Some spectral properties of the neutron transport operator in bounded geometries

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Some spectral propertiesof the neutron transportoperator in boundedgeometriesMustapha Mokhtar-Kharroubi aa Universitg de PARIS V I, Laboratoire d'AnalyseNumerique4, Place Jussieu, 75230, PARISCedex-05, France

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To cite this article: Mustapha Mokhtar-Kharroubi (1987): Some spectral propertiesof the neutron transport operator in bounded geometries, Transport Theory andStatistical Physics, 16:7, 935-958

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TRANSPORT THEORY AND STATISTICAL PHYSICS, 16(7), 935-958 (1987)

SOME SPECTRAL PROPERTIES OF THE NEUTRON

TRANSPORT OPERATOR I N BOUNDED GEOMETRIES

Mustapha MOKHTAR-KHARROUBI

Universitg de PARIS V I Laboratoire d'Analyse Numerique

4 , Place Jussieu 75230 - PARIS Cedex 05 -France

ABSTRACT

Existence and nonexistence eigenvalue r e s u l t s a r e given. We a lso

est imate the leading eigenvalue of the t ranspor t operator i n terms of

some geometrical parameters of the configurat ion space. Some r e s u l t s

r e l a t e d t o the c r i t i c a l i t y Droblem a r e given.

I - INTRODUCTION

This paper dea ls with the point spectrum of the neutron t ransport

ope r a t o r

I, atl A $ = - V.= - 0 (x,v) $(x,v) + K(x,v,v') $(x,v')dv' f 'I% + K$

where K denotes the i n t e g r a l p a r t of A ( t h e c o l l i s i o n operator)

(x,v) E D X V , where the configurat ion space

subset of R3, while the ve loc i ty space V is an a r b i t r a r y open subset

of Ei3. The unbounded operator A is studied i n the Banach space

Lp(D xV) sure) . Its domain i s

D i s an open and bounded

f o r some f i n i t e p ( 1 < p < +m) (with the usual Lebesgue mea-

$r- = 0 3 D(A) = i: $(.,.) E L ~ ( D ~ v ) /v.K a$ E L ~ ( D ~ v ) ,

where r- = (x,v) E a D X V / v is ingoing a t x E a D 1

935

Copyright 0 1987 by Marcel Dekker, Inc.

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936 MOKHTAR-KHARROUBI

For a precise (mathematical) definition of D(A), we may see

[ 1 1 chap. 12, [ 2 ] or [ 3 3 . The usual (mathematical) assumptions are :

o(-,-) E La (DxV) and KE&(LP(DxV))

we also assume the (physical) properties

U(X,V) 2 0 and K(x,v,v') 2 0 a.e (2)

Then the "collisionless" operator T = A - K generates the f o l l o -

wing explicit positive (c,) semigroup [ 4 1

- j:o(x-? v,v)d? S(t) p (x,v) = e p (x -tv,v) if t G s(x,v) I o otherwise

where s(x,v) = inf { s > 0 / x-sv 6! D 3 and p E LP(DxV) . The type of the semigroup S( t) ; t > 0 1 was characterized

recently by Voigt ( 1 4 1 lemma 1 . 1 )

I-- if o @ V

In particular if u(x,v) is homogeneous ( u (x,v) = u (v) ) then

- a if o @ V

r l = - lim inf U(V) if o E 7 v + o V E V

( 3 )

The general properties of the spect.rum, o ( A ) , of A = T + K are,

now, well known ( [ 1 ]chap. 12, 141 , ( 5 1 ) . Let us recall some of

them which are of interest for the present paper.

Under the general assumption :

-1 some power of (A - T) K is

the following hold :

a(A) n {Re > q} consists of

finite algebraic multiplicites (

{ A /Re A = 1 belongs to u (A) ( [

compact (Re h > n ) ( 4 )

(at most) isoleted eigenvalues with

11 th. 12.13, page 277). The line

1 th. 12.15, page 277). If

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SPECTRAL PROPERTIES OF THE NEUTRON TRANSPORT OPERATOR 937

U (A) fl {Re A > n } i s no t empty, t h e r e e x i s t s a ( r e a l ) leading eigen-

value Xo [ i .e Re X < A f o r a l l X E f f (A)] ( f 1 1 t h . 12.16, page 281)

Although t h e general p r o p e r t i e s of u (A) Il { Re A > rl 1 = uas(A)

(asymptotic po in t spectrum) a r e , a t p re sen t , w e l l known, t h e r e a r e

important quest ions which a r e n o t , t o our knowledge, answered up t o

now :

- Can uas(A) be nonempty whatever t h e s i z e of D ?

- Can it be nonempty ( a t l e a s t ) f o r l a r g e D ?

