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On the real point spectrumof multigroup neutrontransport operator inbounded geometriesKhalid Latrach aa Département de Mathématiques, Université deCorse, 20250, Corte, France

Version of record first published: 20 Aug 2006.

To cite this article: Khalid Latrach (2000): On the real point spectrum ofmultigroup neutron transport operator in bounded geometries, Transport Theoryand Statistical Physics, 29:6, 661-680

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TRANSPORT THEORY AND STATISTICAL PHYSICS, 29(6), 661480 (2000)

ON THE REAL POINT SPECTRUM OF MULTLGROUP NEUTRON TRANSPORT OPERATOR IN

BOUNDED GEOMETRIES

Khalid Latrach

Ddpartement de Mathkmatiques UniversitC de Corse 20250 Corte, fiance

Abstract

In this paper we study the real point spectrum of the multigroup neutron transport operator for a large class of unsymmetric collision matrices in bounded spatial domains. Necessary and sufficient conditions for existence, finiteness and sufficient conditions for in- finiteness of the real point spectrum are given. Estimates of the number of real eigenvalues and localization results are also provided. The last part of this work is devoted to the properties of point spectrum which are specific to slab geometry with finite thickness.

I. Introduction

This paper deals with the description of the real point spectrum of a class of linear multi- group neutron transport operators

A$ = T$ -+ K$ N

defined in the Hilbert space X = n Lz(D x K), N 2 2, where i=l

and K = ( K i , j ) (1 5 i, j 5 N ) is the matrix collision operator. Each operator T, is defined on Lz(D x x) by

66 1

Copyright Ca 2000 by Marcel Dekker, Inc. www dekker.com

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662 LATRACH

a* 1 D(T*) = {** E Lz(D x K), t& E L2(D x K); 111 = 0 ) =' where D is an open subset of R" (n 2 2), with boundary dD, V , = {( E R", /(I =

v,}, (1 2 z 5 N), (v j > 0) and := {(z,() E dD x V,, (is ingoing at x E d o } . The function gt(z,t), with (z,() E D x V,, denotes the density of neutrons of speed v,. The entries of the matrix collision operator A' = ( # i , J ) consist of integral operators (with respect to velocities) with kernels K , ~ ( ( , ( ' ) with ( ( , I f ) E V, x 4. The collision kernel t c t J ( [ , t') describes the production of neutrons of speed vi, due to the collision with the host material of a neutron of speed vj. The collision frequency of speed v , is denoted by o=(.). The usual assumptions are:

D is a bounded, u t ( . ) E L"(V,);

{ and E L: ( L z ( D x 4); Lz(D x V,)).

Let A: := inf{oI((); E E Vj} and set A* := min{A:, 1 5 i L. N}. We recall that the streaming operator T, (1 5 z 5 N ) is the infinitesimal generator of the following Co-semigroup on Lz(D x K )

where s ( x , ( ) = inf {s > 0 1 x - s( $ D } . Thus, the operator T defined on X = n D(T,), generates a strongly 1<z<N 1<a<N continuous semigroup V ( t ) . The type of U z ( t ) [as well as that of U(t)] is --oo because these semigroups vanish for large t . Therefore, the spectrum of T is empty. The multigroup transport operator A is defined as a bounded perturbation of T , so it generates a Go- semigroup V ( t ) = e t A . On the other hand, if the restriction of the collision matrix Ti to the subspace n &(K) is compact then V ( t ) is compact on X for large t and (A-T)- lK

is compact on X (cf. [I]).

The problem of determining the spectrum of the multigroup neutron transport operator has been considered by several authors under varying assumptions on the type of medium a.nd the scattering process. We can quote, for instance, the contributions of K. Jorgens [2], E. W. Larsen 131, M. Mokhtar-Kharroubi [l], G. H. Pimbley [4], S. Ukai [5], H. I>. Victory [6], R. Van-Norton [7] and Yang Mingzhu, Zhu Guangtian [8]. Recently, under the assumptions

n &(D x K ) which admits as domain D ( T ) =

I < t < N

2) o4-r) = OZ(t1, (1 I 2 I N); 2 ~ ) ~ t j ( O ( ' ) = ~ z j ( - t ~ t ' ) ~ (1 52, j I N ) ;

zzz) the restriction of K to nl<r5N Lz(V,) is compact and - nonnegative (in the scalar product sense) in n,cz<N - Lz(V,),

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MULTIGROUP NEUTRON TRANSPORT OPERATOR 663

a detailed analysis of the real point spectrum of the multigroup neutron transport operator was given by M. Mokhtar-Kharroubi [l].

