Spectral Analysis (C.I.M.E. Summer Schools, 64)

f(x)

5(0,s) s

P(s)f = .

J 2. 0

f

kj

kj

(t ) cPkJo(t,X)dt I

est UDe resolution de l'identite pour (-~ ). Nous avons ainsi demontre 1e Theor-

L'operateur (_

Ll )

avec domain It 2

(R

n)

admet une

diagona1isation canonique

definie par

(WOk (t ) = j

lim

s-gO

J

f(x) CfkJo(t,x)dx

5(0,s)

ou

sont des fonetions propres generalise.! pour (-I!:.) qui sont

chaque

f

e

2

n

L (R ,

I I) s

f(x) =

lim

J

s~""O

nous avons 1a synche se suivante

- 9 -

G. Bottaro

3.

Le probleme que nous etudions maintenant est celui de construire l'operateur qui diagonalise

-1:1

dans le cas ou le domain de

l'operateur n'est pas constitul de l'espace

Rn

fonctions definies sur tout

, mais seulement sur une partie

[2].

Nous etudierons

dans un premier temps quelques cas particuliers : a)

Pour le laplacien ayant comme domain

~

ox

f (x , •••••• ,x

n

1

n-

l' O)=O}ou Q =J(x •••• x )€ Rn:x > 01 (1 n n J

la transformee de Fourier est bien remplacee par l'operateur unitaire suivante

oU

si

f f C oP (.1L) ou o

b)

D'une

fa~on

1

f(x) F(x. ~ )dx 11. n-l -i Z" x A j=l j j cos x ~ F(x. ~ ) = e

(Uf)( ).) =

n

n

analogue. si le laplacien a comme ensemble de

definition

r-empLacon s L' ope ra t eu r de Fourier par

2

U : L (Q) _

2

n-l )( M)

L (R

- 10 -

G. Bottaro

0\1

t e c" (0.) et

si

o

n-1 mTT

1\

T)

F(x'~i'··· .... n-1' c)

=

Lx.:X j

-i

.12

V;

j=l

e

J

sin

mlT

-£-

x

n

Enfin nous signa Ions l'exemple suivant : Ie laplacien ayant comme

domaine

! fEH 2 (U)

: f = 0

2 = [XER : lx/=a

a

n .Q = R ,L(x , .. ·x ) f Rn : 1 n Nous

n

sur S

Ixl~

a

f

]

0\1

1.

definissons

f Eo cO> (Do) o

en posant pour

(Uf)( A) =

J

f(x) F(x, ;l )dx

!l

0\1

F(x,

A )=

e

( a

J y et

H

11

i(x,~ )

e)H

+

!4k-1 2

~

v(x, A,

(Ixl e)

0 1: );

v(x,

1, C)

=

(0\1

etant des fonctions de Bessel de premiere et t rof s i eme

- 11 -

G. Bottaro

espece.

II est interessant de noter les conditions qui sont verifiees

par

A, e)

V(x,

general.

parce qu'elles seront caracteristiques du probleme

Elles sont:

r, A )

~, ~ )

v(x,

e2 v(x, ~ , p)

=

si

xeS 2

a v(x, ).,

l}./ )

) Ixl

-

i\~lv(x, A, 1'>./ )

= o<JxJ (l-n)/2)

La derniere est connue comme "condition de Sommerfeld", ou de

radiation.

Ces

relations peuvent @tre prouvees par l'utilisation

de techniques elementaires de fonctions holomorphes et de l' expression asymptotique suivante des fonctions de Bessel de troisieme espece H

"

+

(z) = "2-)\ e i{z-\lI7T -1/4Tf)(1

CTT

0(_1_) •

z

I zl

II serait interessant de donner une idee du

Theor.

U est une diagonalisation pour (-

Si l'on suppose pour un moment que f '6

c'OO (!l),

f = 0

o

sur

S

a

(U(-ll )f)( >.)

= lAI 2

J -

f(x)F(x,A ) dx

~

=

sur Le domaine, donne.

U est unitaire on a pour chaque

2 :

=J (_ll) f(x) .n.

/1)

F(x,:\ )dx

\~ 2 (Uf)(

A)

=

=

[f(x)(-l1 )F(x, A)dx n,

(c-ar F(x, A )

=0

sur S 2)' a

- 12 -

G. Bottaro

Pour demontrer que U est une isometr1e(la demonstration est prise en pa! gEe

t1e ians [6])on note que s1

~2

alors (s1

r+1

=

R :ig(x) = (21l")

""

o

-

(.11)

E.. > 0)

(t

n/2

t

On dedu1t que s1

1t R

n,

i2

¢( ~,

f E: C""'(n)

o

f(x) g(x)dx -.

r

1

)n

R

e)

=C271

)-n/2j f(x) [ei(x,A) + v(x,~,

e»)dx,

donc

c$(~.e)

IAJ

2

-

P

Nous ut11isons alors 1'1dentite de Parceval et la premiere identite de la Tesolvante s1

0 <:

ot

<

f' <

~ -e

00

et si

E

est une resolution de

l'1dentite on a

51 nous faisons Ie passage

a la

lim1te, par des techniques d '

- 13 -

G. Bottaro

int~grales &~gulieres

1 -

-2l«EIl.+E Il

,..-0

f"

Le resultat et

~

4.

~tendons

)f,f)-«Eo{+E

J

"'-0

)f,O] =

suit · a ler s en faisant

--':> 00

techniques

on a

0< ~

j c<~I~IZ~~

P

et apre5

IUf(\)1 tJI. ~

2

d

~

•

0

La surjectlvite de l'operateur e st; prouvee par des

•

semblabtes et que nous exposerons dans le cas general.

ce que nous avons dit pour le laplacien

a

l'ext~rieur

1a sphere au cas d'un operateur quelcGnque autoadjoint elliptique

de

a

coefficients constants pour ce qui regarde le probleme exterieur [3] •

On peut demontrer le Theor.

Sin est un domaine

exterieur de front1ere 5 une fois

differentiable,compac~et

M=-L.a ij

lJ

2

~XjOXj

M est l'operateur

, (a

ij

= a j i)

et si on appelle

solution du probleme suivant (M _

v(x,

~2) v(x, ')" , l )

'\.~ V p( A ))

A,

dans

= 0

= -e i(x,'A

a10rs un operateur diagonalisant

)

n. si

x (:- 5

v

la

- 14 -

G. Bottaro

1 I (2 11 ) n

est defini pour les f (: cf"(ii,) par o

'\ 1

(Uf) ( ,,) =

~ f(x)F(x, A) dx

:n.

OU

'I ., vr----:-F(x, ,,) =e i(x,A) +v(x,;\, P (~ »

On a ainsi l'extension naturelle de ce que nous avons dit pour le laplacian

a

l'exterieur de la sphere; la condition de radiati~n peut

d

s' ~crire en utilisant la derivee conormale indiquee par

J Ix)

et la ' fenction r(x) qui est le produit scalaire x . (j";j" ,y) ou y est le point de la variete d'e'quation

p(-1) =r-

ou la normaL.exterieure a la mteme direction et orientation que x. Vu que dans la demonstration du caractere isom~trique du diagonalisant du laplacien nous avons utilise seulement les proprietes qui sont satisfaites par

v

la demonstration peut etre generalisee sans '

aucune difficulte. Nous donnons maintenant quelques idees pour la demonstration de

a

la surjectivite de

U , c' est

g().)=UU* g().).

Sil'onpose

2 n dire du fait que pour chaque g c= L (R )

*

r(~) =g(-\)-UU

g(:I) onpeut

v

definir Le s deux fonctions

J

pn)

et S (x, m

e)

s

J

pP) ~

2 m

r(A) F(x, >.) d

~

r(J) F(x,.l) 2 m

p(A)_ e

2

dA

- 15 -

G. Bottaro qui verifient les equations (M-

On a

t

2

(x,~)

) S

S (x, m

m

t)

-7'

= s (x) dansn;

m

0

en

S (x,f) = 0 m

si

s ---+ U*r = O. m

2 L , parce que

En utilisant encore des techniques d' integrales singulieres on prouve que pour chaque

J

O<.· c«p:

r(A)F(x,':\)dA=0, donc

o(~p(~)~~

J pO) =k

La fonction

rO)F(x,:\)d'A=O

p.p.enk.