- Can it be empty whatever t h e s i z e of

- I n t h e case where uas(A) # e , can we l o c a l i z e the l ead ing eigenvalue

D ?

i n terms of some geometr ical parameters of D ?

- The c r i t i c a l i t y problem : i s t h e r e a s i z e of D f o r which t h e l ead ing

eigenvalue is zero ?

- Can t h e geometry of V a f f e c t Uas(A) ?

The p resen t paper i s an at tempt t o a t t a k such ques t ions i n a s u f f i -

c i e n t l y gene ra l s e t t i n g f o r t h e a p p l i c a t i o n s . Apart from t h e disappea-

rance of uas(A)

t h e f i r s t time by Albe r ton i and Montagnini [ 6 1, a l l our r e s u l t s a r e

new. Even t h e disappearance of oas(A) f o r small D is n o t , a s we

s h a l l s e e t h e r e a f t e r , a general phenomenon.

f o r small bod ie s , which has been pointed ou t f o r

Let A* = i n f u ( - , - ) . It fol lows, from t h e obvious i n e q u a l i t y

II s ( t ) l l G e - ’* t h a t

rl < - A* ( t h e s t r i c t i n e q u a l i t y may hold)

I n o rde r t o avoid some ( u n i n t e r e s t i n g ) t e c h n i c a l i t i e s , we s h a l l

s tudy the spectrum of A mainly i n t h e ha l f - r ay {A/A > - A* }

( in s t ead of { A / A > q 1 ) . We s h a l l denote by

of

P(A) t h e eigenvalues

A

Before s t a t i n g our main r e s u l t s , we need a d e f i n i t i o n

DEFINITION 1 - The r ad ius of a bounded s e t D , i s the r ad ius of t h e

g r e a t e s t b a l l included i n D .

belonging t o { A / X > - A* } .

Our work i s organized a s fol lows :

Sect ion 2 is devoted t o some no ta t ions and assumptions we s h a l l need

i n s e c t i o n 3 . The l a t t e r d e a l s w i th r e s u l t s t h a t can be obtained

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directly, that is to say without comparing our transport operator

to another one : we give a sufficient condition under which P(A)= 0 for small D(th. 1) ; similar results are scattered in the literature

for particular cases [ 6 I examined in full generality as here (although its analysis is simple).

We derive a connection between the leading eigenvalue of the trans-

[ 7 I 181 , but this problem has never been

port operator and that of a bounded operator i n L p ( V ) (th. 2) ;we then obtain, as direct consequences of this theorem, a sufficient condition vnder which P(A) = 0 for all D (cor. 1) and a sufficient condition of subcriticality of A whatever D (cor. 2 ) . These consequences are also illustrated by examples and by pertinent remarks.

We also give an upper bound of P(A) in terms of the diameter

of D(th. 3 ) . The geometry of the velocity space V (or gquivalently

the support of K(x,v,v')) may affect ap(A) ; endeed if V is contai-

ned in a half space (for instance if V is a halfball centered at

zero), then P(A) is empty regardless of D, a(-,.) and K(-,*,.)(th.4).

Although unphysical this result is interesting from a mathematical

point of view.

The effective existence of eigenvalues in { X/X > - A*) is the

main purpose of the last section (section 4). The structure of A in its full generality is too general t o allow a direct treatment. Fortu-

nately one can exploit the positivityof thescattering kernel by using

comparison arguments (i.e. we compare our transport operator to another one whose point spectrum is not empty). This idea is not really new

(see Huber's results cited in [ l ] page 289) but has never been fully

exploited. Endeed, as a prerequisite, we have to select the "good"

transport operators whose point spectra are analyzable in detail, and

next compare our transport operator to one of them. Thus, in L (DxV)

the real point spectrum of a transport operator is completely analyzed

in [ 9 l under the assumption that the collision operator is positive (for the scalar product in L ). This enables us to give in L2 setting :

2

2

sufficient conditions under which P(A) f 0 for large D

(large radius) ; sufficient ones under which P(A) # 0 for all D ; an estimate of the limit of the leading eigen- (5)

value as the radius of D goes to infinity ; a criterion

of existence of eigenvalues for a given D , and, finally, an explicit lower bound of theleading eigenvalue in terms of the radius of D .

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A l l t he se r e s u l t s , a r e s t a t e d i n t h 5.