The aim of this work is to extend [I] to unsymmetric collision operators which can be factorized in the following form h' = M I k M z where k satisfies (1.1) and where A 4 1 and A42 are diagonal multiplication operators

Thus, our assumptions are:

( H I ) t ) KJW = e m qw m'), ei(.) E LYVA and pl(.) E LV,);

8%) & , ( - E , 6') = %(<,<'), (1 i 2 , ; I N ) ;

222) ug(-[) = uz(<) (1 5 2 5 N ) ;

2 ~ ) @i( - t )P i ( -< ) = ei(5) Pi(€) 2 0 w) the restriction of k to n15,<N Lz(V,) is compact and

-

a.e on V, (1 5 2 IN);

nonnegative (in the scalar product sense) in fll<,<N - Lz(V,)

where I C , ~ (E,C) (resp. &j (<,<')) is the kernel of the operator K2j (resp. I?tJ) (1 5 2, ; 5 N ) .

The plan of this paper is as follows. In Section 11, we describe the real point spectrum of the operator A in a bounded geometry. The main result of this section is the derivation of a positive quadratic form associated with the eigenproblem (Theorem 2.1). Indeed, using this theorem we derive necessary and sufficient conditions under which the point spectrum is not empty for large D and we show that the number of eigenvalues increases indefinitely with the size of D. Furthermore, all these eigenvalues converge to a known limit when the size of D goes to infinity (Theorem 2.3). We end this section by estimating the number of real eigenvalues and by localizing them (Theorems 2.4, 2.5). Section 111 is devoted to the analysis of the real point spectrum of the operator A, in a slab of finite thickness where there is an appearence of a continuum in the spectrum (see, for instance, [4], [9]). We discuss only the spectral properties which are specific to the slab geometry.

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664 LATRACH

11. Spectral Theory in L2 for Bounded Geometries

This section is devoted to the fine structure of the real point spectrum of the multigroup neutron transport operator in bounded geometries A$ = T $ + K$ in the Hilbert space

X = n Lz(D x K). Let us now consider the spectral problem N

*= 1

Since a(T) = 0 (cf. [9]), Eq. (2.1) is equivalent to

(2.2) - Now, let p = fi A 4 2 + where fi denotes the positive square root of K . It is clear that p # 0 and (2.2) is equivalent to

(2.3)

where GA := fi Mz ( A - T)-' MIV%.

Conversely, if the spectral problem (2.3) has a non trivial solution p, then 4 = (A - T)-'

Now we shall show the following result which is crucial for our subsequent analysis

Theorem 2.1. Let X E R and assume that ( H l ) holds, then

verifies (2.2), and then (2.1) w (2 .2) cj (2.3).

(2) GA := fi Mz ( A - T)-' MI&? is a compact self-adjoint operator,

(zi) af X > -A*, then Gx is nonnegative and

Proof. Let and ZC, belong to n L,(D x K), I < r < N

where (., .) (resp. (., .),) denotes the scalar product in

and where ( v ) ~ denotes de z f h component of the vector zr. Hence,

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MULTIGROUP NEUTRON TRANSPORT OPERATOR

where V% = { F t , j } (1 5 i, j 5 N ) . In view

Fz,j (1 5 i , j 5 N ) with respect to velocities, we can write

= (9, Gx$)

which gives the self-adjointness of Gx for X real. On the other hand, since k is compact on n Lz(K), applying [I, Lemma 11 we obtain the compactness of ( A - T)-'K on

n Lz(D x K). Next, using 1 < t < N n Lz(D x K) and therefore Gx is power compact on

1 C t S N l C t < N the self-adjointness of Gx, we infer that it is itself compact which ends the proof of ( 2 ) .

(it) We start our proof by showing the identity (2.4) in the case where f i is replaced by a self-adjoint Hilbert-Schmidt operator. By approximating fi, which is self-adjoint com- pact operator on n Lz(D x K), by selfadjoint Hilbert-Schmidt operators, the identity