2 1

u(x,k)

J

ei(x,A)rn)d)

.

pO) = k

verifie l'equation

2 (M - k )u = 0

2

et la condition de radiation.

Pour le laplacien en dimension 3 il existe un theoreme bien connu de Rellich la

(5]

qui assurej_aintenant,que

transformee de Fourier de

u

est nulle, et alors

r , et donc r, est nulle.

Le

theoreme de Rellich a ete generalise par Vainberg [7J , qui utilise pour la demonstration une solution fondamentale particuliere de 1 'operateur.

5. Nous en avons donne

[3J

une nouvelle demonstration, un peu plus

elementaire, qui nous a permis de prouver non seulement que l'equation 2 Mu - k u = 0, quand

u

verifie la condition de Sommerfeld,a seulement

- 16 -

G. Bottaro

la solution zero en tout l'espace, mais aussd , plus

. generalement

Ie Theor.

La solution du pzob Ieme exterieur (M - k

2)u

= 0

dans Q

est unique, si elle satisfait la condition de radiation et si ses valeurs ou les valeurs de sa derivee conormak sont

a

donnees

la frontiere.

Dans un premier temps on peut demontrer Theor.

Si

u

est une solution de l "quation

J

alors max lim R-7>,o ou

lui

2

Mu - k u

2 dQ"

1=

O.

si

u

1=

=0

dansll

0;

~I(y,R)

I(y,R) est une famBl.e convenable dlell~soidescentres en

Une telle relation est demontree en utilisant la formule de Green par des techniques semblables

a

celIe que l'onutilise en general pour

demontrer Ie theoreme de Stokes et par l'utllisation de deux solutions fondamentales de

J y et

e sp ece ,

M_ k

2

Ny etant des fonctions de Bessel de premiere et seconde A Ie determinant des

domain exterieur)

a

x

a i j• et

f

la distance de

y (point du

dans une metrique particuliere liee

a

- 17 -

G. Bottaro l'operateur differentiel.

Si

S

est 1a frontiere du domaine

prouve les deux egalites

D'apres une formule qui donne le Wronskien des fonctions de Bessel, on obtient les faits suivants: ou

max lim R->PO

~

luj2

d

~ f

0

.) I(y,R)

ou les seconds membres des formules que nous avons

ecrite~9

sont nuls

pour tout Y» Dans ee dernier cas on doit prouver que

u = 0 , ce qui est obtenu en

tenant oompte. de l'expression asymptotique des fonctions de Bessel, ce la nous p ermat; de montrer,lorsqueon a complexifie la variable radiale qui est dans les expressions ecrites, le caractere holomorphe au voisinage de 1'.:0 des deux fonctions

u(y) . 1- ( n/ 2) + (klyj) , ce

I YI qui

est

absurde s i

ufO.

e-

L'analyse est rendue plus

difficile, si la dimension de l'espace est pair, car dans ce cas les fonctions de Bessel sont multivoques. meme dans ce cas,l'expression de

u

Alors nous devons prouver que, est univoque. Le theoreme est

- 18 -

G. Bottaro enfin demontre si lIon prouve par un calcul facile, Ie L~~.

Si la fonction

2

u , telle que (M - k )u = 0

dans'o',ou sa

derivee conormale sont nulles sur la frontiere et

s~

la

condition de radiation est verifiee on a

J

l~

R -'!>"'"

~

\u1 2

d 6" = 0 •

I(y,R)

Comme on a dit plusieures fois la diagonalisation non canonique est rendue canonique par l'idee de G~rding.

6.

II ne semble pas inutile remarquer que, de cela, on peut d~duire pour un operateur

A = P

(~,

•••

~)( 0\1

p

est un

polynSme reel elliptique) Ie theoreme bien connu de Balslev [1J ' qui dit que 6"(A) = R(p). On a

va

que l' int~gra]e ddz e cte , image de la diagonalisation canont.que,

a comme domain l'image du polyn8me caracteristique. Alors il n'est pas difficile de prouver Ie Theo;,.

Itant donne un operateur autoadjoint

dans un espace de Hilbert U : H' ~

1:1\

HO) d

A, avec domairw dense

H, et

~

etant la dfagona H s at.fon canonique (A ferme de Neumann, alors A = 6"(A).

cr )

de Von

- 19 -

G. . Bollaro Cette remarque permet aussi de trouver nous avons et u die .

I, ( A)

dans tous les cas que

- 20 -

G. Bottaro

BIB L lOG RAP HIE 1

E. Balslev.

The essential spectrum of the elliptic differential n). operators in LP(R - Trans. Amer. Math. Soc. 116( 1965).

2

G.F. Bottaro. Alcuni risultati di analisi spettrale per l' operatore di Lap 1ace su insiemi non limitati (in corso di pubbli cazione).

3

G.F. Bottaro. Alcuni r isultati di analisi spettrale per operatori di fferenzi ali a coefficienti costanti su insiemi non limitati • (In corso di pubblicazione ).

4

• L. Garding.

Eigenfunction expansion. - Pg. 303-352 de Bers John

Schechte;~artial differential equationstllnterscience New York (1964). 5

F.Rellich.

Uber der asymptotichen Verhalten der honningen von

Au + ~

u

= O.

In - Unendlichen Gebalten Jahr

Deutsche Math. Verein. 53. (1943) page 57. 6

N.A. Shenk

Eigenfunction expansion and scattering theory f or the wave equation in an exterior region. - Arch. Rat. Mech Anal. 21 (1966) pg. 120-160.

7

B.R.Vainberg. Principles of radiation, limit absorption and limit amplitude in the general theory of partial differential equations. - Russian Math. Surve ys, 21 n.3 ( 1966 ) pg.115-193.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C .I.M.E.)

EIGENFUNCTION EXPANSIONS

o

LARS GARDING

Corso

tenuto

a

Varenna

dal

24

agosto

al

2

settembre

1973

Preface. The

titl~

of these lecture s cover an awful lot of ground.

They oould include everything in the convex hull of Pourier series, spectral theory in Hilbert s pace

clas sic ~

and the ser ie s

expansions of classical physics, group repr es entations and quantum meohanios. I have chosen to give a general introduction and then treat the Schrodinger operator and the summation theory

ot eigenfunction expansions of selfadjoint elliptic opcratGrs in more detail. Lars

G~rding

- 24 -

L.

G~rf;ling

I. General eigenfunction expans i ons . Introduct ion.

We s hall gi v e a l eisurely expo s i t i nn of the s pe c-

tral theorem and the t h e or y of general eigenfun ction expansi ons . There will be practioally no pro of s and few references. Th e su bject is well covered in the books by Ber ezanskii (19 65 ) and Maur in

1,. Funotional analysis dict i onary.

To fix th e t erminology and

give an id ea of the prerequisites we s h all s t art by revi ewing t he basio concepts of functi on al analysis. Linear

~paoos.

