spectral theory of the neutron transport operator in bounded geometries

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Spectral theory of the neutrontransport operator in boundedgeometriesMustapha Mokhtar-Kharroubi aa Université de Paris 6, Laboratoire d'AnalyseNumérique, 4, place Jussieu, 75230, Paris Cedex,05, France

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To cite this article: Mustapha Mokhtar-Kharroubi (1987): Spectral theory of theneutron transport operator in bounded geometries, Transport Theory and StatisticalPhysics, 16:4-6, 467-502

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TRANSPORT THEORY AND STATISTICAL PHYSICS, 1 6 ( 4 - 6 ) , 467-502 (1987)

SPECTRAL THEORY OF THE NEUTRON TRANSPORT OPERATOR IN BOUNDED GEOMETRIES

Mustapha MOKHTAR-KHARROUBI

Universit6 de Paris 6 Laboratoire d'Analyse NumGrique

4 , place Jussieu 75230 Paris Cedex 05, France

ABSTRACT

We give a complete description of t h e e point spectrum of the transport operator for positive collision operators : necessary and sufficient conditionsfor existence; necessary and sufficient conditions forfiniteness; estimates of the number of eigenvalues; iocal- ization of the point spectrum. Finally we indicate some open problems.

I. INTRODUCTION

Let A be the integro-differential operator

A@ = -v .--o(v)@(x,v) + K(x,v,v')@(x,v')dv' = T$+K@ ( 1 ) a+ ax V

with classical boundary conditions, arising in neutron transport theory ( [ l ] chap. 1 1 ) .

We denote by x the spatial variable and by v the velocity one, (x,v) E D x V . D is an open, bounded and convex subset of R3 , while V is an open subset of IR3 symmetric with respect to the origin of IR3 . G(-) and K(.,*,-) represent, respectively, the

collision frequency and the scattering kernel. The integral operator

467

Copyright 0 1987 by Marcel Dekker, Inc.

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468 MOKHTAR-KHARROUBI

(with respect to v ) appearing in ( 1 ) and denoted by K is, gener- ally, called the collision operator.

The usual assumptions are

a(*> E L ~ ( v ) , K E .c(L~(Dxv)) for some p ( 1 5 p < - )

O(v) 2 0 , K(x,v,v') 1 0

D(T) = CQ E LP(D~V) / v .2 E LP(D~V) , $ I r - = 01

! The domain of the collisionless transport operator T is :

where r- = ((x,v) E aDx V such that v is incoming at x] T is the generator of the following positive (CO) semigroup :

(3)

where xD(*) is the characteristic function of D [Z].

A is a positive and bounded perturbation of T , so it generates a positive (C,) semigroup T(t) which solves the Cauchy problem,

describing the time evolution of the neutron distribution in a nu-

clear reactor ( [ 13 chap. 12).

In this paper, we study the spectrum, u ( A ) , of the transport operator A,

haviour of the solution of the Cauchy problem ( 4 ) .

which will play an important role in the asymptotic be-

Let us recall the main spectral properties of T and A :

- the type of S(t) = etT is equal to - q , where + m if o ~ V lim inf ~ ( v ) if o E V

v 4 0

The characterization of the type of

to Voigt ( [ 2 ] lemma 1 . 1 ) . Moreover, if

etT for the general case is due

0 E v , then

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SPECTRAL THEORY OF THE NEUTRON TRANSPORT OPERATOR 469

- Q(T) = { A / Reh S -q)

( [ 3 ] Theorem 1 . 1 p a r t ( c ) ) .

= aa,(T) (= approximate po in t spectrum)

Under the general assumption

Some power of (A- T)-’K i s compact (Reh > - 11) ( 5 )

t h e following hold

- a ( A ) il {Keh > -T)} c o n s i s t s of i s o l a t e d eigenvalues with f i n i t e

a lgeb ra i c m u l t i p l i c i t i e s . The l i n e {A / Reh = - T)} belongs t o u(A). ( 1 1 1 page 2 7 7 ) . Using semigroup techniques much more may be proved. For ins tance :

- a(A) fl { A / Reh 1 - r ) +E} i s f i n i t e (VE > 0) . The leading eigen-

va lue i s s t r i c t l y dominant. (We r e f e r t o [ I ] and [3] f o r t h e d e t a i l s )

The general p rope r t i e s of t h e t r anspor t equation i n t h e most

general contex t a r e described i n 131.

We denote by u (A) (= asymptotic po in t spectrum) t h e spectrum of a s A i n t h e region {A / Reh > -r)} .

Much f i n e r r e s u l t s concerning u (A) a r e obtained f o r p a r t i - as c u l a r homogeneous and i s o t r o p i c ( i . e . a(-) and K(.,.) depend on-

l y on t h e modulus of v e l o c i t i e s ) cases :

- For the monoenergetic t r anspor t opera tor i n s l a b geometry,

is real, f i n i t e and never empty (Lehner and Wing

- In s l a b geometry with continuous energy,

but empty i f t h e th ickness of t h e s l a b i s small (Mika [5] 1967).

- I n bounded th ree dimensional geometry, (3 (A) i s r e a l , f i n i t e

but empty, i f t h e diameter of D is s m a l l (Albertoni and Montagnini

161 1966). This work was extended by Ukai ( [ 7 ] 1967) t o s o l i d and

l i q u i d moderators wi th , e s s e n t i a l l y , t h e same r e s u l t s .

uas(A)

[4 ] 1955)

uas(A) i s r e a l , f i n i t e

as

F ina l ly , l e t us mention a paper by Larsen and Zweifel 181 de-

voted t o a more general t r anspor t opera tor than those usua l ly stud-

ied i n Transport theory.

To our knowledge, no p rec i se r e s u l t s ( l i k e , f o r ins tance , those

obtained i n 141, 161, [7 ] ) have been proved f o r an i so t rop ic and (or )

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non-homogeneous scattering kernels. Our aim, in this paper, is to

describe the real point spectrum for. which we give a complete ana-

lysis in the region {A E R / A > -A* = -inf a(-)} ( * I . In fact, if

0 E 7 , the assumption (7) on the collision frequency ensures that A* = q . In particular, we generalize the known results [61. Some

(partial) results concerning the complex eigenvalues may be found

in E91.