(2.4) will remain true for fi. Indeed, let F = (Fi ,]) (I 5 i , j 5 N ) where Ft,, E L(L2(Vt),L2(VJ)) are Hilbert-Schmidt operators with kernels Fz , l ( ( , [ ' ) E Lz(V, x Vj). Thus we have

i s i < N

( F M 2 ( A - T ) - ' M I F $,$) = C ( F e , j P j ( A - TI)-' ej Fj,k $ k , $ e ) e .

e,iIk

On the other hand,

(Fe,jPj (X-Tj)-' Fj,k $ k ) ( X , t ) = . ( . . E ' )

dt' F e , j ( l , t ' ) P j ( < ' ) 6 j ( < ' ) [ / e- (A+uJ("))s d5

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LATRACH 666

Let S be the unit sphere of R” and let t’ = vjw’, (w’ E S ) . Taking account of the fact that d<’ = (vJ)(”-l)dW’ one gets

The change of variables x‘ = x - sw‘ yields

So, F,,,P, ( A - Tj)-l O j FJ,k is an integral operator with kernel

where $, and $& are extended by zero outside of D. Now, by using the convolution theorem together with the Parseval identity we get

where

N i ’ 3 3 , k ( x , <, t”) e-’= d x 57;,3,k 1 and

On the other hand, after some calculations, we get

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MULTIGROUF' NEUTRON TRANSPORT OPERATOR 667

which proves (2.4) in the case where f i is replaced by a Hilbert-Schmidt operator. Now, by approximating fi by a sequence of self-adjoint Hilbert-Schmidt operators ( F n ) n E ~ we obtain

In view of the evenness of u3( t f ) , 6 3 ( [ f ) p 3 ( t f ) , i = 1,2 , ..., N , and (fi $)(w,[') with respect to tf, the imaginary part of (Gxg, 4) vanishes identically, and therefore the identity (2.4) follows. Q.E.D.

Throughout this section, pk( A, D) denotes the kfh positive eigenvalue of the compact self-

adjoint operator Gx on n Lz(D x K ) (each eigenvalue is repeated according to its multi-

plicity). On the other hand, by the equivalence between the spectral Problems (2.1), (2.2) and (2.3), it is obvious that the real eigenvalues of A are obtained by solving the equation p k ( A , D ) = 1 (for k = 1,2, ...).

N

i= l

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LATRACH 668

MA4 =

We now turn our attention to the bounded part of the mdtigroup transport operator which we denote by B. Thus, the matrix operator B is defined by

where the z t h component of B, i = 1, ..., N is given by N N

(BG), = -o404*(0 + c K , * J > 4 E nwk) j=1 k = l

As for the continuous model (see [9], [lo]), we are going to link the spectral properties of

A to those of the spectrum of B. To this purpose, consider the spectral problem

B$ = A4 ( A > --A*), 4 E n Mv,) ( 2 . 5 ) 1<,<N

The problem (2.5) may he written in the form

For technical reasons, we suppose

This allows us to use the transformation ql(<) = d(X + oz(())6'F1(()&(<) $z(<), i =

1, ..., N and to obtain do = M A I? M A v(<)

where

0

Therefore, the spectral problem (2.5) becomes

(SAY)(O = 4 5 ) where Sx = MA k M A , 'p E n L 2 ( x ) and X > -A*. In view of (Hl), the operator

SX is nonnegative and compact. We denote by I(Sx(( its norm. Then we can easily verify that I(Sx/( is continuous and strictly decreasing in X > -A* . We denote by a,(A) (resp. u p ( B ) ) the point spectrum of the operator A (resp. B ) . Thus we have:

1<i<N

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MULTIGROW NEUTRON TRANSPORT OPERATOR 669

Theorem 2.2. Assume that (Hl) and ( H 2 ) a r e satasfied, then

u p ( B ) n { A > - A * } # 0 if and only if lim l lS~ll > 1. A--X'

I n thzs case, the unique A0 such that IlS~,ll = 1 is the greatest eigenvalue of B . 0 Remark 2.1. Assumption ( H 2 ) is technical and can probably dropped. It is used solely

in Theorems 2.2 and 2.3. 0 Definition 2.1. We mean by the size of D the radius of the greatest ball included in D .