A seminorm in a l i n ear s pa c e _L • {f. g •... } wi t h

scalars C is a non negative fun c t i on f-. p(f) whi ch is homogehe ous so that p(af) = lalp(i) for scalars a an d s at i sf i e s the t ri an gl e inequality. It is a norm i f p (r )=0 => f=O. A coll ection aeminonna mak es L a l i n e ar

topolog i~

(p} of

s pa c e wh er e t he neighbour-

hooda of a point f are defined by a finite numbe r of equa ti ons p(g-f)
p~and

{ql

of semi n onns give the s ame t opol oglY

(open sets) if and only if t h ey are equivalent in t he s en s e t h a t every p is •< a pos itive line ar combinat ion of the q's an d

17. A sequence

{fn lin ia aaid t o t end to zero,{fn)"O

conver se~

i f n..,_=>

p(;f'n)~O for every p, it haa :!Ih:eiiUJIlit f if fn-f.O and it i s a Cauchy sequence if f -f :+ 0 when n and m"C'O. The s pa ce L is said n m to be(sequentially)complete i f every Cauchy s equ en ce has a unique

,

.

limit. A Frechet apace is, by definition, a c ompl et e linear s pac e collection of aeminorms. A Banach s pa oe i s a completespace with just one nonn. -

- 25 -

L. Gdrding

-1

Let x= (x , ••• ,x ) be co or di n a t es in I n ,~~~ the 1 n

Examples.

en· ,~=<~ _A J 1 •••~ ...1n

gradient and xe( = %1 ••• x

spaoe ~

n

.

m onom ~a1s .

co nsis ts of all C" func t ion s f such that

The Schwar t t

supl x~(x ) l <

OOfor all~and ~. Provi de d wi th the c orresponding s emi norms i t I

is a Fr ech et s pa c e . I ts mai n virtue i s that the Fourier transf orm

ar:JP~~iS

a con tinuous linear bij ection with a c ont inuou s in-

verse. All bounded con t i nu ous f unct i ons f f r om X

= lin

t o m. con fl t i -

tute a Banach s pa c e wi t h t h e norm cu p /f (x ) / . When dfl f 0 is a me~' sure on X we put I lfll p = (

If (x) IPdr( x))1 / P

< < Ge an d f i s a Bai r e f un ct i on . The s pa c e of' all Bair e func wh en 1=p= t i ons f su ch that

Ilf II p <

Ban a ch s pac e wi t h norm I If Hi l ber t s pa c es .

(RJ

II

n odul o those with I lf II p =0 i s a

whi ch we ~~l l deno te by L P (X , ~ .

A Hi lbert s pa c e i s a Banach s pa c e H wi th a n orm. .

IIfll def ined by a s cal ar produ13t (f ,g ) with t he usual lin earity pro perties

su ch that

I If II

= (f ,f) 1/2 • Th e prime example i s

the s cal a r prd odu e t

Ix f(~)mr fd . Her e f and g a re c on pl ex value d bu t t h zy c ould alc o h av e v al u es i n a Hi l ber t s pace h v:i t h sc a'l ar- pr odu c t ab

, e .G . th e a pa c e

')

l~

of. s eque nce s a=( a , :O-2' ••• ) wi t h " components (V = OO i nc l u ded ) 1 2 2 VIi t h Ia 1 = rl ~ 1 < 00 and ab = ~~k' More gen er ally , l et

l.

'J (d t ~~i ng

be a mul ~ ipl i cityffun ct i 'J n on X , i .e . a IJai r e f un ct ion the

Ta1~s

1, 2, ••• an d OCand, cons i der func t ions '

l7i th Bai re comjion errt s cuc h theo.t

- 26 -

L . Gar-ding

Take them m~dulo tho se for whl ch the int eg ral vani~ h es ~nd u s e the scalar product (1). Then we ge t a Hilbert ~ p ace de note d by

L2eXtJ'~ which we shall call a s pec t r al s pac e for sh or t bec au se it occurs in the spectral t heor em below. Put t i ng

H~

2

=l~ ~ )

we

can write

as a kind of dire ct sum of the s eque nc e 'space s

H~.

In all this we can replace X by a loc ally compact s pa c e exhausted by ·a countable union of compact s e ts an d we shal l use t he term spectral s pa ce also then. The Hilbert spaces considered s o are separ a bl e , i.e. the y have dense countable subsets and we shal l c on sid er no others. A collection

~eJ

of unit vectors su ch tha t .(;' p => eoL,J. e"

.O_.'i s said to· be an orthonormal set.

i.e. (eJ..,e/l )

Orthonormal s et s in s e pa r a bl!e

spaoe. are countable and every maximal one i s al so a basis, i.e. eTerr t is the sum

let,,,)e~ where

Duals. Distributions,

The dual L'

1: lef,~12

=If''.''~ of

< ...

a linear to polo-

gioal space consists of all linear continu ous funct i ons

f~ < f',f>

f~c tion/

t ... L to II. When L is a Hilbert space, an y su ch -. is a s c a l ar product, t. .ef,g), for some g

L. The du al':!' of

.Y'

is called

the space of tempered dist r i bu tions. Any lo c ally int egrabl e f uncN) tion tex) on ~ whi ch is Qelxl f or s ome N as x~ .. gi ve s rise to the' tempered distribution g-. jifeX)geX)dX • In can be oonsidered to be a part of

~I.

particular,~

In some instanc e s we s ha l l

use the el ements of di stribution t h eory . The du al B' of a Ban ach

- 27 -

L.

space B with norm

II f" II

111' 11

G~rding

is a Banach space with norm

= sup 1<1", s» I

when

I Il' I I<1.

When B is a Hilbert space and =(f,g) then 11f t I/=llgll and one may identify B' wi t h

B via the identificat ion of 1"

and g.

The dual L' of a general space L can also be pr ov i ded wi t h s emi norms in various ways but we do not go into that. Linear operators.

Let TIL1~L2 be a linear operator ( f unc t i on )

from one linear space to another. The kernel of T, Ker T, con s i s t s of all l' in L 1 with Tf=O and ~L1 is called the range R(T) of T. Both are line ar s pac es . When L,=L 2 and

E:L~L,

i s t he i dentity

operator and Ker(T-zE)~O we say t hat its el ement s are e i ge nf'un ct ions of T and z. a: i s an eigenval u e of T. ',/' h en z if:; no t an ei genv alue, the r e s olvent (T_zE)-1 i s a line ar operator f rom R( T- zE) to L 1•

~h e n

L

1

and L2 are Ban a ch s pa ces then

~L,"L2

is

continu o~

if and only i f -Del > l iT I I = s up I ITf 112 ~h e r e

wh en I If 11

1

~

1

the ind ice s i ndic a t e 'no rms in B 1 and B • Such op 8r ato r s 2

ar e als o a ai d to be bounded.

According t o a theorem by Bana ch,

a b ijection between Banach sp aces is bounded both ways if it i . bound ed one way . When II 1'fl12 = IIfl11 e~ d

f or a l l

f , T i s bounded

s ai d to be i sometric. A unit ary opera tor i f:; a · line ar i sometric

bij ec t ion between Hilb ert s pa c es .

~

linear opera.tor T be t ween

Ban a ch apa e e c Ls s ni d t o be compa c t i f it map e bonnded s eque nce s i n to s equenc e s wi t h Gon e conv ergen t su bs equ enc e .

~he n

n is a

E a~a c h

apa ce and 1~' : ~B bounded linear, t he Ge t of su ch z 't"'e t ( T- zl! )- ' Lc con t Lnuouc Lc r-n op en se t in a:

c ~ ·. l ' c a.

til'. r-e r oLv en t c o t o=- .T .

- 28 -

L. Gardi.t1g Its complement is compact and called the spec*rum of T. When T is compact, its spec trum

is discrete outside the origin whlre

it consists of eigenvalues z of finite multiplicity dim Ker(T-zE). In particular we have the Fredholm alternative: z';'O is not an eigan1 . value <=) z is not in the resolvent set <=) (T-zE)- is bounded.

t e~

When B1 and B are Hilbert spaces, the sum lTI 2

2

=1[IITekI12 where

is a complete ( maximal) orthonormal s et , is independent

of this set. Operators with /T/< e-are said to be Hilbert-Schmi4~ 2 2 operators. They are comptet. If B1 = L (X1"1)' B2=L (X then 2,J'2) T is Hilbert-Schmidt if and only if (Tf)(x) = jK(X,y)f(y)dfl1(y) where

JJ IK(x,y) 12d/'2 (X)dft (y)

< po.