In Lp s e t t i n g (1 Q p < +m ) t h e r e i s a c l a s s of :ransport opera-

t o r s whosepoint s p e c t r a can be completely analyzed. This c l a s s i s t h e

one of s epa rab le s c a t t e r i n g k e r n e l s K(v,v') = f ( v ) g(v ' ) (f E Lp (V),

1 1 g E Lp*(V) * - + -

f a c t we s h a l l prove only those r e s u l t s which a r e of i n t e r e s t f o r our

p re sen t paper

s epa rab le model (see those s t a t e d i n t h e no te [ l o ] ) 1 . Thus by compa-

r i s o n with sepa rab le s c a t t e r i n g k e r n e l s , one can t a c k l e t h e quest ions

(5) i n

= 1 ) . This a n a l y s i s i s c a r r i e d out i n s e c t i o n 4. I n ' P P*

[ a l though many o t h e r r e s u l t s can be obtained f o r t h e

Lp spaces ( 1 Q p <+ m ) . The r e s u l t s are s t a t e d i n th . 6 . Before c l o s i n g t h i s i n t roduc t ion , l e t us comment b r i e f l y on t h e

i n t e r e s t of uas(A). It i s twofold :

- Its knowledge i s necessary (and s u f f i c i e n t ) f o r va r ious s t a t i o n a r y

t r a n s p o r t theory problems

- Its knowledge i s necessary (but not s u f f i c i e n t ) f o r t h e s tudy of t h e

time asymptotic behaviour of t h e semigroup etA which so lves t h e

Cauchy problem :

governing t h e t i m e evo lu t ion of t h e neutron d e n s i t y i n a nuc lea r reac-

t o r

only p a r t l y determined by t h e one of A ([ 111 p.44). Recently Voigt[ 41 d e r i -

ved a l a r g e class of t r a n s p o r t o p e r a t o r s f o r which

t h e time asymptotic behaviour of

t h e author [ I d . On t h e o t h e r hand it i s p o s s i b l e t o s tudy t h e t i m e

asymptotic behaviour of et *i0 ( f o r smooth i0 ) d i r e c t l y by s tudying

only t h e gene ra to r A (without any knowledge on t h e spectrum of

( [ I ] chap. 1 1 ) . Endeed, t h e spectrum of etA is, gene ra l ly ,

Uas(A) determines

eU. This c l a s s has been enlarged by

e tA)

[ 1 2 l .

t Op(A> ( e t A) = .tOp(A) u io] P F i n a l l y l e t us n o t i c e , s i n c e e

( [ 1 1 1 th . 2.4, p. 46) , t h a t our r e s u l t s on t h e p o i n t spectrum of A

a r e t r a n s f e r r a b l e t o t h e p o i n t spectrum of

opinion, t h e l a t t e r cannot be s tud ied d i r e c t l y : t h e only way t o eet precise results on u p ( e t A ) ( l i k e , for i n s t ance , those obtained he re

et A, and moreover, i n our

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9 40 MOKHTAR-KHARROUBI

o r i n 19 1) i s t o s tudy Up(A) ! The reason t o t h i s f a c t i s simple :

t he generator A i s known while t h e semigroup etA i s not !

2 - NOTATIONS

This s e c t i o n i s devoted t o some no ta t ions and assumptions we s h a l l

use i n s e c t i o n 3 . Let Ko(v,v') = sup K(x,v,v') and UO(v) = inf U(X,V)

xE D xE D

we denote by KO t h e i n t e g r a l ope ra to r , i n Lp(V), a s soc ie t ed w i t h

t he kernel G ( v , v ' ) , and assume t h a t :

(7) KO E l (Lp(V)>

some power of : K,(v,v') q ( v ' ) dv'

- - %p i s compact (h>-h*)

Let us denote by B t h e bounded ope ra to r i n LP@)

p j - o0(v) V(v) +

and by i? t h e c losed ope ra to r i n LP(DxV)

Ko(v,v') V(v') dv' = B V (V E Lp(V)) JV

- 7 K (x, v, v 1 $(x ,v ' ) dv' K $

K ' ( X ~ V ) E Lp(D xv)} Ivl Fina l ly we denote by P(B) (resp. P(A)) t h e eigenvalues of B

( r e sp . of A) i n t h e half-ray { A / > - A* 1 .

3 - DIRECTS RESULTS

Theorem 1 : Let be bounded i n Lp(D x V). Then P(A) = 0 i f

d II;(II < 1 (where d i s t h e diameter of D).

Proof : A E P(A) if and only i f 1 i s a n eigenvalue of ( h - T)-' K

( h - T l - 1 +(x,v) = T;Tj d s e r.7 e $(x-sw,v)ds

1 - T;T !: U(x-? w ,v)dT k s(x,w) - 0

where w = f$ . L e t ex be the ope ra to r

g, -+ ex g,(x,v) = / V I (x-T)-' K + (x,v) .

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941 SPECTRAL PROPERTIES OF THE NEUTRON TRANSPORT OPERATOR

* It i s easy t o s e e t h a t

x l!eX1! S d i f h > - A . Thus, due t o

( ,i-T)-' K = 8 , we g e t 11 ( k-T)-l K < d 11 11 which proves the

theorem. Q.E.D.