0 Remark 2.2. The quadratic form (2.4) will play a basic role in the sequel. Nevertheless, we need another version of (2 .4) taking account of the size of D which does not appear explicitly in (2 .4) . Let C be the unit ball of R" and C,j = dC ( d > 0 ) the ball centred at zero with radius d. It is easy to verify that the eigenvalue problem

is equivalent, after the change of variables x' = dx ( x E C), to

Consequently, theoperator GA on n Lz(C , j xv ) has the same point spectrum as G(X, d )

on n Lz(C x K). As in Theorem 2.1, we verify that the quadratic form associated

with G(X, d ) is given by

l j z < N

l < t < N

Lemma 2.1. Assume that ( H l ) is satisfied, then

ple(X,d) < (ISxII V k , V d > 0 and VX > -A* . Furthermore, YE > 0 we have :

lim pk(X,d) = l lS~l l uniformly o n {A > -A* + E } d-+oo

Proof. We first observe that the operator Sx ( A > -A*) may be written as

sx = M A it MA = (MA dz) (MA dz)*,

0

(2 .7)

0

which implies

On the other hand, Eq. (2.6) gives

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Next, the use of the Parseval identity yields

which proves the first part of the lemma

The proof of the second part is based on the max-min principle ([ll, p. 128 1 ) . Thus, let Sk be a k-dimensional subspace of &(C) and let $(.) E n &(V,) such that l l $ , l l = 1.

The max-min principle yields Iq tSN

min tllvllL2(c)=1t V E S k )

(G(A d ) ‘p c3 i , ‘p @.I Pk(A,d) 2

min Id(w)lZ dw - - { / ! V I I L Z ( C ) = ~ % V E S k ) J,,,

where v@$, denotes the function (z,() --t y(z)+((). In view of the compactness of the unit sphere of Sk, the minimum in Eq. (2.9) is reached at a function ‘p(A,d) with Il’p(X,d)ll = 1. so.

Again, by the compactness of the unit sphere of Sk, {‘p(A, d p ) p E N } has a converging subse- quence which is also denoted by {‘p(X, dp)pEN}. On the other hand, the use of the Parseval identity together with Fatou’s lemma, implies

N and therefore, since is an arbitrary normalized function of n Lz(V,),

,=1

liminf pk(A,d) 2 l lS~l l . d-+m

Then Eq. (2.7) allows us to state that we have a limit. Q.E.D.

Theorem 2.3. Let d be the radius of D and suppose that ( H l ) and ( H 2 ) are satisbed. Then, u p ( A ) n { A > - A * } # 0 , for large d if and only if op(B) n { A > - A * } # 0 , the number of real eigenvalues of A increases indefinitely with d and these eigenvalues

0 converge to Ao, the greatest eigenualue of B , as d t +co.

Proof. As a consequence of (2.4) we have

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MULTIGROUP NEUTRON TRANSPORT OPERATOR 67 1

Next, the Parseval identity and Eq. (2.8) give

IlGxll < llSxll for all domain D. (2.10)

If u p ( B ) n { A > - A * ) = 0, then (cf. Theorem 2.2) llSxll 5 1 for X > - A * and then IlGxll < 1 for X > - A * and for all D , i.e. o(A) n { A > -A* } = 0 regardless of the size of D. Conversely, assume that u p ( B ) n { X > -A* } # 0 which is equivalent to assume that lim llSxll > 1. Let d be the radius of D. We may assume, without loss of generality,

that D 3 dC = c d . We are going to prove the following x-x.

V k E N, VE > 0 and VM > - A * + E , pk(X, D ) -+ llS~l1 (2.11) { as D -+ Rn uniformly in X E [-A* + E , MI

where D + R" means that the size of D goes to infinity.

As previously, by the max-min principle, [ l l , p. 1281, we have pk(X, D ) 2 p k ( X , Cd). So, to prove (2.11) it suffices to consider pk(X, c d ) instead of pk(X, D ) . On the other hand, p k ( X , C d ) = pt(X,d) and clearly pk(X,d) increases with d (see the quadratic form (2.6)), hence, owing to Dini's theorem, it suffices to prove that pk (X , d) --t llSxll as d --t 00 for X fixed. This assertion follows from the second part of Lemma 2.1. Now, the use of (2.10) implies (2.11).

