of the neutron transport operator for a homogeneous finite moderator, J. Math. Anal. Appl. 18 Ž1967., 297]314. 6. R. Van Norton, On the real spectrum of a mono-energetic neutron transport operator, Comm. Pure Appl. Math. 15 Ž1962., 149]158. 7. J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Mh. Math. 90 Ž1980., 153]161. 8. Zhang Xianwen and Liang Benzhong. On the transport equation with integral boundary conditions and it’s conservative law, Appl. Math.-JCU 11B Ž1996., 33]42. 9. P. F. Zweifel and J. Ohlmann, Spectrum of a Vlasov-Fokker-Plank operator, Progr. Nucl. Energy 8 Ž1981., 145]150.