Theorem 2 : If P(A) # 0 , then P ( B ) # 0 and the leading eigenvalue

of A i s l e s s t h a n o r e q u a l t o t h a t of B . Moreover the l a t t e r i s l e s s

than o r equal t o - A* +r(Ko) (r(K ) = s p e c t r a l rad ius of KO).

Proof : L e t x ( > - A * ) be t h e leading eigenvalue of A . There e x i s t s

lp E L P (D x V) such t h a t (1 -T)-' K l p = q i . e .

,-;(x,v) e-xs e - I z o ( x - ~ v , ~ ) d 7

so IP(x,v) I j:(x7v) e

ds I K(x-sv,v,v') lp(x-SV,~ ' ) dv'=lp(x,v) V

-6 +ao(V))S ds j;o(v,v') lp(x S V , ~ ' ) dv' I - I

Extending lp(x,v) by zero outs ide of D , and in tegra t ing over D , we

Ko(v,v') 11, (v') dv' = s I) ( J , E LP(V)). x

Hence r(Sk) > 1 . On the o ther hand, thanks t o (7), it follows from the Krein Rutman

theorem t h a t r(SX) is an eigeiivalue of Sx depending continuously on A[ 13 1 . Now, s ince

r ( s X 1 = 1 ,

l i m r (S ) = 0 , t h e r e e x i s t s ho >% such t h a t 3, ++m

i . e . 0

Ko(v,v') q0(v ' ) dv' 3 q0 E Lp(V) such t h a t Jv Ao+ uo(v) = $,(v) (8)

XoQo . This proves the f i r s t p a r t of the theorem. F ina l ly i .e. B$o =

(8) implies

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A s immediate consequences of theorem 2 we have the two following corollaries :

Corollary 1 : For homogeneous scattering kernel and- collision frequency P(A) = 0 for all D if P(B) = 0 . Examples :

2 In L

1 In L

Endeed the

Remark 1 :

2 K(v,v') dv dv' setting P(A) = 0 for all D if G 1

above inequalities ensure that P(B) = 0 . The converse of corollary 1 is, generally, not true (see -

Remark 6 below). However in LL setting, if V is symmetric with respect to zero and if the collision operator is positive (for the scalar product in L ) then the converse of corollary 1 is true [ 9 ] . 2

Remark 2 : If KO is quasinilpotent (r(K ) = 0) then P(A) = 0 for all D . For instance if K(x,v,v') = 0 for I v I > I v'l , then K is quasinilpotent, so we find again a result of Jorgens ( [ 1 4 ] th 6.3).

Corollary 2 : For homogeneous scattering kernel and collision frequency

the transport operator is always subcritical (i.e. P(A) c

for all D) if B is subcritical. [A E R / A <g

2 K(v,v') dv dv' Examp 1 e s

In L2 setting B is subcritical if I I (and A* > 0). In L1 setting B is subcritical if sup

(and A* > 0). Remark 3 : In LL se of corollary 2 is true [ 9 ] 9

setting with positive collision operator the conver-

d -(- A*+r(Ko) + uo(v))

Let a(d,v) = 1 - e 17 and Kd(v,v') =a(d,v) Ko(v,v')

we denote by r(d) the spectral radius of the integral oper-

ator Kd , in Lp(V), associated with the kernel Kd[v.v1).

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* Theorem 3 : Let d be the diameter of D . Then P ( A ) c { X / h < - A + r ( d ) )

Proof : Let X(X > - A * )

the proof of theorem 2, we have be an eigenvalue of A . As at the beginning of

s(x,v) -(A +uo(v))s I Ip(x,v)l QJ e ds Jt0(v,v') 1 ~p (x-sv.v')I dv'. 0

Using the inequality s(x,v) <-& and integrating over D , we get

we know (by theorem 2) that A < -A* + r(K ), whence

(X+X*) $(v) d IV Kd(v,v') $(v') dv' = Kd ' which yields r(d) = r(Kd) > X + A * i.e. ?, < -A* + r(d). Q.E.D.

Remark 4 : r(d) may be estimated explicitly. For instance, in L setting we have the following result.

1

1 Corollary 3 : Let us consider the transport operator in L (DxV). Let

1 1 q

where q* is the conjugate of q (- + 7 = 1)

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(where H(.) i s extended by zero o u t s i d e of V ) . Taking E such t h a t

* * c3"*= , i.e, = dq l3+q , t h e r e s u l t fol lows. Q.E.D.

Remark 5 : Clea r ly t h e above e s t ima te i s only i n t e r e s t i n g f o r small d

For l a r g e D t h 3 and t h 2 g i v e b e t t e r e s t ima tes .

Theorem 4 : We suppose t h a t V is contained i n a ha l f spacedefined

by a plane con ta in ing zero, (and i s bounded). Then P(A) = fl regar-

d l e s s of D , o (.,.) and K(. , . ) . Proof : Without loss of g e n e r a l i t y we may assume -

3 v c I v = (Vl'V2,V3) E IR / v l 2 0 I .