We say that T has a

Hilbert-Schmidt kerner. Adjoints.

When B"B 2 are Banach spaceSt any con tinuous

h as a bounded adjoint T':B 2'. . B,'

defined by

T :B,~B2

=

where f is in B, and g in B2'. When B" B2 are Hilbert s pa c es , the Hilbert adjo int T~ map T.T~

:B2~B,

is defined by (Tf,g)=(f,T~g). The

is antilinear and we have TH* = T. The main point abou~

this adjoint i s that R(T)

.l

Ker T

;where!, means orthogon al to. I's ome t r-Lc maps T : B ,.~B 2 ar e characte.rized by

T"

= E,

and unitary ones by T~T=E1' TT~ = E2

E, and E2 are the identities on B, and B2• In fact, if

I Ifj I~

where

I ITfl I~ H

for all f, then (Tf,Tg)2 = (f,g)1 for all f,g. When T = 2 T , T is said to be se l f ad j oi nt and we have I !(T-zE)fl 1 I I ( T- Rezr) 11 2 +(Imz) 21If 11 2 s o that its c,pect rum i s on the real 2=p axis. Operators P su ch that p and r= p ar-e c all ed (ortho c on al )

- 29 -

L. Gl\rding

projectiob8 and

h~e

the char a ct er i zi ng pr oper t y t hat the Hi l ber t

space H they operate on i s t he di r e c t sum

H1~

H2 of H1=PH and H2=

(E-P)H and that the se s pac es are orthogonal. For this on e ~rit es H=H 1 • H 2

and calls H1 and H orthogonal 2

comp leme ttt s o ~ n e abh,

1

other. More generally, if S is a part ?f H, the s et S

of all f

orthogonal to all g in S is a closed linear su bs pac e call ed the orthogonal complement of S. By Hahn-Banachs theorem,

~

i s t he

closed linear span of S. 2. Symmetric and selfadjoint unbounded

op er a~s.

relations in a Hilbert space H, i.e. linear '

I

ranges of t h e projec tions S. f . f 2 .... f 1 1

Consider linear

sub s e t s · S ~ o f2 H.H .

The

and f 2 are call ed the '

domain D(S) and the range R(S) of S re s pectively. By S- 1 i s mean t the set of pairs f

2.f 1

with

f1.f2~ S.

Wh en f 1=O => f 1. f 2=O ,

5 is called a function and we write f 2=Sf 1 for f1.f2cs. The subspace of D(5) defined by Sf=O is then called the kernel of S and 1 is denoted by Ker S. Clearly, Ker 5 = 0 <= > 5- is a function. The closure of 5 will be denoted by

[5].

When S is a closed function

, then D(5)aH <=> S is bounded ( special case of Banach's t h eor em) . The adjoint S· of S consists of all g1.g 2 such that

f1~2.S => (f1,g2)=i~2,g1) • This extends the definition of adjoint for bounded operators and

l.. of

means that all g2--g1 constitute the orthogonal compl ement S S in n.H and hence SlIBI = S U that

is the closure of S. It is cle ar

(5. 1). = (S·)-1 and that S·

is a funct ion <=> D(S) is de ns e

in H. Hence the adjoint of a closed densely defined funct i on is also a closed densely defined function. Densely defined linear

- 30 -

L . G~rding

linear functions wi l l be call ed lin ear oper ators for shor t . A lin ear operator A i s said to be symme t r i c i f AC:A~,

Definition

selfadjoint if A=AH and essentially s el f adj oi n t i f its closure i s selfadjoint • .That A is symmetric means' that (Af,f) is r eal f or all f in D(A) and when A,E are s ymmet r i c we s ay that A

f

B i f (Af,f)~ (Bf,f)

fo r all f in D(A)() D( B). Example. Diagonal operators. Let A be mul tipl i ca t i on by a Baire f unc t i on a(1;; )on a sp ectral space H

2

D

L

(X.)!. V)

" " . for '-almost all 1;. Then A wi t h D(A)

and assume t ha t \a ( l; ) I

{1;afwH}

closed and dens el y defined an d A* is mul ti plic a t i on by H

D(A

is

arrr

on

= D(A). We say that A i s a diagon al operator . When A=A~

)

and B=BH

ar e suc h op erat ors , A ~ ~B mean s that a(I;)~b(l;) al mos t

everywhere with re s pe ct toJ4 . Example.

If A i s clo sed an d densely de f i ne d , D(A) i s a Hilb er t

s pac e wi t h the s cal ar produ c t (( f ,g)) = (f, g ) +(Af, Ag ) and he n c e the re i s an 3: H~ D(A ) su ch that (f , e)=(( Sf ,g ) ) f or all f,H, gED(A). Cl early E S

-1

f

3

f

0, 3=3* , (E+A!EA)S=E and h ence

A~A =

- E i s selfadjo i n t wi t h domain D(A) .

Example. Diff erenti al opera tors. positiv e dens i t y '

an d II = L

2

Let Ab e a C ~ m::mifo ld Viit h a

(Jl,! )

onli. squar e integrable with re spect t o x so t ha t p

=t

f

,.ax

t he s pac e ,of compl ex funct ions

r . If

w
i:; Lebe agu e mca su r-e , a diff eren ti:>.l ope r2ltor

\c (x )r)iL , D

= ~/i ) x

wi th smo oth coeffi ci ent s and i t s f ornal ad j oint

I~

=r:r

!:l. oLJx)

- 31 -'

L. Gltrding

(Pf,g)=f,~g) when f and ~ are

then have the property that

sufficiently smooth and their p~oduct has c ompact Gu~po rt . To e et OIl

operators out of this, put e.g. A=P wi t h D(A) = C o~. Then A is

P-

densely defined and AW =

W

wi t h D(A ) = all f i n H su ch t hat

P-f belongs to H when taken in the s en s e of ditributions . This st11l does ·not say much about AW but whe n P i s elli pt ic in the sense that its characteristic polynomial p(x,d =

~ifi al4(x)~.(

has constant degree m and it s pr i n c i pal part Pm (x,d =

L:.

a~xk·

Itt/=m never vanishes wh en ~ Io is real, t he s i t ua t i on i s simple r . Then by one version of Weyl's lemma,

Pu.1t° (=>

u ~

'St m

where ~k is t~e space of distribu ti ons wh os e de rivative s of order ~k are locally square integrable. Applying t hi s to ~ we see that n(AW ) c'1(m • Hence in this case we a t least know t hat the elements of n(AW ) look locally. Wh en

.Jt

i s compact, ~=H,

and tlds solves the problem, but if Jl is not compact we have failed to oharacterize how the elements of n (AW) behave far away. To this orucial question there is also no general answer, but special oasee, e.g.JQL- ~

and P=D 1

2+

••• +dn

2 + V(x) ar e of

n considerable interest in quantum mechanic s. Finally, ifJr.l= R and dx is Lebesgue measure and P(x,D)=p(n) has constant coef f i eients

Je-ix~f(x)dX

cients all questions are answered -by the Fourier transform

lJ.f(~)

.. (2lt)-n!2

- 32 -

L. Gc'l:rding which gives a unitary map L

2(](n)""L2(](n).

In

fact,a-1p(D)~ =

pee) is then diagonal and P(D)f is square integrable if and only if p(e)dr-' f is square integrable. The theory of symmetric operators depends on the Lemma.