For mathematical reasons, we investigate the spectrum of A in Hil-

bert space setting, that is in L2(D x V ) . The collision operator K may be viewed as an operator Kx in J(Lz(V)) depending on a para-

meter x E D . Our key assumptions are :

\ o(v) = a(-v) . The positiveness of K (for the inner product in Lz ) was suggested

to us by Ukai's paper [ 7 ] . Indeed the isotropic model used by Ukai

[ 7 ] verifies this property. However, to our knowledge, nobody has

made full use of this property (even in the isotropic case). We do not know if this assumption is satisfied by anisotropic models used

in physics. In this paper, we make this hypothesis for mathematical convenience : the existence of a square root of K enables us to

symmetrize the eigenvalue problem and allows for very precise results.

This work is organized as follows :

In Section 11, we deal with the homogeneous and anisotropic case for

which we prove the following results :

1) We give a necessary and sufficient condition under which the

point spectrum is not empty for large (in a suitable sense)

(theorem 2). D

2) Under this condition, the number of eigenvalues increases in- definitely with the size (in a suitable sense) of D , and, moreover

(*I If 0 f! v , spectral results in the full real axis are presented in section VI, at the end of the article.

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all these eigenvalues converge to a known limit when the size of goes to infinity (theorem 2).

D

3) We give the necessary and sufficient condition by which CJ (A) is not empty whatever the size of D may be (theorem 3 ) . P

4 ) We give a sufficient condition implying the finiteness of

Up(A) (theorem 4 ) .

5) A sufficient condition implying the infiniteness of Up(A) is also given (theorem 7)

6 ) For isotropic scattering kernels, we give a necessary and sufficient condition by which up(A) number of eigenvalues (theorem 5 and 6 ) .

is finite and calculate the

7) We give a criterion for deciding whether CJ (A) is not empty

for a given domain D . This criterion involves two geometrical parameters of D (theorem 9).

P

8) We give a lower boundforthe number of eigenvalues. Upper and lower bounds of all the eigenvalues are also given (theorem 10 and 11).

In section 111, we show how almost all the previous results may be extended to the non-homogeneous case. Finally, the section IV pre- sents some open

Besides ( 6 )

i Kx

problems.

we assume

is compact in L(Lz(v) )

(7)

Re& 0.- If 0 E v , one easily verifies that the second part of

(7) implies the equality A* = . This is a simple consequence of the characterization of given above.

DEFINITION 0.- The 2mmpah t ope,x&oh A uliee be h a i d t o bc ha,i%opiic .id ~ ( v ) = U(U) and K(x,v,v') = K ( x , v , w ' ) whehe v = I v I and V ' = Iv'I .

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472

11. THE HOMOGENEOUS CASE

MOKHTAR-KHARROUBI

1 . A preliminary remark

Let B be the bounded part of the transport operator

v E L2(V) Bcp = -u(v)cp(v) + I K(v,v')cp(v')dv V

Let 0 be the closure of the set {-~(v),v E V} . It is well known that U(B)/O consists of a discrete set of eigenvalues (Weyl's lema) because K is compact. These eigenvalues are real since K is selfadjoint. We denote by 0 (B) the point spectrum of B . Let us consider the spectral problem

P

BQ =A,+ (a > -A*) Q E LZ(V)

After the change of functions becomes

cp(v) = Ja(V)+%J(v) , the problem

Sh its greatest eigenvalue. It may be easily seen that strictly decreasing and continuous function of the h parameter. We may state :

is positive compact thanks to (6) , (7) . Its norm I ISh l l is then

IIShlI is a

THEOREM 0.- up(^) n fhih

In this case, the unique eigenvalue of B . 2. The eigenvalue problem

Let -r) be the type

> -A*} # 0 if and only if lim I/SAII > 1 . h+-h*

h such that I I S - I I = 1 is the greatest h

and the corresponding bilinear form

of the semi-group e . tT

tT -tCJ(v) e $(x,v) = e $(X- tv,v)X,,(X- tv) $ E L2(D x V)

where % is the characteristic function of the set D . From the obvious inequality Ile I( I , it is clear that -q S -A*

(see also remark 0 ) .

tT

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SPECTRAL THEORY OF THE NEUTRON TRANSPORT OPERATOR 4 7 3

(A - T)-l exists if Reh > -n and is given by :

where s(x,v) is defined by : x- s(x,v)v E aD Now, the spectral problem :

is equivalent to :

(A- T)-lK@ = $

Let fi be the positive square root of K , and Q = . It follows from (9) that :

Conversely, if cp satisfies ( l o ) , one easily sees that

JI = (A- T)-I d ? ~

The bilinear form

verifies ( 9 ) , so (8) ++ ( 9 ) cs3 (10).

will play a basic role in this paper. This paragraph is devoted to its study.

Being compact in L2(V), h? may be approximated in the uniform topology by a sequence (H } of Hilbert-Schmidt operators. We denote by Hn(v,v’) the kernel of Hn . n

(H Q,$) = lim (H&$) = lim (Hn(h-T)-lHnQ,$) n -+- n+- h

because Hn 4 d? in f(L2(DxV)) as well. Now, thanks to the convexity of D , whose kernel is :

H; is an integral operator,

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474

where Hn(-,*) is extended by

MOKHTAR-KHARROUBI

zero outside of V x V . So

= dvdv' dd(x,v) Nn(x- x',v,v')cp(x',v')dx' J v x v J D J D A

= F ' dx$(x,v) ' Nn(x - xt ,v,v')cp(xt ,v')dx' J v x v dvdv' h3 a3

where a ( * , * ) and c p ( * , . ) are extended by zero outside of D . Using the Parseval identity, we may write

;1 1

f i n I

,v,v')cpp(x' ,v')dx (w). dvdv' /$ $(w,v) [ k3 Nh( * - x' (H:VyW = ' J v x v

A

I n To apply the convolution theorem to NA(* - x',v,v')~(x',v')dx' (w)

we need : Nf;(*,v,v') E L1@13) (for fixed (v,~') 1, i.e. ReX > -A*. (12)

From now, we assume (12) . It is the reason why the results we shall prove hold, only, in the region {A > -A*}.

and

hence

where

Hn HC being the adjoint of

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= IIHn@- f l@ l l by the Parseval identity, L 2 W X V ) Lz (DxV)

llHn6- xiJ11 1 ReX+A* , so passing to the is bounded by - I 1

and (iV" . +Q(vll) + A

limit in (13'), we obtain :

For real A. , we may write (HA@,@) in the form

which shows that the imaginary part of (HA@,@) vanishes identically

since both ~(v") and $(w,v'l) are even with respect t o v" (we

use the assumption ( 6 ) ) .