We assume ( fo r s i m p l i c i t y only) t h a t K ( . , . , . ) i s bounded.

Let A (A > - h a ) be an eigenvalue. Thus 3 IJJ E LP(D x V) such t h a t :

$(x,v) ~ J:(x,v) e-As e-J$(x-Tvsv)dT d s 1 K (x-sv, v ,v ' )+ (x-sv ,v ' ) dv'

V

so - s (x ,v ) -(A+oo(v))s

0 J V ds KO(v,v') \$(x-sv,v ' ) 1 dv' I $(x ,v ) i J e

Let H(v) = sup Ko(v,v') .

We extend IJJ by zero o u t s i d e of D x V and i n t e g r a t e over V

V ' E v

We put 9 (XI = 1 I + (x,v) 1 dv . It may be e a s i l y seen V

t h a t :

dx'

(H(.) i s extended by zero o u t s i d e of V)

so r(Hh) > 1 .

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On the other hand, there exists c > O such that H (x-x') Q x and, moreover, H (x-x') = 0 if (x-XI), = x -XI < O x 1 1 IX-xIj

because of the assumption on V , whence :

CX(X1 > XI1) = L(x,x') . 2 H (x-x')<

x Ix-xll

Denoting by L the integral operator in Lp(D) associated with the kernel L(x,x') one sees that r(L) > 1 , which is impossible since L is a quasinilpotent operator of Volterra type. Q.E.D.

Remark 6 : Clearly the geometry of V has no effect on P(B). For ins-

tance in L setting, if K(v,v') = K(v',v) and 0 (v) = U then - u + 11 K 11 is an eigenvalue of B , so P(B) # 0 regardless of V . This example shows that the converse of corollary 1 is, generally, false.

2

4 - COMPARISON RESULTS

In this section we assume the additional assumptions :

V D is convex (for simplicity only)

is symmetric with respect to zero ( 9 )

K(x,v,v') = K(v,v') u(x,v) = u(v) (for simplicity only) (10)

(11)

Remark 7 : If is equal to the type rl of the semigroup {S(t) ; t 2 0 } (this follows

trivially from ( 3 ) ) . 7 '(v) - A * is bounded in the neighborhood of zero

for the physical examples (see those given in [ 7 ] ) .

0 E 7 , then the first part of (11) implies that - A *

The essence of comparison techniques is the following : let - - K(x,v,~')> 0 be anotherscattering kernel, with K(x,v,v') less than K(x,v,v'), and be the transport operator T + , then, under ( 4 ) ,

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one proves [ 131 t h a t P(A) # 0 i f P(;;> # 0 and t h a t t he leading

eigenvalue of A . Thus i t

s u f f i c e s t o f i n d t h e "good" t r anspor t ope ra to r whose P(x) is

analyzable i n d e t a i l . There i s a c l a s s of t r a n s p o r t ope ra to r s which i s

well adapted t o the

f o r general LP spaces.

x i s less than o r equal t o t h e one of

L 2 s e t t i n g , and an another one which i s S u i t e d

(I) L2 ANALYSIS

I n h i l b e r t space s e t t i n g , our assumptions i s - There e x i s t s a kernel K(v,v') > 0 such t h a t :

2 product i n L )

- 5

The r e a l po in t spectrum of A = T + K i s completely analyzed

i n [ 9 ] . This enables us t o prove t h e following (very d e t a i l l e d )

theorem. Before s t a t i n g it, some p re l imina r i e s a r e necessary :

i n the present s e c t i o n t h e n o t a t i o n D ->IR means : t h e r ad ius of

D goes t o i n f i n i t y .

3

- Let Sx be t h e symmetric ope ra to r :

and IIzi l be i t s norm i n l (L2(V) ) . I p i i m 11<11 > 1 , we s h a l l

denote by A t h e unique 1 such t h a t IISAll = 1 . - A+ -x* -

We denote by H t h e (not necessa r i ly bounded) ope ra to r

l e t s(v)=- s (x ,v ) dx and

y = - I d i s t ( x , aD> dx mes (D) D

( 1 2 ' )

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(where d i s t (., a D ) is t h e d i s t ance func t ion t o t h e boundary a n ) . I n the casewhere H i s a bounded ope ra to r , we introduce t h e i n t e g r a l

ope ra to r H , i n L (V) , wi th t h e ke rne l

Clear ly y11ii11 Q II ii II ( s ince s (v ) > Y-).

- -- - - 2 -

H(v,v') = t/s(v) K(v,v') t /s(v).