On the other hand, let 17 < Xo. Obviously, (2.10) and (2.11) yield

1 < p k ( X , D ) 5 pl~-i(X,D) L: . . . I pi(X,D) VX E [-A* + 6 , 171

for d large enough and p l (X , d) 5 1 for X 2 Xo.

9 < v1 < XO and p t ( a 2 , d ) = 1,

{ This shows that there exist q, (1 5 i 5 k) such that

i.e., there exist k eigenvalues of A between 7 and Xo. Since 17 (< Xo) is arbitrary, this achieves the proof. Q.E.D.

Let us now define the operator :

i --t ( M $1 (0 = Q(f)lii(O where mmm 0

0 dmmm Q ( E ) =

In what follows, we shall investigate the relationship between the eigenvalues of A and the singular values of ( M a ) . We denote by P (1 5 rn L: w) the number of eigenvalucs of ( M f i ) * ( M f i ) (the square of the singular values of ( M G ) ) .

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612 LATRACH

T h e o r e m 2.4. Let 6 k be the k t h eigenvalue of ( M f i ) * ( M fi) (the square o f the singular values of ( M e ) ) . Let rl < 1 and let d be the radius of D. If the hypothesas ( H l ) holds, then:

( i ) A has at least w real eigenvalues,

(ii) any solution Xk of the equation pk(X, D ) = 1 satisfies

0

N

Proof. Let X t R, then Gx is a compact selfadjoint operator on f l L z ( D x Vt). Lct

k ( k 5 zil) be a finite integer, the max-rnin principle yields p k ( X , D ) 2 pk (X , S ) where S is the greatest ball included in D , p k ( X , D ) (resp. p k ( X , S) ) is the k t h positive eigenvalue

of G x in n Lz(D x Vi) (resp. is the k t h positive eigenvalue of G x in n L2(S x x)). Let

E = ( 4 E Lz / ( M f i ) * ( M f i ) 4 = 0) be the kernel of (M f i ) * ( M fi). By

the general theory, the dimension of the orthogonal subspace , EL, to E in n L 2 ( K ) is

w. Let $ = (Il$ll~,(s) = 1) and Sk be the k-dimentional subspace of E l

spanned by the k first eigenfunctions of ( M fi)* ( M fi);,. Again, by the max-min principle we have

1=1

N N

*=1 *=1

N

2 = 1 1

JZqq

p k ( X , S ) 2 min tllull=1, u E m ' S t 1

( " X U , u ) = ( G x 4 X , k ) , u ( X , k ) )

V(X, k ) Jq (because of the compactness of the unit sphere of 4 8 Sk) where .(A, k) =

with p ( A , k ) f Sk and / / p ( X , k)ll = 1. So, we may write

P ~ ( A , D ) 2 P ~ ( X , S ) 2 ( G X U , ~ ) 1 C (PI ( A - Ti)-'Ot Ftj~jr F ~ , I ~ I ) L ~ ( s ~ v , )

N

J = 1

Let s" be a ball of radius d"= qd with the same centre as S. It is clear that

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MULTIGROUP NEUTRON TRANSPORT OPERATOR 673

- Observe that if x E S, then T(I,() 2 - (' - 7 ) d . Hence

(2.12)

because ( ( ~ f i ) * ( ~ f i ) ' p , y ) 2 bk, ( y E ~ k ) .

Finally, the use of Eq. (2.12) leads to

which completes the proof of ( 2 ) .

The assertion (zi) follows immediately from (2.12). Q.E.D.

Remark 2.3. If the functions @i(()/?i(<), i = 1,2, .., N satisfy ( H 2 ) , we can take for P 0

Our next task is to estimate explicitely the eigenvalues of A as in the continuous model (cf. [ 9 ] , [lo]). For the sake of simplicity, we shall assume that a, ( ( ) = ut (1 5 i 5 N). Let Vk be an arbitrary k-dimensional subspace of L z ( C ) , and {yl, ..., yk} be an orthonormal basis

E , ( t ) . of V k . We define E , ( t ) =

Note that I ( + ? , ( ( L ~ = 1, so s,(t) < 1. For t large enough we obtain ~ ( t ) < 1.