Hence

It is clear that HA is a positive operator for A > -A* . Rematrh 1.- Under (5) some power of HA is compact, so HA itself is

compact, since it is positive.

Thus :

THEOREM 1 .- HA h a p o h a w e compact o p e h a t o h in L 2 ( D x V) (for

a > -A*) ,

The remainder of this paragraph is devoted to some lemmas we shall use

in the sequel.

Let

and

EA = Ker HA = {@ E L 2 ( D x V) / HA@ = 03

E = Ker d? = {Q E L z ( D x V) / &$J = O}

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P k o o ~ . - cp E EX implies (EAcp,cp) = 0 , and then & @ = 0 thanks to (15) ; hence a cp 0 , that is cp E E . Conversely, let cp E E , i.e. Xcp = o . Clearly h? 0 , so, thanks to ( 1 4 ) , we have

(HX(P,Q) = 0 VQ E L2(D x V) VX > -X* . In particular, taking V k > -A* . Q.E.D. Remahh 2.- Let C be an arbitrary convex and bounded subset of EZ, and Ck = kC (for k > 0 ) . The spectral problem

Q = H cp , we get IIHX(P112 = 0 , i.e. (0 E EX a

HXcp = arp cp E L2(Ck x V) (16)

is equivalent, after the change of variables x = kX (X E C) , to another one

where k enters as a parameter. Thus the point spectrum of HA in L2(Ck x V) is the same as that of H(h,k) in L2(C x V) , Arguing

as above, the quadratic form associated with H(X,k) is

Let us denote by pn(X,k) the nth eigenvalue of H(X,k) eigenvalue of H(h,k) is repeated according to its multiplicity). Let /IS 1 1 be the norm of the operator are, now, in position to state

Lem 2 . -

(each

Sx (previously defined). We A

(1) pn(h,k) < IISXII Vn , wc , VX > -A*

(2) VE > 0 , Vn lim pn(h,k) = I ]Shl l uniformly in k+-

{A t -A*+€} .

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SPECTRAL THEORY OF THE NEUTRON TRANSPORT OPERATOR 47 7

Ptlaad .- A s a consequence of ( 1 7) , we have

If we denote by M the operator

SA is nothing else but Using the Parseval identity, we deduce from ( 1 7 ' )

MKM = (M&)(M&)* , S O IIMXlY = IIShll .

(H(A,k)$,$) < IISAII 1 1 $ ( 1 2 which proves (1 ) .

Let, now, $(*) € L2(V) be such that II$IL = 1 , and Sn be a n-dimensional subspace of L2(C) .The max-min principle ( [ l o ] ,

chap. 8) yields the estimate

2 ( V )

where cp @ 9 denotes the function (x,v) + cp(x)@(v) . The minimum is assumed at of the compactness of the unit sphere of the finite dimensional space Hence

cpn(A,k) E Sn with Ilqn(A,k>Il= 1 because

'n *

Let k ---)a , There exists a converging subsequence of cp (A,k )

in L2(C) (the unit sphere of Sn is compact) denoted ( t o sim-

plify) by cpn(h,kp) . Gn(X,kp) malized function. Using Fatou's lemma, we get

P p+w * P

converges in L2@3) to some nor-

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Now, J, being an a r b i t r a r y normalized f u n c t i o n of L2(V) , we

f i n a l l y g e t

Thus

lirn pn(h,k) t I ISh/ l . k-tm

l i m pn(A,k) = I IShl / s i n c e w e have shown t h a t p,(h,k) < IISAII . k - m

The uniform convergence w i t h r e s p e c t t o h E { h / h t - A * + € } may

be e a s i l y checked. Q.E.D.

L e t us d e f i n e t h e t h r e e fo l lowing o p e r a t o r s :

1 Note t h a t - & i s c l o s e d . J;-

Note t h a t v% - i s dense ly def ined . I4-T

1 D(K) = {V E L2(V) /- E L2(V) , - K (w) E Lz(V)} m m \m

Lemma 3.- The t h r e e fo l lowing a s s e r t i o n s are equiva len t

1) - fi i s a bounded o p e r a t o r ; m 1 2) & - i s a bounded o p e r a t o r ;

3) K i s a bounded o p e r a t o r .

J;

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Phoo6.- The proof is based on the fact that = & is the adjoint ( 1 . 1

of the densely defined operator 8 - 1 and K = (#&y(&&) 4-7’

we omit the details.

3. Existence. Non existence. Finiteness. Infiniteness. Estimates of

the number of eigenvalues

D E F I N I T I O N 1 .- The ha&uh 06 a bounded a c t D h the hadiub 06 ,the glrecLtent b a U included i n i t .

THEOREM 2.- CJ (A) n {Rex > -A*] # 0 404 eUJLge D ( = eUJLge hadiub) P

i d a n d o d y i d u (B) n {a > -a*} # 0 . In ttkin c a e , .the nwnbe4 06 heal e i g e n v d u u 06 A incheaben i n d e 6 i n i t d y w i t h t h e hadiub 0 6 D , and a l l t h u e eigenvduen convehge t o 0 6 B , a ,the hadiuh 06 D g o a A? i nd in i t y .

Pmv6.- Let k be the radius of the domain D , and C be the unit

ball of R3 . Without loss of generality, we may suppose that k C = C c D . k Let pn(A,Ck) be the nth eigenvalue of HA in L2(Ck x V) , and pn(A,D) be its nth eigenvalue as an operator in L2(D x V ) . Let sn be the n-dimensional subspace of L2(Ck x V) spanned by the n first eigenfunctions of HA in L2(Ck x V) . Let Sn be the n-di-

mensional subspace of L2(D x V) obtained after we have extended by

zero to

the max-min principle ( [ l o ] , chap. 8), we may write

P

t h e g t e a ; t a t &genv&ue

- D x V , the functions of Sn , outside of Ck x V . Using

Thus

pn(a,ck) I pn(X,n) Vn , VA > -A* . But p (h,C )

H(h,k) in L2(C x V) (see remark 2). Applying lemma 2, we obtain

is nothing else but the nth eigenvalue pn(A,k) of n k

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where D -B IR3 means : the radius of D goes to infinity.

Now, if u (B) n {A > -A*] = 0 , then IISall < 1

so pn(A,D) < 1

spectrum is empty, whatever the size of D . The same holds for the complex point spectrum as is well known for the transport operator

VA > -A* (theorem 0), P

VA > -h* , VD , V n , and therefore the real point

[ill.