IVI

We a l s o put :

& (R) = I,,, <Rl&)I dw (R > 0 is a r b i t r a r y )

1

where q ( . ) i s t h e f o u r i e r transform of a n a r b i t r a r y func t ion

q ( . ) E L2(lR3) such t h a t : IlqII = 1 and the support o f q i s L2 (B3)

included i n t h e u n i t b a l l of B3 . (Note t h a t (I (?, I1 2= 1 by t h e L

Parseval i d e n t i t y , so & ( R ) < I ) . F i n a l l y we s h a l l denote by X(D) t he

leading eigenvalue ( i f it e x i s t s ) of A i n LL(D x V).

We a r e , now, i n p o s i t i o n t o s t a t e t h e

Theorem 5 : Under (12) t h e following hold :

1) I f l i m 11 > 1 , then P(A) # 0 f o r

Moreover l i m h(D) > 3; . ( In p a r t i c u l a r X+ -A*

D +lR3 -

s i z e i f X > 0 and A* > 0).

l a r g e D (= l a r g e radius).

t h e r e e x i s t s a cr i t ical

5

2) I f H is notbounded then P(A) # 0 f o r a l l D

3) Let 0 (.) be constant and H be bounded. For a given D,P(A) # 0 -

- i f 11 ?i 11 > I ( i n p a r t i c u l a r i f y II HI! > I ) .

4) Let k be t h e r a d i u s i f D , u(v) be constant (u(v) =u) and V be bounded (vmax being t h e g r e a t e s t speed). Then :

and if , 2 R Vmax P(A) # 0

[ ~ - E ( R ) III ZI

5) I f V is bounded away from t h e o r i g i n then t h e r e a l pointspec-

trum of A i s no t empty.

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- Proof : t h e above r e s u l t s a r e t r u e for

they remain t r u e f o r A = T + K : f o r 1 ) [ resp. 2 ) , 3 ) . 4 ) , 5)] see [ 9 1 t h 2 [ resp. t h 3 , t h 9 , t h 10, t h 15 1 . .

A = T + ? [ 9 1 . By comparison

Remark 8 : An important question emerges from the previous theorem : How

t o know whether a (given) c o l l i s i o n operator i s comparable t o a pos i t ive

one ? ( the p o s i t i v i t y i s understood i n the sca la r product sense). We do

not t r e a t , here, t h i s (somewhat d i f f i c u l t ) problem. However one can i l l u s -

t r a t e t h i s , with some examples :

- Let V be bounded and K(v,v') > C > 0 on V x V . Then, c l e a r l y , - K(v,v') = C v e r i f i e s (12).

- - One can, a l so , construct a p o s i t i v e operator K i n terms of K i t s e l f . - 2 For instance, i f K(v,v') = K(v',v) = K(-v,v'), one can take K = aK

( for some pos i t ive constant a). This amounts t o assume

a K(v,v")K(v",v')dv" < K(v,v') ( 12") V

2 [ Introducing H(v) = K(v,v') d v ' , it i s c l e a r t h a t (12") is f u l f i l l e d

i f one has : a J H(v)H(v') Q K(v,v'). (we r e c a l l t h a t

the physical examples [ 7 1 ) I . . IV

H(-) i s bounded for

Remark 9 : Similar r e s u l t s hold f o r nonhomogeneous s c a t t e r i n g kernels

(see ( 9 I ) . The assumption t h a t D i s convex i s not , r e a l l y , necessary :

Endeed the leading eigenvalue i s an increasing funct ion of D , so

(since D contains open and convex subsets) t h 5 remains t r u e f o r now

convex D . The only change concerns 3 ) where s(x,v) (which is used f o r

the d e f i n i t i o n of H ) i s replaced by t h e one corresponding t o any convex

subset D of D . Similar ly dist(0,aD) (which i s used f o r the d e f i n i t i o n

of y ) i s replaced by d i s t ( * , a E ) . . -

-

In general Lp spaces our assumption i s t h e following :

There e x i s t s a separable kernel K(v,v') = f (v)g(v ' ) 2 0 'L

such t h a t

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2r 1 K(v,v') < K(v,v') , f (.)E Lp(V) and g(.)E Lpsi(V) (L + - = 1) P P*

h(v) = f(v)g(v) = h(-v) V v E V (13)

h(.) does not vanish identicaly.

2, 2, 2, Our main task, now, is to analyse P(A) where A = T + K . The results are presented (for the sake of clarity) in five lemmas. Next, by com-

parison arguments, we shall easily obtain our main theorem (th 6 ) where a detailled analysis of P(A) is given.

let us consider the spectral problem

(14) is equivalent to

One follows an analysis similar to the one of monoenergetic model cases [151 [I63 1171 . Multiplying (14') by g(v) and integrating over V , one gets

(14") .