If ACA

w

follow~ng

and A is closed, then

D(AW) = D(A)~ Ker(A~-iE)lKer(A*+iE) wh e r e the sum is direct. Proof. Clear that

0

write (AW+iE)f = 2if

and +

H = Ker(A*-iE)&R(A+iE). Vlh en fE D(A*)

+(A+iE)f

0

in this decomposi tion. Can be

wr i tten as (A* +iE )(f-f -f )=0, i. e. f=f +f +f , f E Ker(A* +iE7. + 0 0 + If f =O, then f+ +f_'ii D(A) s o that (A-iE)(f++fJ=-2i=!:,_ J..Ker (A* +iE) => f =0 s o t h a t f =f =~ and f =0. Hence the sum i s direct. + 0 The lemma h as the follo wing corollary wh i ch we s h a l l use

wi th ou ~

reference: if A~A* i s densely def i ne d end cl os ed , t he c onditions

a r e equivalent. If A

A*

i s densely d efined, the c ondit i ons . in H (i) A is e s se self adj o int ( ii) Ker ( A~~iE ) =O (ii i) R ( A~iE ) d en s e /

a r e eq uival ent. In all thi s vie 'may of course r eplace i by ~i 'f/here

~ 3.

10 ~he

i s any r eal number. s pe c tr al theorem. A s e t

S =

~ A}Of

c omnu t i.ng ope r a t or s on

a Hil bert s pa c e H i s s a f.d t o be di f'. gonalizable i f t ho r i s a s po c t ... re.l s pa c e L

2

~J - 2 -1 2 = L (X,j4.Y) ~nd a unitary map U : H-? L such that UAU

is d LagoneL for every A in S. Whe n 3 ccn c a.c uc of jUl: t we h ave the ::r e c t r al ( i)

OYIO

op er:>.tol,

the or em of Hi l be r t and von II:eu..T1ann :

A=A~ => A d Lagona'Lf.z ab'l e pi t h Jed l , UAU-1=e •

Here e i s the co ocd inr.te of R. There i o a l e o a unici ty re sult:

- 33 -

L. G~rding any two spectral s pa c es that achieve this hav e equavo.Lon t measuresjt.and their mutipli city functions Yare equal alm ost everywhere. The th eorem i s usuaaly given in a weak er form symbolized by the formula

(ii)

A=

J1. dE)'

called the spectral resol ution of A. Her e the

E~

ar e c ommu t i ng

orthogonal projections in H uniquely det ermin ed by A such t hat

).1'

=>E~. E,. , .\.... => E,,-.E, ~ ... --=>

E,.""'" 0

and t he formul a

should be interpreted so that fED(A), g6 H => (U,g) The func t ions • ., (E....f,g) are of bounded variat i on over the real axis. The conne~tion with (i) i s given by the formula -1

Eli = U

O~

U

wher~ ~~(~)=1 when ~~~ and 0 otherwise. Bi t her f ormula (i ) or (ii) allows us to define a commutative ring of, e.g., bounded and continuous functions 1'(A) defined by

~(A)

=

u-~du

or (%(A)f,g) = J~())d(Elof,g) • There is also e.g. a version of the s pec t r al theorem dealing with a norm closed commutative ring S of bounded op erators A such that 1 Sa = S. It is diagonalizable with X = Sp S, UAU- = a(~). Here Sp S is the Gelfand spectrum of S represented by all homomorphisms S).A ..~(A)E4.. with IdA)!

logy making the a(~)

~ IIAII provided with the weakest topo-

= ~(A)

continuous.

- 34 -

L . Garding

4. Eigenfunction expans ions.

Consider (i ) 'an d ass ume t h at}4i s

a disorete mea sur e co n cebtr'at e4 t o a dis cre t e se t { I;} • Then , if gj(~) .i S t h e jl t h c ompone nt of g(I; ) , t he expansion g

='t F

g j(l;) e jl; ,

defines a basis{ ejl;} of

1~j~V ( I;)

2 L (X,f' 't/) with paiI'\'1i se orthogonal

elements. In particular f 'H

' ~f

=

-,.=-

t;j

Uf j (I; )U- \ jI;

so that the value s Uf j(l;) of Uf are s i mpl y t he co ll ec ti on of

s e~je "1 J, (I; )=

ooefficients of f relat iv e t o the comple te ort hogon . al

u-1ej~lin

H of eigen f u nc t i on s of A. Now in c ase of a ge ne r al mea-

sure, the Uf j (I;) still ex i s t almo st -ev e r ywh er-e , but the ei genfunot ions are miss i ng. The exampl e A =d/ i dx an d U t he Fouri er transform, wh er e the ei genf unc tions i f an y t h ing so uld be the exponentials .iXI;, indic at es that we shou ld l ook fo r the mi ssing eigenfunctions ou t side t h e

Hilbe~t

s pac e H bu t al s o in s ome way

conneoted wi t h it. Once this thought has caugh t on , t he r e are several ways open t o

co nst~ct

e i genf unc t i ons .

Suppose for i ns t anc e t ha t G i s a dens e linear subspac e of H and that G i s itself a Hi lbert s pa ce wi t h a dif fe r ent s c al ar pr o-

duct such that t h e inj e ct i on Then i t

~j

gCG

G~H

is a

Hi l b~ rt-Sc hmi d t

op erator.

i s a compl et e orthono rmal set in G we hav e

=> g =t(g"j )~j

and

L l lfj

II~ <

00

•

The indioes G and H i ndicat e the spac e wh er e the scalar produo ,s ~d

the norms ar e taken. Then the sum

- 35 -

L . G~rding is finite almost everywhere on X and

Ug(e) =!.(g,fj)GUfj(l;), IUg(l;)I Henoe, for almost all

e,

g-'lJg~(1;)

(1)

s h(l;)llgII G•

the coef f i c i ent s

=
are continuous functions of

g~ G.

ej(I;)6-GI ,

If also AG CG whe r e A is dia-

= a(I;)
gonalized to a(l;) by U, then ~~J~j(~»

for all

g in G and almost all I; in X so t h at (2)

Alej(l;)

where A': G '~G I

= a(l;)ej(l;)

is the adj oint of A: G~G. Hence the e j (d E GI

are a sUbstitute for the eigenfunctions in the discrete case. Our construction i s made more us'ef'u L if we replac e G by the intersec... tion of count ably many Hilbert s pa c es G wi t h Ialbert-Schmidt imn

beddings into H. Then it is eas ier to achieve that AG<:G

f or a

g i ven collec tion of op erato r s t ha t are diaGonalized by U. It is a ls o pos sibl e t o pa ss t o inductive limit s of such s pa c es G. In this way on e pr oves e. g. the follo win g r e Gutt due to K. Maurin. Say that a di~tributi on in~n

is of ord er k if loc ally a sum

of der i vat i v es of ord er =< k of loc ally sq u are integrable functi ona. Then e .(I;,x) are dis t r ibut ions of order J

a t mo st

tn/2]

+1•

.In parti cular, if A i s rm e:ct en s ion of a C"'dif f ,-,rf'nti al oper a tor

P(~ , D )

on

c~~

and A i s

di '-'eon~lized

t o a (; ) , then for

almos t ali ; ,"'i th r e '::?eet to)A, P(x,D)ej ( e, x) = a ( ; ) e j (l;, x ) in the di str ibuti on ~e n B e . If, in ad~ i ti ,-,n , p (x , n ) i s ell iptic,

- 36 -

L. G1I.rding

then by Weyl' s lemma , t h e eig enfunctions x ~ e .( ~ ,x) a r e in

c·c.ft)

J

for almo s t all e. Furt h er r esul t s of this ki nd can be

found in the book s by Berezanskii (1965) and Mauring (1 968). Their we ak point i s the non-uni queness of the s pa ce G an d the v agu e nature of the ei ge nfunc tion s . But i n the c lass i cal cas e of differe n tial opera t or s i n on e v ari able, the e igens pa c es are f i n i t edim en sional and the gen eral r esul t i s hel pf ul . To be more .p r e ci s e , suppos e t hat P ha s ord er m an d t ha t Jt is an int erval an d l et be a s el f -ad j oi nt ex t en si on of P. Let s1 ( z, x), ••• , sm( z, x) a basis for

~h~

s ol u t i on s of Ps= zs,

z~~

A

be

, which behav e s nic ely

i n z an d x , Put

(Uf)k(~) =~k(~'X)

and let L

2(R,r)

f( x) dx , fE Go ~ ,

be t he Hilbert s pac e of f un ctions F: RO+ cr ·::ith

the s c alar pr oduct (F , G) = wh e r-e the m.)tm ma t rix

2: Fj ( d GkCe )d fj k ( d f(d

=

(r

jk( ~ ) ) i s hermi t i an and i n cre as es

wi t h e. Then , by the ? pect ral theo r em 2nd the r e su lts jus t s t a t ed , there i s a

f

ou c h that U:

L2 GJl) ~ L2(R,

r) is unitary and dL a g orie.Lf. z ee

t o mult ipl i cati on bye . The pr obl em tha t

A

r cn ~ in ~

i~

then t o com-

pu te t' . Thi s i s don e by t h e Ti t chmar sh-Lodaira f o ~u1a f or ~hi ch we r efer t o Maurin ( 1968 ) . It sett le s ev er y t hi ng in pr i nc iple . If,

.