Conversely, let u (B) n (A > -A*] # 0 and x (> -A*) be the

greatest eigenvalue of (arbitrary) and n be an (arbitrary) integer. Since

Pn(Ao,D) + //She// > 1

D -+IR3

P B . We have IIShll > 1 Vh < h. . Let ho < x

It follows, for sufficiently large D , that

Consequently, there exist hl, ..., An such that

because pp(A,D) + 0 and p (h,D) is continuous with respect to X .

Thus A, , ..., An are eigenvalues of A . Q.E.D.

P

Remahh 3 . - It is not difficult to give sufficient conditions under

which u (B) n (A > -A*} is empty (or not empty).

R e m h 4.- There exists a critical size (that is a size of D for

which the fundamental eigenvalue of A is zero) if and only if

l l S 0 l l > 1 (and h* > 0 ) . Recall that (h-T)-’ , also, reads

P

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V where w = - and s(x,w) i s defined by x-s(x,w)w E a D .

(A+u(v ) ) s cp(x- sw,v)ds . Let oh be t h e opera tor : cp+ e -

Oh i s defined f o r A E [-A*,-[ wi th IIOAII 5 d (A 2 -A*) d being

t h e diameter of D . THEOREM 3.- 0 (A) fl f,A > -A*} # 0 doh & D id M d o d ! j id ,the

IVI

[ S (x,w)

J O IVI

P opehatah E nvX bounded.

Ph0od.- I f t is bounded, thanks t o l e m a 3, we may wr i t e HA i n

the form :

The opera tor GA i n L2 (D) defined by

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Ga@(x) = I Ga(x - x')&(x')dx' D

MOKHTAR-KHARROUBI

is analyzed by the author in [12], and verifies :

which is nothing else but (18).

n u s lim p,(a) = lim l l ~ ~ l l = +a VD . Q.E .D.

R e W k 5 . - If K is bounded, then, thanks to lemma 3, Ha has a

limit in the uniform topology as h-+ -A* . The limit denoted by HA*= (am, '0 -a*\JITi '1 &) is, thus, a positive compact operator

in L 2 ( D x V) . Let N be the number of eigenvalues exceeding one

of the operator €Ia, . THEOREM 4 . - Id K 0 bounded, Xhen Ahe .Ou~nApoht o p e h a t o h A h a aA ~ e ~ X N e i g e n v d u a . 1 6 1 0 n o t rn eigenvdue a d H-A* , then A h a a &kLte numbeh 0 6 e i g e n v d u u .

Pmo6.- (cf. [I31 page2361 where

Pn(h) is the nth eigenvalue of HA (A 2 -A*) . Thus there are (at least) N curves p (.) crossing the line p(A) 1 . These curves give rise to (at least) N eigenvalues of A . Let {A } be an infinite sequence of eigenvalues. They must accumu-

late at -A* [ll]. Being an eigenvalue of H (Vn) , 1 must be an

eigenvalue of H-a* . Q.E .D.

Rematlh 6 . - The hypothesis concerning H-A*

a+ -a* a+ -a*

Ipn(A) - pn(-h*) I I I /Hh-H-A*(I

n

n

An

(theorem) is probab ly ,

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notnecessary. Infinitely many eigenvalues will exist only if some curve p (.) crosses the line p(A) = 1 infinitely many times ; this is, probably, not true. In the isotropic case, the curves de-

crease strictly with respect to h [ 7 ] ; this yields more precise results.

THEOREM 5. - I n Rhe &5otz.opLc cane, .the &andpohA opehatoh A h a a

&ihXe numbm 0 6 L g e n v d u u 4 aMd onRy . id ,the opmatoh

bounded. I n Rkii cane, A han exactly N eigenudueb.

Remah 7.- In 161, the point spectrum is shown to be finite under some assumptions. One, easily, verifies that those assumptions imply the boundedness of K . Of course, the authors do not assume the posi-

n pn(*)

&5

-

tiveness of K result is that

Let L(x,x') =

, but only its selfadjointness. The novelty of our we obtain exactly the number of eigenvalues.

( b is the greatest speed).

THEOREM 6.- L& ud Comideh t h e hi6tz.opk~

Tb Ib K(u,V')2dudw1 < 00 . Then the number of eigenvalues of A is

bounded above by

#&I

10 Jo

The proofs of the last two theorems (concerning the isotropic case) are in the appendix.

Rewk 8.- Note that the disappearance of the point spectrum for small bodies is equivalent to the finiteness of the point spectrum (at least in the isotropic case).

Let us come back to the anisotropic case. For

define its Radon Transform by cp E L' (V) , we

dLp(w,s) = cp(v)dv w E S 2 , s E R . Jo.v=s

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lim R1KqI2(w,s) = +m s + o

va E S2 ,

Rhen u (A) n Ia > -a*) 0 .in@~Lte ~ O / L D . P

P h o U 6 . - Let Sn be a n-dimensional subspace of L2(D) and

dJ E L2(V) [ l o ] , we may write

be such that IldJlk = 1 . Using the max-min principle (V)

u(A,n) assumed (this is possible because the unit sphere of

Let h = K @

being a normalized function in Sn , at which the minimum is Sn is compact).

( h ( - ) Thus

is extended by zero outside of V ) .

Let p = S and ds = (h+u)dp h + U

Now we choose II, as follows

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Let A. +-A* = -u , u(hi,n) converges (in fact a subsequence) to

u(n) in L2(D) (thanks to the compactness of the unit sphere of

i--rm

Sn ) , and fi(Ai,n) converges to fi(n) in L2(R3) with 11 u(n) 11 LZ m3) = 1 , On the other hand,

(by assumption). Hence, using Fatou's lemma, we get :

i.e. lim pn(A) = +- Vn , VD Q.E.D.