Extending h(.) and o ( . ) by zero outside of V and inverting the order of integration in (14") , we obtain (after the change of variables x' = x-sv and thanks to the convexity of D) the following integral equation

2r Conversely, one easily checks that E P(A) if (15) admits a nontrivial solution. Thus one is led to only consider the spectral problem (15)

in LP(D) .

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On the o ther hand

so % mapps Lq(D) i n t o i t s e l f o r a l l q ( 1 Q q Q + , and i s

compact i n Lq(D) f o r f i n i t e q because D is bounded (cf [18]

page 74). Its spectrum is then t h e sane i n a l l

Hence it s u f f i c e s to study the s p e c t r a l problem (15) i n L2(D). - .-

Following the idea of lehner and wing [151 we obta in

Lq (1 < q < + w ) .

due t o the eveness property of (I (.) and h ( .) . Hence :

which shows t h a t 5 is a p o s i t i v e compact operator i n LL(D) . Lema 1 : P(A)# % 0 f o r a l l D i f and only i f 1, i y d v = + w .

Proof : l e t k be the radius of D and S C D be a b a l l with

radius k . It may be e a s i l y seen t h a t

-

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so

where b(x,v) i s defined by : x-b(x,v)v E as f o r x E S . Now, l e t 2 be a b a l l concentric with S and 7 k be i t s radius

(with r < 1 ) .

Clearly

On the o ther hand, thanks t o the f i r s t p a r t of ( l l ) , w e have

l e t t i n g X .+ - X* i n (19) and using (ZO) , Fatou's lemma y ie lds

Thus, f o r every D there e x i s t s A(D) > - A* such t h a t

"%.CD) L2(D)

value of 5 ( for instance) i n LP(D) , which shows t h a t X(D) i s

the leading eigenvalue of 1 i n Lp (D x V). Thus P ( 1 ) # 0 f o r a l l D . This proves t h e f i r s t par t of the lemma.

is the leading eigen- '5 'L2(D)

II = 1 . But, due t o ( l a ) ,

Conversely, l e t us suppose t h a t J, F , d v is f i n i t e :

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where d is the diameter of D . The last term may be written

-X 1 - e but - < c < + m for x 2 0 , whence X

’L Thus P ( A ) = 0 for small D (= small diameter). This ends the proof of

the lemma. Q.E.D.

?J

Lemma 2 : P ( A ) # 0 for large D ( = large radius) if and only if

h(v)dv > 1 . In this case l i m h (D) = r where r i s D + R3 lim * Jv - x + - x

’L 1 . (A (D) being the leading eigenvalue of A ) .

Proof : Combining (16) and (19) one can write

where r < 1 is arbitrary. This shows that lim IINxII = 1, h(v)dv . D+ R3 L (D)

Thus:equation “ N x i = 1 has a solution x ( D ) , at least for L (D)

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l a rge D , i f and only i f l i m h(v)dv > 1 . Moreover, it i s A + - I, 0 W+A -

c l e a r t h a t A ( D ) goes t o as R goes t o i n f i n i t y . Q.E.D.

The following lemma gives us a e x p l i c i t lower bound of the leading

eigenvalue A ( D ) of 1 . l e t $ E L (R3)be such t h a t : Il$II = 1 and

the support of $ i s included i n t h e u n i t b a l l of R .

2

3 L

- 2 1

Let E(R) = I$(w)I dw where $ is t h e Fourier transform of $ . J , w j > R

Then :

Lemma 3 : Let V be bohnded (Vmax being the g r e a t e s t speed) and

o ( . ) be constant (u (.) = a ) . Fina l ly l e t k be the radius of D . Then :

2 'max P(x)* 0 i f k > 1 F - m -EW

L (V)

and

l = P . 1 h(D) > - O + 7 [llhI$ ( l - € ( R ) ) + k

Proof : without loss of genera l i ty w e may assume t h a t D 3 Ck = k C - where C i s t h e uni t b a l l of R3 . Cleary llNAu

On t h e other hand t h e s p e c t r a l problem

2 i . e . ~ ( x - x ' ) y , ( x ' ) d x ' = a ~ ( x ) y, E L (C,)

'k J

becomes, a f t e r the change of var iab le x' = k y ' (and x = k y) ,

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(22)

It follows from (23) that

, an easy computation shows that F(8,k) = 1 If > Vmax (I-E(R)) IIhIf

?, so IINII > 1 . Thus the leading eigenvalue of A ' L ~ ( D )

II = 1 exists and is bigger than 8 . Q . E . D . II%(D) L2(D)

defined by

In the following lemma, we give a (practical) criterion of existence

of eigenvalues for a given domain D . Of course we shall assume that dv is finite (otherwise there is nothing to prove due to J" hCv)

lemna 1). Recall that jv h(v)s(v)dv 2 7 Jv% dv (see(l2')) .