Theorem 2.5. Let d be the radius of D and suppose that a,(() = u, = u (i = 1, ..., N).

I

the number of positive eigenvalues of K .

k

IQl(w)lz d w (1 I i 5 k, t > 0) and s( t ) = L>* 1=1

then the equation pk(X, D ) = 1 admits (at least) a solution XI; 2tv, I f d > (1 - E ( t ) ) \ l f i M 2 & l l

where v, is the greatest speed. 0 Proof. From

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674 LATRACH

we infer that

where y~(X,cl) E V k and I I v ( X , d ) l l ~ ~ = 1. On the other hand, k k

IG(A,d)(W)l* dw L ( 1 - 4 t ) ) . L<i G(X,d) = x a I ( X , d ) G I with o j ( A , d ) = 1, j = 1 )=I

Thus, $ being arbitrary with norm 1, we obtain

I

= F(X).

Finally, we easily verify that the equation F(X) = 1 admits y as a solution if d > 2 t u, , and then there exists Ak > y such that p k ( X k , D ) = 1. Q.E.D.

(1 - E ( t ) ) I I ( f i M 2 f i ) I I

111. Spectral Theory in Lz for Slab Geometry

In this section we consider the multigroup neutron transport operator in slab geometry

N i

A, $z(z,O = -5 VI& (z,O - U * . t ( O $ 4 X , O + . , l c ~ , 3 ( ~ , S f ) i , ( . , E ' ) d € '

(3 1) ) = I - 1 1 $d-a,t) = $*(a , -0 = 0, E E i0,11, = 1,. N

where z E ( - a , a ) (2a is the thickness of the slab ) and < E (-1,l).

In what follows, we set

~ ~ 4 0 = & ( F ) and u ~ % , , ( < , € ~ ) = G , J € , C ) . Now, Eq. (3.1) may be written in the form A = T + K

matrix with

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MULTIGROUP NEUTRON TRANSPORT OPERATOR 675

and K,,* denotes the integral operator with kernel E c , J ( ( , ( ’ ) .

For the sake of simplicity, we suppose that

5,(() = i = 2, ..., N and

{ 51 < 5, for all i = 2, ..., N .

The operator A is defined on the Hilbert space X = [Lz( (--a, u) x (-1,l); dz d() lN

We recall that (see [l, Lemma 5 and Theorem 81):

o(Ti) = {A E C , ReA 5 -Zi} ( i = 1, ..., N ) . o(T) = { A E C , ReA 5 G I } .

If K ZJ compact on [L2(-1,l)lN, then the line { A E C, ReA = -?I}

is included in the spectrum of A. Our assumptions are:

( H 3 )

‘1 ‘ZJ((>(‘) = @t(() ‘ Z J ( < ? ( ‘ ) P j ( ( ’ ) ? @i(.)? P I ( , ) E LoO(-l?l);

22) %J(-t,(’) = z i j ( t , ( ’ ) , (1 5 i, j 5 N ) ;

221) @ , ( - ( ) P i ( - ( ) = &( ( )P i ( ( ) 2 0

zv) the restriction of

a.e. on (-1,l) (1 I i i N ) ;

to [L2(-1,1)IN is compact and nonnegative (in the scalar product sense) in [Lz(-l, 1)IN

- where ntj ((, 6’) (resp. El, ((, (‘)) is the kernel of the integral operator K;, (resp. K,J ) (1 5 i, j 5 N ) . The resolvent of Ti is given by

where p E Lz[( -a ,a) x (-1, l)] and ReA > -oi.

Let & be the nonnegative square root of Ir‘ and Gx := fi M2 ( A - T)-’ M I fi ( A > -?I) where M I and Mz are defined by

-

( M I : x --+ x,

where

O ( ( ) :=

0

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As in the preceding section, set fi = ( F z , 3 ) ( 1 5 i , j 5 N ) , we get N

(GA), ,~ = F t % k P k ( A - T k ) - ' @ k Fk,]. k=I

So, Gx may be expressed as N

Gx = C G: k=1

where

Using (3.3) together with the eveness of B k ( < ) P k ( ( ) and a$(., () (with respect to 0, and arguing as in the proof of Theorem 2.1 we obtain

Moreover, the Fourier transform with respect to x yields

where ( f i y ) k is the kih component of (fi y ) and b(w, <) = -& 1' ip(z, [) E-" dx.