Rem& 9 . - We have seen (theorem 2 ) , how the spectra of A and B are related. If we restrict ourselves (for simplicity) to a constant collision frequency, u(v) = u , u (A) is obviously not empty for large D . On the other hand, for bounded i?. , u (A) = 8 for small D (= small diameter). The following theorem gives us a criterion to decide whether u (A) is not empty for a given domain D . Let s(x,v) be defined by x- s(x,v)v E aD . Physically, a particle initially at x and moving with the velocity -v reaches the boundary a D of the domain, at time s(x,v) . Let dist(x,aD) be the distance from x to the boundary aD of D . Clearly,

a+-a

P P

P

I s(x,v) 5 - ( d is the diameter of D ) . dis t (x, aD) IVI IVI

We define : s(v) = - s(x,v)dx and Y = - dist(x,aD)dx. mes (D) J Y d

I v I obviously -

Let K(v,v') = K(v,v') and be the corresponding integral operator in Y(V) . Note that

s(v) 5 - IVI *

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( F due to theorem 3 ) .

is assumed to be bounded, otherwise there is nothing to prove

THEOREM 9.- Fa& a given domain D

1) ap(A) = 0 id dllKll < 1

2) up(A) # B id 11E11 > 1 . In pcVdhUem, up(A) # 0 i d

Y l l K l l > 1 . Phoo6.-

1) is a part of the proof of theorem 3. Let us consider 2). In the

proof of theorem 3 , we have established the inequality

Hence

but (see the proof of theorem 3) h = I& (PI2 SO

lim llHhll 2 1- & cp(v") 12dv" . a+-u 1"

Thus lim pq(x) > 1 which proves 2). Q.E.D. h + - U

Remmh 10.- Theorem 8 involves two different geometrical parameters :

the diameter of D and the parameter Y = mes(~) j, dist(x,aD)dx

Remmh 1 1 . - Let (for example) V be bounded and K(v,v') I C then

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((:I1 2 [I dv” which shows tha t the V J

point spectrum has t o do with the value of - mes (V)

appearance of the

which may be viewed a s t h e mean time, f o r t he p a r t i c l e s , t o reach

the boundary a D of the domain D .

4 . Localization of t he poin t spectrum

L e t k be the rad ius of D and C be the u n i t b a l l of lR3 . Without l o s s of gene ra l i t y , we may assume t h a t = kC c D . Let

pn(A,D) be the nth eigenvalue of HA i n L2(D x V) and pn(h,Ck)

be i t s nth eigenvalue i n L2(Ck x V) . Let Fn be the vec tor space

spanned by the n f i r s t eigenfunctions i n L2(Ck x V) and rn be

Fn the n-dimensional space obtained by extending the func t ions of

‘k

by zero t o D x V , outs ide of C k X V .

The max-min p r inc ip l e y i e lds

= Pn(h,Ck) 9

L2 (CkXV) pn(h,D) 2 mi: (HAW,@) = min (HhVsV)

QE Fn L2(DxV) q E F n I I W I I = 1 IIVII = 1

so Pn(X,D) 2 Pn(X,Ck) . On the o ther hand, p (h,C ) = p (h,kC) i s nothing e l s e but

p n ( h , k ) t h e nth eigenvalue of H(h,k)

mark 2 ) .

Thus

n k n i n L2(C x V) (see re-

P~(A,D) t Pn(hsk) Vn . L e t S be an a r b i t r a r y n-dimensional subspace of L2(C) . We

choose an orthonormal bas i s i n

L e t $i be the Fourier transform ( a f t e r extending i t by

zero outs ide of C ) and Qi(R) = rf di[&i(w) I2dw . Note t h a t

I1

sn : w,, ... ,M .

I w l l R

l i m E ~ ( R ) = 0 . R + w

n

i- 1 Now we choose an a r b i t r a r y R such t h a t C E ~ ( R ) < 1 .

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4aa MOKHTAR-KHARROUBI

Rematrh 12.- If the curves p (.,D) decrease s t r i c t l y (as f o r t h e

i s o t r o p i c case [ 7 ] ) , the root of

nothing e l s e but t h e nth eigenvalue of t he t r anspor t opera tor . In

the general case, t h i s i s not , a p r i o r i , t r u e and theorem 10 only

a s s e r t s t h a t the nth curve pn(.,D) of Ha cont r ibu tes t o Up(A)

i n the region { A / h 2 p) . P h 0 0 d . - Let us come back t o (17")

n pn(h,D) = 1 i s unique and i s

Hence n ' IGn(A,k)(w)12dw 5 C ci(R) .

l w l 2 R i= 1

We have ( see (17"))

$(*) i s an a r b i t r a r y func t ion i n L2(V) such t h a t I I Q l l = 1 , so,

using (19), we ge t n 1 1 - Z: Ei(R)IIIKII(U+h)

p,(A,k) 2 i=* = f (A) . R2 %ax + (a + A ) z

k2

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An easy calculation shows that f(@) = 1 provided that

Thus Pn(P,D) 2 pn(P,k) 2 f(p) = 1 , so there exists h 2 fi such that pn(hn,D) = 1 . Q.E.D.

Remacrh 13. Assume that {LJJj} (functions of class CN with compact support in C ) .

la1 aA aaQj (w) = i that i2 = -1 ) , so

n

1 I j I n belong to Cr(C) (N > 1)

A w Qj (w) (la1 I N) ( i is the complex number such

for all la1 SN . Hence

But there exists r > 0 depending (only) on N such that :

r(l+ 1~12) I sup Iwa12 for all w E n 3 (see [141, page 3), N

la1 I N

dw which gives us an explicit Hence E. (R) I C. (N) r J ’ J Iwl2R ( 1 + lw12)

estimate of E.(R) . Remanh 14.- The following theorem is intended to give us estimates of the distance between two eigenvalues of the transport operator. Thus, combined to theorem 10, it will give us upper and lower bounds of all the real eigenvalues. But an important difficulty arises: to number the real eigenvalues of A ? O r equivalently, how to charac-

J

how

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MOKHTAR-KHARROUBI 490

t e r i z e t h e nth eigenvalue of t h e t r a n s p o r t o p e r a t o r ? The answer

is easy i f t h e curves P (-,D) decrease : t h e nth e igenvalue of A

i s t h e unique h,

p ( . , D ) e x h i b i t a genera l behaviour, t h e r e i s no s imple ways t o num-

b e r t h e e igenvalues of A , Thus w e r e s t r i c t ourse lves t o t h e case

where p ( . , D ) decrease , f o r ( t h e f i r s t p a r t o f ) t h e fo l lowing

theorem

n such t h a t p (h , D ) = 1 . Now, i f t h e curves n n

n

n

THEOREM 1 1 .- 1) 16 t h e w v p n c+,(-,D) deckcue (& pa.mXcLLeatt in t h e AibXxa-

pic c u e ) , we have - A n 2 (hn-l + h*)2 1 1 - ~n(hn-~ ,D> 1 / I K ] / - ~ Wheke hi Rhe i.th ~ g w v d u e 06 A .

2) L& x1 be nhim)2ee ; .then : h l - h 2 2 ( X ~ + A * ) ~ [ ~ - P ~ ( X , , D ) I I ( K I [ - ~ .