2 e. g., P=D +V(X) with. a r eal pot en tial V we are i n a cas e trel".ted by H. Weyl in hi s t i ne f eatur es,

c 1 a ss i c ~1

e.e.

2 a pe r fr om 1910. I t has many interes -

that/if .Jl, i s the wh ol,e line and V is ~,ma1 l in

t h e sense of havd.nr : compa ct ::uI' ;;J or t th en P i s di ag on e.1ize d by a u ·.i t :>.ry map '::hose main ;;Jart i s cl ose t o the Pour- Lr r- tr['.n sform. ;'! e

- 37 -

L . G!rding

shall not elaborate t his eni gmat ic st at ement h er e . It wi l l be olArified in the second part of the se l ectures wher e we treat the corre sponding case in X3•

- 38 -

L. G~rding

II. Spec tral t heory of t he Schro di nse r

o p e r ~to r

Introduction. The on e pa r t i cl e Sch rod i nGer ope rat or or H = H

o

where H

o

Ha~ i lto n i an

+ V

is the L&pleceopp~raior in 1 3 an d V a r e al pot enti al,

is the basic object of non-relativistic, quantum mechanic s . The operator H is diagonaliz ed by the Fouri er t r an s f orm and we s h al l o prove that if V is small f ar away, then H i s diagonali zed by a unitary map oloce to the Fouri er transform. We sh al l als o deal with scat tering theory. Its cl as sical coun t erpart i s t he s t u dy of particles entering the pot ential f i el d wi thou t being captured by it. Our narrative is based on t wo papers by Ik ebe (196°,1965) and a seminar exposition by Horman der of t h e f i rs t of t he m. It is just

a simple introduct ion to a well develpped field of ma t hema tical physics. The theory of one particle ·Hamiltonians i s wel l covered by B. Simon . (1970) who. also gi v es a long list of r eferences. Two papers by Kuroda (1973) should be added to t his list. The extensiun to many . particle Hamiltonians was pioneered by Fadeev (1963) and has many open prbblems ( see e.g. Hepp (1969)). Some int ere sting deTelopments in scattering theory of two ap rticles are due to BalsleT and Combes (1971) and Simon (1973).

1. The Laplace operator.

2

A = D1 \+D2

2

2

+D and l et H be the o 3 i 2 2 3 corresponding selfadjoint operator on L =L (R ). If F is the o Put

Fourier transform defined by

(1.1)

Fou(~)

=

Jre-iX~u(X)dX

-1 then F oH0 F 0 is multiplicati on by

- 39 -

L. Gllrding

.( 1. 2)

"f II;; 14 IFoU(I;; ) 12dl;;

that~

'Not e

wi t h domain

=

< ~,

Since Jr(1+ 11;/4)-1 dl',;

<

~e

00 •

Schwa r t z s pa ceJPis dense in

it fol lows fr om (1.2).that

Ho•

Fou is

integrable and h ence u(x)

= Jeixl',;F 0 u(l;)dl; I

bounded and unifo rmly c ontinu ou s when uE: D(Ho). Later we shal l n e ed a more pre cise r esult that fol lows from Sobolev' s inequalities, namely that to every e>O there i s a Ce>O such that

I l u ,KII_ ~ elIAu,L I I + ce l lu,LI I

(1.3)

wher e K and L are co n centr ic bal l s of rad i i 1 an d 2. Her e 2 ( 2 /Iu, KI/ = ) K'u (x)! dx and t he l eft s i d e of (1. 3) i s def ined an al ogou sly. Th e s olu t i on 2 u=u o of t he Schr odinger equ a t ion Dtu=-Hou, u(O,x)=f (x)S L i s e i v en by the f ormu l a

(1.4)

uo( t,x ) = e-itHof (x) = f e- i t II',; 12+i XI; :J ;) dl',;

"'"o

wi t h f

f::~

(1.5)

:::

::(" X) . 0 ,gn ' 1'1- 3/ 2jI,-i(X-r)2/ 4' f (y )dY

wh e r e c i s a c ons tant. We shall n e e d two pro pe riies of u

ly tha t

t.., Of

and tha t if

A

f

o

=>

u ( t ,x ) .... 0 we akl y in L o

o'

n?me-

2

=F f has compact su ppor-t , t her e i s a K su ch that 0

Ix /f Klt / =>

u o( t, x ) =

Q( /x /-n) ,

all n.

To pr ove ( 1. 7 ) we ' shall UGe the Me t h od of the s t a t i 1'J nan-.. phase,

J

no ting t hat ~( t/ 1',;1 2+xl',;) = O ~hen 1; =- x/ 2t . ?u t t i nc l;=n- x/ 2t we get

" 2/ 4t u ( t , x ) = e-~x o

If

I x l~K /t 1 ~7i th

"t

e~ n

2~ f (n-x/2t )d n.

a l arge enough K

0

v-e

e an take

I n ; ~ IX/4 t l

i n t he

- 40 -

L. G£rding A

integral. In fact, n-x/2t is bounded on the sup port of f o i t n2 i t n2• 1 e The n , by intePut LT} =. n1t/i so that t- Inl-2L e

n

grations by parts

[u (t,x) I

~

[uo

(t , X)g(i1

It

r n flLnn ;

(TJ-x/2t) Idn=O( It 2, To proTe (1.6), note that if gCL then o

\!jt

0

r n Ix/t I-n)=o ( [x rr,). -

= Jeite2~(e)l;JfY de.

When f,~ are in ~ and vanish close to the oriGin, t he int egral o 0 1 is Q(ltl- ) ( Use L once) and the re st follows by a density

e

argument. 2. The SchrOdinger operator.

Consider H = H +V = o

4

-v

wher e V

is real and

w.

shall see that H is selfadjoint on D(R)=D(Ho)' that

(~.2)

and that the function

is uniformly continuous when

fe D(R).

To prove this we shall use

(1.:S). According to (2.1), Ilvu.,KII ~ C Ilu,KII_ and a summation OTer balls covering X3

(2.4)

so that (1.3)

gives

IIVul1 ~ e IIAull +cellull.