Hence, - a

It follows from (3.7) that Gh is nonnegative on [Lz ( -a ,a ) x (-1, 1 ) I N . Note also that GA

may be decomposed as Gx = G: + G; is continuous in X on [-GI, +m[

while G i is only defined on ] - F1, +co[.

we now consider the closed operator

N N

G: where k=Z k = 2

( d ) k defined by

( G ) k : $ - p@(fi$)k (5) i" D(m ( d f ? ) k ) = {$ E [Lz ( -1 ,1 ) IN such that m(fi$)k (0 E L ~ ( - l , l ) }

where hk( . ) = @ k ( . ) P k ( . ) , k = 1, 2, ..., N .

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MULTIGROUP NEUTRON TRANSPORT OPEFUTOR 677

T h e o r e m 3.1. u p ( A ) n { A > GI} = 0 f o r small a zf and only if the operator

(#)(&?)I is bounded . 0

N Proof. As it was noted above, we have Gx = G : + C G:. Each operator G:, 2 5 k 5 N ,

is continuous in X E [-Z1, +m[ and satisfies k=2

11G:Il 5 llm l l ~ k l l L ~ ~ - l , l ~ I l (X - Tk)-’l l

5 11k11 l l h k l l L m ( - l , l ) 11(-g1 - Tk)-’ll.

Accordingly, since uniformly on [-:I, +a[ for k E { 1,2, ..., n}.

Next, axwme that Z := ( @ ) ( f i ) l is bounded. Then, G: extends for X = -a1 and G I- is given by

lim 11(-Z1 - Tk)-’l/i~(~~[(-~,~)~(-l,~)l) = 0, we get lim llG:11 = 0 a -0 a-0

- 0 1

An easy computation shows that IlG;[[ 5 Gf;, for X 2 -ZI and that lim IIG:;,II = 0.

Hence, /IG:Ilr(~czrc-a,a)x(-l,l)l)~ < 1, for X 2 - G l , if a is small enough which proves t,hc first part of the theorem.

Conversely, if 2 is not bounded, then there exists + = ($1, ...,$ N) in [L2(-1,1)IN such

that 241 4 Lz(-l, 1). Assume that Ilill = 1 and consider the function p(x, E ) = -

It is clear that

a-0

i (0 6.

IlGxll L (GAP, P) L ( G P , PI.

Now, using (3.6) (with k = 1) and letting X + -Z1, Fatou’s Lemma gives

lim- IlGxll 2 4“ 1’ lZil(E)IZ dF. (3.8) X--a, U l

Therefore IlGxll = +a, since (due to choice of $) the right hand side of (3 .8) is

infinite. Hence, there exists 1 such that IIGyII = 1 which implies that 1 2 -l?1 is an eigenvalue of A. Q.E.D.

The second part of the proof of Theorem 3.1 shows that the unboundedness of 2 implies that up(A) n { A > -(TI} # 0. This conclusion fails if the operator Z is bounded. In particular, we can determine a critical size a0 of the slab such that up(A)n { A > - G I } = 0 for all a < ao.

lim- x--v1

Note that, in general, the expression of dk is unknown. So, it is more convenient to give an equivalent statement to that of Theorem 3.1 making use of k directly. To this end, we introduce the following unbounded operator (m)k11( m)

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Observe that (m) Ell (m) = (& )* (G ).

Similarly, we introduce the operator R k R = ( f i R ) * ( G R )

where R : y + R(()p with

Corollary 3.1. If (m) k l l (m) is n o t bounded, t h e n O p ( A ) n { A > -?I} is n o t

e m p t y for d l a . If R k R is bounded, t h e n u p ( A ) (l { A > -gl} i s e m p t y for s m a l l a . 0

Proof. By ( H 3 ) , it is obvious that KtJ (1 5 i , j 5 N ) , are nonnegative operators. Let H,, = 0 if i # 1 or j # 1 and HI1 = lill. Since the leading eigenvalue increases with the collision matrix, we have o p ( A ) n { A > -Z1} # 0 if op(A") n { A > -?I} # 0 where A" = T + H ( H = {H$,} ) . On the other hand, hy Theorem 3.1, op(A") n { A > -?I} # 0

for all a if and only if # is not hounded, and clearly the latter operator is

bounded if and only if m kll is hounded. This proves the first part the

corollary. Next, assume that R k R is hounded, then R f i is hounded and consequently Q.E.D. ( 6 ) l is bounded. This together with Theorem 3.1 completes the proof.