Remmh 15.- Suppose t h a t an upper bound of An-, i s known. A lower

bound of

we could hope t o compute ( a t least numerical ly) @(hn-~,D) which is

t h e nth eigenvalue of t h e p o s i t i v e compact opera tor H i n

L2(D x V) , and t h e r e f o r e 1) gives us an upper bound of An . Now,

us ing t h i s upper bound of An , t h e same process gives us an upper

bound of h+l and so on, so an upper bound of h1 ( t h e g r e a t e s t

e igenvalue) enables u s t o estimate a l l t h e r e a l e igenvalues . Presum-

a b l y , t h e accuracy of t h e s e estimates becomes bad as n i n c r e a s e s .

Note t h a t an upper bound of t h e f i r s t e igenvalue in terms of t h e diam-

eter of . D , f o r genera l t r a n s p o r t o p e r a t o r s i n L’ spaces , i s

given by t h e au thor ( [ 1 2 ] , theorem 3).

Pmo$ 06 Rheokem 1 1 .-

i s given by theorem 10. With t h e s e estimates of An-,,

An- 1

H~ = f i ( A - ~)- l f i so H~ is c l e a r l y an a n a l y t i c f u n c t i o n

of A . Moreover, HA i s s e l f a d j o i n t f o r real h and K e r HA i s

independent of h > -A* (lemma 1 ) . These condi t ions imply t h a t :

- m ( * , D ) are a n a l y t i c wi th r e s p e c t t o X ,

- t h e r e e x i s t s an a n a l y t i c and normalized eigenvector c o r r e -

sponding t o t h e e igenvalue @(h,D).

(See [ I 5 1 theorem 3.9 and remark 3.11, page 3 9 2 ) .

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Thus w e may w r i t e :

(we have w r i t t e n &-,(A) i n s t ead of h ( h , D ) t o simp1Lfy the nota-

t i o n s .)

D i f f e r e n t i a t i n g (20) with r e spec t t o h w e g e t

Mult iplying (20’) by cp(n,X) ( f o r t he s c a l a r product i n Lz ) , w e

o b t a i n

t h e s e l f a d j o i n t n e s s proper ty of

It fol lows from ( 2 1 ) t h a t

[& ( A - T)- l & I = -& [ ( A - T ) - l I 2 tf? (because dHh d dh dh NOW - = -

The mean va lue theorem g ives

Now, the f a c t t h a t %(An) =

theorem.

1 and ( 2 2 ) imply t h e f i r s t p a r t of the

It is easy t o show t h a t ( h - T ) - I K and 5 have the same po in t

spectrum. I n p a r t i c u l a r p l ( h ) = IIH 1 1 of (A- T)-’K . On the o the r hand r [ ( A - T)-IK]

U

i s equal t o the s p e c t r a l r ad ius A lim 1 1 [ ( A - T ) - l I n ] ] lIn

n 3-

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and it is easy to prove, by induction, that with respect to h . Thus p l ( h ) = ~u[(A-T)-~K] decreases with re-

spect to h . In fact it decreases strictly. Indeed : let a < @ be such that pl(a) = PI(@) then p l ( h ) = p l ( a ) VA E [a,@] . Thus the real number pl(a) # 0 is an eigenvalue of HA VA E [a,BI which contradicts Gohberg’s theorem ( [ 1 1 theorem 11.4, page 258) because

lim IIHA/l = 0 . Thus pl(A) decreases strictly with respect to A , A + - so Al is characterized by the relation &(A7) = 1 . On the other hand, A1 is simple (for A) if and only if pl(A1) is simple for HA, , so A2 is the greatest h such that pz(A2) = 1 . Now the first part of the theorem is true for &(.> decreases. Q.E.D.

1 1 [ ( A - T)-lKInII decreases

n = 2 without assuming that

111. THE NON HOMOGENEOUS CASE

We consider in this section the general transport operator ( 1 ) .

The general ideas are the same as those of the homogeneous case. The

spectral problem (8) is still equivalent to (10) because K admits a square root. Now HA = &(A- T)-lh? is selfadjoint. Indeed :

because ~ ( v ) , & cp and Jl are even with respect to v a

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which shows that (5). We denote by %(A) E be the operator

HA is selfadjoint. It is also compact if we assume the nth positive eigenvalue of HA . Let

As f o r the homo eneous case, if lim I I A = H - ~ * 7&-!-) *(-" &) exists in the uniform topology.

E is bounded in L2(D x V) , then

A. -* -A* 7 47 -A m . _ . -

H-A* is then compact and selfadjoint. Let N be the number of eigenvalues exceeding one of H-A* . As f o r the homogeneous case, we have :

THEOREM 12.- 16 .Lh bounded i n L2(D x V) , A has, at Least N

e igenvdued. Zd 1 A not an &Lgl?nvdue 06 H-AL* , u (A) .h ~.hi..te. I d IlKlld < 1 then

P up(A) - 0 .

For the sequel we restrict ourselves to two kinds of non homogeneous scattering kernels :

(a) There exists a convex subdomain Do C D such that K(x,v,v') = Ko(v,v') VX E Do ;

(b) K(x,v,v') = a(x)Ko(v,v') ,

We begin with the case a). Let &(A) be the nth positive eigenvalue of the restriction of

HA to L2(Do x V) . Using the max-min principle, we get

i j , (A) 5 *(A) vn (23)

The properties of &(A) are known (from the previous section), so

the following theorem is a simple consequence of the results obtained in the homogeneous case.

THEOREM 13.- Under (a) the following hold

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We c o n s i d e r , now, t h e c a s e b)

L e t {Hn] b e a sequence o f H i f b e r t - S c h m i d t o p e r a t o r s approximating

& i n t h e uniform topology. L e t Hn(v,v ' ) be t h e k e r n e l of Hn . Thus fi = l i m &-, where fin i s t h e i n t e g r a l o p e r a t o r

n +a

On t h e o t h e r hand, ?h(h.- T ) - l & is a n i n t e g r a l o p e r a t o r i n L 2 ( D x V)

whose k e r n e l i s

= N:(x-x',v,v') , SO

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Now, arguing a s in t h e homogeneous case (see t h e arguments used t o

prove (15) ) , we e a s i l y ob ta in :

I n t h i s case, HA is a p o s i t i v e compact opera tor i n LZ(D x V ) . THEOREM 14.- Under ( 2 4 ) t h e following hold

Rhen a p ( A ) # 0 hegandeei~ 0 6 D and a(.) ;

2) l e X a(*) be constant and KO be bounded & L ~ ( V ) . LeX

p h o O 6 . - K(x,v,v') 2 &Ko(v,v') so t h e g r e a t e s t eigenvalue of T + K

i s bigger than t h a t of t h e homogeneous t r anspor t opera tor T + a o K o

[161. Thus 1) and 2) a r e s i m p l e consequences of t h e r e s u l t s obtained

i n the homogeneous case . The proof of 3) i s t h e same as t h a t of theo-

rem 7.