This proTes (2.2) and that IIHul1 + Ilull and IIHoul1 + llull are equiTalent norms on D(H Hence H is closed an d the fact that o). u,TeD(H ) => (HU,T) .. (Au,v) + (Vu,v) = (u,Rv) o shows that ac..B'I • That H=ii rfJ can be seen as follows. If 0

A:f

is real, then

- 41 -

L . G!rding

(H+~E)(H +~E)-1 = E + V(H +~)-1 o

0

and, according to (2.4), Ilv(HO +i"E)-\11

if"

(e: +

s e ] IHO(Ho+i»:)-1 u II +

OJ I)./)Ilu"

~

0e

I I(Ho+i"E)-\1 I

<

Ilu 11/2

is large enough and E<1/2. Hence R(H+:I,\E) = L

2

s o that H is

indeed selfadjoint. The statement (2.3) follws sinc e (2.3) gives Ilv(ei(t+s)Ho

_ eltHo)fll

~

cll(eiSHo _

E)~fll+cll~isHo_E)fl

3. The wave operators and the scatterine matrix. The wave operators W+ are defined by W+f = lim

eitHe-itHo f

t ...~

wh en the limits~Exis~. If they exist for f in a dense part of L 2, they exist for all f since the operator product on the right, = e i t He-itH0 is unitary and then the wave operators are i sometric. Since (U(t)f,g) = (e-itHof,e-itHg) = e- i t (e-itHof,g) when g is an ei genfunction of H, Hg=~, (1.6) shows that

wh er e N is the span of eigenfunctions of H. Hence, if H h a s discrete s pe ct rum , W+f exists only when f=O. In ord er for W+ to exist on L2 , the operators Hand H must be s o cl os e that the o scalar product

(U(t)f,g) tend s to zero only exc ep t i onally. If

the wave operators ex i st on L (3.3)

2

I

then (3.1) shows i~~ediately that

eit~, = W eitHo +

+

for all t . The te chnic al t erm for thi 8 i s that both intertwine e i t H ~~d eitHo

W+

an~

W·

- 42 -

L . Ghding The wave operators have a quantum mechanical i nt er pret Qti on. The time dependence of states f,gErL~ in the free (kinematical) c ase wi th Hamiltonian H and in the perturbed cas e wi t h Hamf.L t on i an o H are giTen by t

~ e-HHof, e -itHg.

AocOrding to

(3.3),

the wave operators map state s of th e first

past respectively. The classical analugues of the s t a t es W+f are the states ( position and momentum) at t ime 0 of a particle comi ng with in or going out a given rectilinear motion f or l arg e t ( cor r esponding to f). In view of this one u ses the expressive notation

f~Rt

W+f.

fhe soalar product

defines the scattering operator or S- matrix

S="'W + Its matrix elements can be measured experimentally theoretical importance. We note that by virtue of with eitHo

Bn~

it has als o

(3.2), S

c ommu t e ~

and henoe with all funct ions of the free Hamiltonian

Ho • Starting again from (3.~) and noting that H i s digonaliz ed by a the Fourier transform

F = F Wo -

~o

=>

we get

.itH F~

= F~

- 43 -

L. Gdrdihg

=>

W_ unitary

e i t H i s diag onalized by F .

This formula s h ows t hat there i s a clo se c onnection between t he wave operators and the di agonaliz a ti on of H and we s hall explo:l!t this later. For the momen t we only g i ve a condit i on on t he po t en tial V tha t guaran t e es that t h e wav e operato r s ex is t . Lemma 1.

(3.5)

If V satsif i es (2.1 ) and

Joc;-312

(J

IxI
v (x) 2dx) 1/ 2 dt

<-"

in parti cular if V = Q (l x l- 1- £), £>0, t h e wave opera t or s exis t . Proof . We fir s t prov e t hat f 6 D(H) implies

(3.6 )

u '(t) r = ieitHve- i tHo r

and

(3.7) II (u( b ) -U(a))f ll= l l .fabu 'Ct)f·d tll~

-C llve- i t Ho r

l]

dt

'tH The jpectral t h eorem shows t ha t t ~ e ~ f i s un iformly continuous and that

d ei tHf

i Hei t Hf dt s o that

(ei ( t+ S)H _ ei t H)f = s i Hi t Hr + 2 ( s) and anal oe ou sly for:Ho • Hence . U( t +s )f _ U( t )f = e i( t +S)H( e-i ( t+S)H o e i t H.~ s He - itH0 r +

_ e-i t Ho )f +

- itH0 s ( s ) = se i t H(.) ~V e

r + 2 (s ) •

Si nce t he de riv ativ e U' ( t )f i s cpnt inuo u s we c an i nt egrat e it and thi s prov~ s wav e

(3. 6)

o p e r ~ to rs

j

and

exist .,.

--

(3.7).

Hen ce , i n ord e r to prov e that the

i t suf f i c e s t o

I lv-i t Ho f lldt

<

when f bel on~s t o s on e dense pa r t of L ar'gumerrt gives

t ha t

sh o~

~ x I >1 V ( x ) 21 x ,- 4dx

2

<.QtO

•

N OTI

( 2.1 ) and a s i mpl e

- 44 -

L . Ghding

so that by (1.5) Ilve-itHo fl1 2 ~ cit

,-3 IX~Kltl

r

( V ( x)2 dX +

Here, by (1.7), the last term is Q(t4-n)

v(x)2Ixl- 4dx = Q(t 4-n)

Ix1'>K It I for all n when ~oe (fI"O • Taking e.g. 1l=8 this gives 0 IIVe-itHo fll Cltr 3/2 ( f . . V(x)2 dx)1/2 IxT
s

Consider the resolvents G (z)=(H ~ZE)-1 o 0 and G(z)=(H_zE)-1 when Im z ~ O. A direct computation shows that 4. Green's function.

Go (z) has the kernel

Go(~;YfZ)

= eiofilx-yl/41tlx_yl

, Im 'h

> O.

We shall investigate the properties of the kernel G(x,y;z) of G(z) and state them in Lemma 2. ~t Go =,G o (z), G = G(z), Imz ·

4 o.

Then

G0 = G +G VG = G + GVG 0 0 b) The operators VGo,G~.,GVGo are Hilbert-Schmidt operators. In particular,

G(x,y"z) = Go (x,y,fz) + Halbert-Schmidt kernel 0) GIL~L12 , L2 ... L 1 provid ed V.. L2 • oc comp d) Le'" :B .. all bounded continuous ,(x), \ . those which are

o at ....

Then

Im z~O .. > GoV :~:B. . compact and z"Go(z)VfltB_continUous Re z>O, 1m z ~O .. > Ker(E+GoV) in \ . vanishes provided V = Q(lxl- 2- E ) tor some E)O.

- 45 -

L. G£rding Proot.a) is obtained by multiplying

the identity H-zE

Ho~ZE+V from the left by G=(H_zE)-1 and from the right by Go =

lr

(H or conversely. To prove b) note that o-ZE)-1 V(x)2I G ,2dx dy o(x,y"z)

< cae

also with V(y) in the integral. Hence VG

o

and G V and hence also 0

GVGoare Hilbert-Schmidt operators. c ) Since Go:L

_

-

L

~

..

, V:L-.L

2

2 are bounded . operators so are VG 0 :L"~ L and 2" 2 2 G _G = GVG : L--",* L and G( z ) : L-' L L"C L and, by duality, o 0 loc 2 G(;}: L . ... L1. To prove d) note that . comp and G:L 2_L2 or

f6B

=> (GoVcr)(x) =

Since V(y) =

~Go(X,y"Z)V(y)cr(Y)dY.

Q(/y/-2-e)

we get absolute convergen~e, continui ty

e

in x and Q( Ixr ) as x . . . . . Hence GoV:B-.B..

is compac t by the

\I

Arzela-Asco11 theorem. Also,the function z-.GoVf "

>

tinuous when lm z = O. If Ira z>O then G V: B~L =0 => G oG-

1,

B_is con'

,

so that (E +G V)(I~

0 0 1

= 0 =>'=0 so t h a t Ker(E+GoV) = 0 in that case.

I f Imz=O this al so holds since

=>'= Q(lxl-

2

2e)

crEE,

=> ••• => f= Q(/xl-

(E+GoV) f= Q(lxl-

n)

e)

for all n and a r e sult by

Kato ap plies. 5. The perturbed e i genf un c t ion s . a r e bounded e i g e nf un c t i on s of

The func ti on s

Awi t h

eigenvalues

cro(x,~) = eix~

~ 2,

A,o =~ 2'0.