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Lemma 4 : Let u (.) be constant (0 (v) = 0 ) and Then :

# dv be finite.

P(& + 0 if J h(v)s(v)dv > 1 (in particular if Y J, dv > 1). V

Proof :

Hence

. [

* Letting X + - X = - u and using Fatou's

lim ll%II 2 J h(v)s(v)dv > 1 . X + - h L (D) V

"ds ]dv . J:x'v)e lemma one gets :

This ends the proof of the lemma. Q.E.D.

Lemma 5 : Let V be bounded away from zero. Then the real point spectrum of A is nomempty whatever the size of D . Proof : In the present situation 17 = - - , (and then u (1) reduces to the point spectrum). is selfadjoint for h E R , but not necessary positive for However its kernel is nonnegative, so its spectral radius (its norm in the present case) is its greatest eigenvalue. On the other hand, it may be easily seen &hat lim 1 1 ~ 1 1 = + - , so there always exists 1 such that h + - - L(D)

IINXll = 1 . Q.E.D.

%

X < - I* .

L (D)

Bemark 10 : The first part of (11) is only used in (a part of) lemma 1.

we are, now, in position to study the general homogeneous transport operator

A lir = - v . ~ a$ - u (v)$(x,v) + K(v,v')$(x,v')dv' in LP(DxV)(lG p< + -) JV

with the usual purely absorbing boundary condition.

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956 MOKHTAR-KHARUOUBI

Theorem 6 : Under (13) t h e fol lowing hold :

1 ) I f l i m h(v)dv > 1 then P(A) # 0 f o r l a r g e A + - h

- D ( = l a r g e r a d i u s ) . Moreover l i m X(D) >x , where h i s def ined by

D + R 3

1 . ( I n p a r t i c u l a r t h e r e e x i s t s a c r i t i c a l s i z e i f

- h > 0 and A* > 0) .

2) I f 1, 9 dv = + then P(A) # 0 f o r a l l D .

3) Let u ( . ) be cons t an t and dv b e f i n i t e ; then P(A) # 0

i f 1 h(v ) s (v )dv > 1 ( i n p a r t i c u l a r i f 7 I v F d v > 1 ) . V

4 ) Let u (.) ' b e cons t an t ( ~ ( v ) = U ) and V be bounded (vmax being

t h e g r e a t e s t speed) . Let k b e t h e r ad ius of D . Then :

5) I f V i s bounded away from zero, t h e r e a l po in t spectrum of A

i s never empty . - Proof :

previous lemmas.

By a comparison argument, t h e proof i s immediate from t h e

REFERENCES

(il H . G . Kaper, C.G! L e k k e r k e r k e r a n d J . H e j t m a n e k - S p e c t r a l m e t h o d s i n l i n e a r t r a n s p o r t t h e o r y . B i r k h g u s e r Verlag, Sasel (1982 1.

- _

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SPECTRAL PROPERTIES OF THE NEUTRON TRANSPORT OPERATOR 95 7

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P [3] M . Cessenat - ThBor&mes de trace L Dour des espaces de fonctions de la neutronique. C.P.Ac&d.Sc. Paris t 299, s‘rie I, n016 (1984).

[4] J. Voigt - Spectral properties of the neutron transport equation. J MATH ANAL APPL V o l 106, nO1 (1985).

_ - 151 J. Voigt - Positivity in time dependent linear transport

theory. Acta Applicandae Mathematicae 2, 311-331, (1984).

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[7] S. ukai - Eigenvalues of the neutron transport operator f o r a homogeneous finite moderator. J. MATH ANAL APPL 18 ( 1967 ) 297-314.

[8] J. Wika - Time dependent transport in plane geometry. Nucleonik 9 (1967).

191 K. Mokhtar-Kharroubi - Spectral theory of the neutron transport operator in bounded geometries. To appear in the Proceedings of the meeting ” Les mathbmatiques de la cin6- tique des gaz”, Paris, June 1985.

[lo] A. Mokhtar-Kharroubi - Theorie spectrale de l’op6rateur de transport dans le cas anisotrope dGgBn6r6. C.3.Acad.S~. Paris, t 297, s6rie I, (1983) 405-408.

kl] A . Pazy - Semigroups of linear operators and applications to partial differential equations. Applied mathematical Sciences, Vol 44, Springer Verlag (1983).

[14 M. Mokhtar-Kharroubi - The time asymptotic behaviour and the compactness in neutron transDort theory. Preprint.

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k4] K. Jorgens - An asymptotic expansion in the theory of . neutron transport. Comm Pure and APPL MATH 11 (1958) 219-24

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V.J.T. Marti - h e r Anfangs und eigenwertprobleme aus der Neutronentransporttheorie. Zamp 18 (1967) 247-259.

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Some spectral properties of the neutron transport operator in bounded geometries

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