Let p,(X) be the nch eigenvalue of Gx.

Theorem 3.2. We s u p p o s e t h a t there ex is t s 'p E [Lz(-l, 1)IN s u c h t h a t

lirn )m(fiy~)~ (()I = +w. T h e n lim- &(A) = +m f o r all a > 0 a n d all (-0 A - - q n 2 1. Consequent ly , A h a s i n f i n i t e l y m a n y real ezgenvalues. 0

Proof. We first note that (3.4) and the fact that the operators G: are nonnegative imply that p,(A) 2 ph(A) for all n where p i ( X ) is the nth eigenvalue of G:. Let V, c Lz(-a ,a) be an n-dimensional suhspace and V, = V, @ p with ))pJ/ = 1. By the max-min principle [ l l , p. 1281 and the compactness of the unit sphere of V,, we have

-

P ~ ( A ) L min (G:$, $1 = (Gi$J(A), $(A)) {ILEG", llILll=1)

where +(A) = .(A) p with u(A) E V, and I ~ U ( A ) ( ( L ~ ( - ~ , ~ ) = 1. Thus,

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MULTIGROUP NEUTRON TRANSPORT OPERATOR 679

( A + 6) gives The change of variables ( = ~

I W b 1

Again, by virtue of the compactness of the unit sphere of V,, if A t + -?I then u ^ ( X , ) has a converging subsequence in Lz(R). If we denote the limit by G , then clearly 11211 = 1 and Fatou's lemma shows that lim- p,(X) = +cm. Q.E.D.

A - - q

Corollary 3.2. We suppose that there ezists 111 E [Lz(-l, l)]"' such that

lim lh1(()(kp)l(()l = 00. T h e n A has infinitely m a n y real eigenvalues. 0 (-0

Proof. Let 'p = mfi$. It is clear that y satisfies the assumptions of Theorem 3.2. Q.E.D.

References

1. M. Mokhtar-Kharroubi, Spectral theory of the multigroup neutron transport operator. Europ. J . of Mech. B Fluid, 9, 197-222 (1990).

2. K. Jorgens, An asymptotic expansion in the theory of neutron transport. Comm. Pure and Appl. Math., XI, 219-242 (1958).

3. E. W. Larsen, The spectrum of the multigroup neutron transport operator for bounded spatial domains. J. Math. Phys., 20, 1776-1782 (1979).

4. G. Pimbley, Solution of an initial value problem for the multi-velocity neutron trans- port equation with a slab geometry. J . Math. Mech., 8 , 837-866 (1958).

5. S. Ukai, Real eigenvalues of the monoenergetic transport operator for a homogeneous medium. J. Nucl. Sc. Technol., 3, 263-266 (1966).

6. H. D. Victory, On the convergence of the multigroup approximations for rnultidimen- sional media, Annali di Mathematica pura ad applicata (4), CXL, 179-202 (1985).

7. R. Van-Norton, On the real spectrum of a monoenergetic neutron transport operator, Comm. Pure and Appl. Math., 15, 149-158 (1962).

8. Yang Mingzhu, Zhu Gangtian, Multigroup theory for neutron transport, Scientia Sinica, 22, 101-114 (1979).

9. M. Mokhtar-Kharroubi, Spectral theory of the neutron transport operator in bounded geometries, Trans. Theor. Stat. Phys., 16, 467-502 (1987).

10. K. Latrach, Description of the real point spectrum for a class of neutron transport operators. Trans. Theor. Stat. Phys., 23, 595-629 (1993).

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11. G . M. Wing, An introduction to transport theory, New York, John Wiley and Sons Inc. 1962.

Received: 20 October 1999 Revised: 10 May 2000 Accepted: 22 June 2000

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