Remahk 16.- I f Ko(v,v') = Ko(u,u') where v = IvI and V ' = Iv'I

(and a(v) = a (v ) ) t h e theorems 5 and 6 may be extended t o t h e non

homogeneous case : K(x,v,v') = a(x)Ko(v,v') . We omit them.

I V . OPEN PROBLEMS

1) I n t h i s paper, t h e pos i t i veness of K i s very important. I f n

K(x,v,v') = C a. f . (x ,v ) f . (x ,v ' ) and {a..3 is a p o s i t i v e rna- i j = 1 i j 1 3 1 J

t r i x , then K is a p o s i t i v e opera tor . It i s an i n t e r e s t i n g problem

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to give general conditions on K(x,v,v') by which K is a positive operator.

2 ) The extension of our results to selfadjoint collision operators (without positiveness) will be useful.

3) The eveness property of the collision frequency and the scattering kernel with respect to each velocity variable was crucial in our analysis. In fact we cannot remove it completely. Indeed, let V be a ball and K(v,v') 3 0 if v or v' belongs to (some) half of V , then 0 (A) is always empty (even if K is a positive operator) weaker condition K(x,v,v') = K(x,-v,-v') .

P ( [ 1 2 ] , theorem 4 ) . A good problem would be to assume the

4 ) The study of the complex eigenvalues is still open. See 191 for partial results.

5)What might be the physical meaning of the positiveness of K ?

V. APPENDIX

Let us consider the isotropic case

K(v,v') = K(W,V')

( v = IVI 9 w' = Iv' I ) a(v) = u(v)

Let V = {v/ IvI < b

The spectral problem sphere of IR3 [ 7 ]

rh

I-] and I = ]O,b[ . (8) becomes, after averaging over the unit

(h+a(u)) Ix-x' I

= S . E L q q" E L2(D x I )

where S ( V , V ' ) = VK(V,V')W' . Thanks to ( 6 ) , S is positive in L2(I) , so (26 ) is equivalent to

where q = & .

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One verifies 171 that

where w = IwI . Let & ( k , D ) be the nth eigenvalue of the positive compact operator ih . A real number a is an eigenvalue of L2(D x V) ) if and only if it is an eigenvalue of On the other hand, similarly, (h-T)-lK and HA have the same point spectrum, so & ( h , D ) is nothing else but the nth eigenvalue h ( h , D ) of HA (in L 2 ( D x V) ) .

The curves &(A,D) decrease strictly with respect to h [ 7 ] , so

the number of eigenvalues of & ( * , D ) (or p n ( - , D ) ) crossing the line ;(A) I 1 .

( A - T)-'K (in - HA (in L 2 ( D x I ) ) .

A is exactly the number of curves

Let S be the operator

s is a bounded operator in L 2 ( I ) if and only if is bounded in L2(V) . PROPOSITION 1.- i i~ bounded .id and on ly .id a 5.inite wmbm 06

CWLVeh

Ph00d.-

exists

1 If 2 is not bounded, - will be unbounded, so there

6 E L 2 ( I ) such that 1141L = 1 and Jr

dw = +m r Id5 i ( w ) 1 2 W '(1) JI

Using the max-min principle, we have

where q(X,n) belong to the unit sphere of a (fixed) n-dimensional subspace of Lz(D) .

V O a(w) -A* is Note that 2 co (0 < c < +m) because a(w) - A*

bounded. Let h -B -h* . Fatou's lemma yields P

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l i m &(A ,D) 2 4n dwltj(n)(w)12 arctg cw q ( w ) ' 2 d = + m

hp+-A* JIR3 JI

where s ( n ) = l i m c ( A ,n) i n L2(R3) . Thus l i m &(A,D) = +m V n .

Conversely, i f 6 i s bounded, i s a l s o bounded, s o the number of

curves &(.,D) = p n ( . , D ) c ross ing the l i n e p (A) 1 i s equal t o

the number of eigenvalues exceeding one of

and Theorem 5 a r e proved.

Ptroud 06 ;theohem 6.-

I f rb Ib K(W,W')zdydy' < +m , then FA i s of H i lbe r t -Schmidt type

and

h+-A* P P -+=

H-A* . Thus Propos i t ion 1

(iA)z = 6 E S E & = & F 6 where h A A FA = E h S EA .

Jo J o

L (x , x ' ) 2dxdx ' .

Thus (HA)2 i s of H i l b e r t - Schmidt type and

Hence

f o r a l l A 2 -A* . In p a r t i c u l a r f o r h = -A* t h e number of eigen-

va lues &(-A*,D) exceeding one i s bounded above by C &(-h*,D)2 which proves t h e theorem. Q.E.D.

Remahk 17.- We have omitted the d e t a i l s i n the proof of theorem 5,

because t h e same kind of ana lys i s has been widely used i n t h e previous

sec t ions .

m

1

V I . FURTHER RESULTS

A t t h i s time of wr i t i ng , t he author r ea l i zed t h a t o ther i n t e r -

e s t i n g r e s u l t s may be proved.

of A , only i n the region {A/ A > -A*} . But, i f 0 $? , then

Indeed we have s tudied the eigenvalues

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a ( A ) reduces to isolated eigenvalues ; so we have to study the eigenvalues of A also in the region {A 5 -A*}. The collision operator K i s assumed to be positive and compact in

LZ(V) ; we denote by N ( 1 I N 5 a) the number of positive eigenvalues of K . THEOREM 15.- 16 0 B v , ,then f i e 2mnhppaJz.t opehato4, h a at L e a t N he& e & e n u d u a .

Phpaud.- We have seen, in the beginning of Section 111, that

3 = 6 ( A - T ) - I & is self-adjoint for

since -q = -W if 0 7 . Let n be a finite integer such that 1 < n s N . Let R be the radius of D , and S be a ball included in D whose radius is R . Let &(A) be the nth positive eigenvalue of 3 in Lz(D x V) . Using the max-min principle, one easily sees that

A > -n , i.e. for all real ?.

P n ( A ) 2 &(A) , where &(A) Let E = Ker K = {u E Lz(V) / Ku = 01 . It is clear that the dimension of the orthogonal subspace E' to E in L2(V) (for the scalar product of Lz(V)) is N . Let S, be a n-dimensional subspace of

( IlJlll E' . Let $(x) = - &Em mensional subspace of LZ(S x V), defined by

is the nth eigenvalue of HA in Lz(S x V) .

-.. = 1) , and Sn be the n-di-

LZ(S>

1

% = $ @ ~ n

Using the max-min principle, we may write

mes J sxv '0

where s(x,v) is defined by x-s(x,v)v E as f o r x E S .

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Thus

Letting A -OD, cp(h ) has a converging subsequence (because the P P P - m

unit sphere of Sn is compact). Let o be its limit. It is clear that 0 E Sn and lema in (28), it follows that

I/wl/Lz(v) = 1 , so & f 0 . Hence, using Fatou's

lim p n ( A ) = +m . Thus the transport operator has, at least, n real eigenvalues. Q.E.D.

THEOREM 16.- l&t &I r < 1 a n d f k t R be fithe h.acliu.h 06 D . We abwne f ia t V d bounded

A+--

be ,the nth po&L-Lh eigenvdue 0 6 K . LeA (Ivl 5 vmax< += .id v E V) .

Then any doLeLLtion X n 06 die e q d o n &(A) = 1 AUZ%&LU

Pm06.- Let us come back to (28). The n-dimensional subspace Sn we choose is the one spanned by the n first eigenfunctions of K . Let

be a ball concentric with S , and whose radius is TR . It is clear that

On the other hand

where amax = supo(.) .

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from which (29) follows. Q.E.D.

Remahh 18.- The previous theorems show that there is a close connec-

tion between the eigenvalues An of the transport operator, and

those of the collision operator K . Moreover, in the case where K has an infinity of eigenvalues, then (29) shows that the speed of

convergence of An to -m is controled by the speed of convergence to zero of the eigenvalues of the collision operator K . Remmk 19.- A question arises from theorem 15 : can we assert that

the transport operator has a finite number of eigenvalues if N is

finite (or equivalently if K is of finite rank) ?

V I I . CONCLUDING REMARKS

1) The reader may wonder whether the second part of ( 7 ) is

necessary. In fact it is used only two times in this paper,

as may be easily seen by reading carefully the proofs : in

(a part of) the proofs of th. 3 and th. 5. The author conjec- tures that the results remain true under the weaker condition

2) The spectrum of A is investigated, here, in three dimensional geometry. In fact, exactly the same techniques apply to the

n-dimensional geometry (n L 1) . 3) The boundedness of D is not necessary. Similar results may

be proved for unbounded D , provided that the additional assumption lim l l ~ ~ l l = 0 is made.

1x1 +m I: ( L 2 ( V ) ) X E D

4 ) However, the convexity of D seems essential, since it is

used in the derivation of the basic relation ( 1 5 ) (or (17)).

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% F i n a l l y , we s h o u l d notice t h a t t h e s e l f a d j o i n t n e s s of

d o e s n o t r e l y on t h e c o n v e x i t y of D ( c f . sec t ion 111).

1.

2.

3.

4.

5 .

6 .

7 .

8.

9 .

10.

1 1 .

1 2 .

1 3 .

14.

15.

16.

REFERENCES

Kaper , L e k k e r k e r k e r , Hejtmanek - S p e c t r a l methods 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 z u s e r V e r l a g (1982) .

J . Voigt - S p e c t r a l p r o p e r t i e s o f t h e n e u t r o n t r a n s p o r t e q u a t i o n , J . Math. Ana l . Appl . , Vol. 106, n o 1 (19E5) .

J . V o i g t - P o s i t i v i t y i n t i m e dependen t l i n e a r t r a n s p o r t t h e o r y , Acta App l i candae Mathematicae 2 , 311-331 (1984) .

Lehner and Wing - On t h e spec t rum o f an unsymmetric o p e r a t o r a r i s i n g i n t h e t r a n s p o r t t h e o r y of n e u t r o n s , Comm. P u r e Appl. Math. 8 (1955) .

J. Mika - T i m e dependent t r a n s p o r t i n p l a n e geomet ry , Nuc l ron ik 9 (1967) . A l b e r t o n i and Montagnini - On t h e spec t rum of n e u t r o n t r a n s p o r t e q u a t i o n i n f i n i t e b o d i e s , J . Math. Anal . Appl. 13 (1966) .

S . Ukai - E i g e n v a l u e s o f t h e n e u t r o n t r a n s p o r t o p e r a t o r f o r a homogeneous f i n i t e m o d e r a t o r , J. Math. Anal. Appl. 18 (1967) .

La r sen and Zwei fe l - On t h e spec t rum o f t h e l i n e a r t r a n s p o r t o p e r a t o r , J. Math. Phys. 15 (1974) .

M . Mokhtar-Kharroubi - Some remarks on t h e complex e i g e n v a l u e s o f t h e t r a n s p o r t o p e r a t o r , p r e p r i n t . M . Wing - An i n t r o d u c t i o n t o t r a n s p o r t t h e o r y , John Wiley and s o n s i n c . ( 1962) . Vidav - E x i s t e n c e and u n i q u e n e s s of non-nega t ive e i g e n f u n c t i o n s of t h e bolzmann o p e r a t o r , J . Math. Anal . App l . , 2 2 (1968) .

M . Mokhtar-Kharroubi - Some s p e c t r a l p r o p e r t i e s of t h e n e u t r o n t r a n s p o r t o p e r a t o r i n bounded g e o m e t r i e s . Submi t t ed t o T r a n s p . Th. and S t a t . Phys.

R iesz e t Nagy - Leqons d ' a n a l y s e f o n c t i o n n e l l e . Academiai Kiado, Budapest (1954) .

Vo-Khackhoan - D i s t r i b u t i o n s , a n a l y s e de F o u r i e r , o p g r a t e u r s aux d6 r ivGes p a r t i e l l e s , Tome 2 , V u i b e r t , P a r i s ( 1 9 7 2 ) .

Kato - P e r t u r b a t i o n t h e o r y f o r l i n e a r o p e r a t o r s . Sp r inge r -Ver l ag ( 1984) .

M . Mokhtar-Kharroubi - ThSse de 38me c y c l e , F 6 v r i e r 1983, Univer- s i t 6 d e P a r i s V I .

Received: September 15, 1986 Revised: J a n u a r y 20, 1987

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spectral theory of the neutron transport operator in bounded geometries

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