We s h all c on s truct perturbed eigenfunct i ons ~(x, ~) sati sfying

i n the dist ri bution s ens e fr om wh i ch a unitary map F wi l l be cons t ru c t ed that diagonalizes

when t his operato r has no

H= ~ +V

d i s cret e s pe c t rum . Heuri stically

vre

have

- 46 -

L . Garding

'f(x,~,z) .. (~2_z)G(z) fo , Im z => O,and put

q> (x,/;) = Cf (x,/;,/;2) The most convincing

r easo~

•

for thi s choice are the manipul ations

with Parseval's formula in section 7 below, in particular the formula (7.6). Lemma 3. The functions (S.2) are bounded and continuous when ' Im B

~

0 arid

the functions (S.3) satisfy (S.1) in the distribution

sense. Proof'. We have and

'f"

E

G(zHA-z)
= e_EX

cf = G~z)(A -z)
2

'f'o(x,/; )E:L

•

Then b7 Lemme. 2. a,

Cf: . 'Go (z)G(z)-1 <:p

80

E: ..

(Ei
o

that

(E+G (z)V)(
a:.J, 0

I

o

10

0

10

this giTeB

(E+G V)(m _ o

I

CJ) )

10

so that an appeal to Lemma

..

~

~G

0

V to

10

finishes the proof. Later we shall

use (5.4) in the form (S.5)

~O .. > Go(~2+idCf(x'~)~
100al17 uniformly when e~O.

-rex,/;)

2

- 47 -

L. GAl'Qing 6~

The discrete spectrum of H and the discrete part of its

diagonalization. By Kato's theorem, H h as no positive eigenvalues and we shall see that the spectrum of H is discrete below zero • . Assume the contrary. Then there is a of unit vectors f

such ' that Hf ..., Af

n

n

:and hence G(z)f ~ (). _z)-1 f n n

n

A <0

and a sequence

• Then (H-z)f ..., ()-z)f n

n

wh en Im z =+0. But G(z) and G (z) 0

differ by a compact operator ( Lemma 2.b ) so that also G (z)f o n /' ~ (tft -:a)-1 f n • If f denotes the ordinary Fourier transform this means that ((A -z )-1

A<0,

,A

.

fn...,Oin L

2,

_(l;2_ z )- 1);

n

(d~o

in L 2

s~

that, since

a contradiction.

-L
be a maximal orthonormal s e t of 2 2 eigenfunctions of H and define F d: L ~ 1 by (Fdf)n = (f'4Pn) In t h e s equ el , let

by

(AlP ) = In

H (!)

In

= I). cp • Then, if N i s t h e span n In

is unitary and (6.2)

IIFdf l12

=

f

d(E~f,f)

(I ~O

wher e ) Ad~ i s the s pec t r al re s olut i on of H. 7. The Poiss on ke rnel and Parseval's f ormula.

The Pois s on k er n el

for the upper half-plane is

an d we have

in the

d i str ibu~i on

s ens e .

~e

list t wo mor e pre cis e v er s i ons of

(7.1 ) whi ch wi l l be u s ed l ~t er. Fi r s t

- 4H -

L. Garding

(7.2) .

~-lt 0 => SP~-'X,~)~(oI.)cJc(

11'

~ )'X-(?')d«' (,,)

and C\' i s of bounded v ari at i on . Se condl y ,

i f 1J:C: (7.3)

dCf ()..)

~-1t 0

=>

5P (1;2_d, ~)f(ol, P,I; )dl;dJ~

I f ( 1; 2, ?, d dl;

i s continu ou s an d boun ded on Ii )li~f 0}XR3

t(~.~,I;)

an d

vanishe s wh en <X i s l arg e en ough , We l eav e the proofs t o t he r e ad er and

l~st

solTent

= r)dE~ ~ith

r e-

G(z), namely

0'"

(7.4)

when t£L (7.5' when

t wo consequ enc es for the oper ator H

2

=>

0

J~3t"11 I G(ol_i~) f I I21--(ri.) dd.. ~ j1-C~)d(Etf, f )

, and<j..€'Co(R)

= S't-<1; 2)I (:f,
<';J(H)f,f)

te Co (1(3)

and

and

«« C (1(+).

/Y

0

(The index 0 mean s compa c t sUPPor t )

Here (7.4) i s an immediate cons equ enc e of (7. 2 ) and the i dent i t y

S~1t-1ILG(ol-i~)fll~(O()d~ 5 =

p (oL-l'f) d (E>.f, f ).

The proof of (7.5) s t a r t s f r om Pars eval' s fo rmula I/Gc;)f/l

2

= SIFoGG)f(l;) / 2dl;

where F o is the Fou r ier trans form s o t hat F GG)f(1;)

o

= )( 'GG)f(X) ct)\0 (x,ddx

•

By Lemma 2.0 ( and thi s is a ver y crucial point) we may move G(Z ) from f to

fo

(7.6)

l'oGG)f(1;) = (1;2_z)-1 )f(X) f( x, l;,z) dx ,

in the int egr al sO that by (5. 2),

Combining Lemma 3 and (7. 3) gives t h e desired r es ult (7.5).

- 49 -

L . Garding

8. Diagonal izati on.

(8.1)

J

Putt i ng

f( x, c;)f(x )dx

we hav e a ma p

and we shall prove The or em 1.

rr

V € L 2 is r e al an d Q( l x l~ 2- e ) for s ome e)O, t he

nlpsure :of , (8.21 ois a unitary map H so that,j- HI -1

3

f r om L

J

2

'2.

is multiplication by

on

t hat diagon al i z es ~2 k

•

'l

2

and bY/Ion 1 •

Corollary. The linear span N of the ei ge nfunctions of H is als o the k ernel of F and FF- = E, F-F = the proj eQtion on L

2

e N.

Proo:f. Th e f ormul as (6. 1) and (7.S) show -Tto be i so met ri c and , by Lemma 3,

Hence we get the theorem by obv ious closure pro cesses provided 2

2

we kno w that F is sU; jec t i v e , FL =L • To pr ov e this cruc i al r e sult we f irst note that H i s the cl osure of its -cr-e s t r i c t i on t o

cO" o

Hen c e (8.3) holds when :f.t.D(H) s o tha t

<X- e

C (11) , g = ryA H)f o

=>

FP(H)g(1;) = p(;;2)Fg( ;; )

for every polynomial P. Hence

(8.4)

when'X~ Co (If) and also,

.by an obv ious cl osure argumen t wh en

~ is

continuous and bounded. Next we sh all prove Lemma 4.

FW

= F

o

2 2 which implies the desired r esul t that FL =L • Her e W is one of the wave operators of Section 3. Note the connec t i on with ( 3. 4) .

- 50 -

L.

The

G~rding

of the lemma depends on, the following formula -itH connecting F -F with the unitary operators e itH and e o, o proo~

(8.5)

°

(Fo-F)f(e)=lim

= i

0

o

where

f~

(8.5)

of

5~x,e)v(x)Go(~2_iE)f(X)dX=

f Feit~e-itHo f(~)dt E

.....

#)

Co

• In

~act,

according to Section 2, the las t member

equal~ ....

~(d/dt)FU(t)f(e)dt

= FW_f - Ff o and this proves the lemma. The first equality

(5.5) Vex)

if

J

follows from

we note that G (~2_iE) can be moved to the product o

() (

'fC;x:,d.

(8.6)

(8.5)

Next we shall see that -..., 2 ) ( ) Et+i tEL -i tH0 f () ~)V x Go e -iE f x dx = i oFe "v« I; d~ •

S

This suffices to establish the second equality (8.S) ' s i nce the last integral of this fornula is absolutely convergent in the L 2 norm. To prove (8.6), note that (8.4) shows that its second member equals

_po

eEt+i~2FVe~itHo f(~)dt

iJ o

J

whi ch we can write as an absolutely convergent int egral i

1.DO

F)V(x) eEt+iI;2(e-itHof)(X)dX dt •

Performing the i n t egr a t i on wi t h re spect to t first we cet -~

i

Jo

·tH · !'2 e Et +~~ (e~~ 0 f)(x)dt = i

j

-~

(o,,2 .2)t o " e E+~~ -~n +~~nf (n)dndt =

0

_~2)-1 eixn;o (,1)dn

=j(iE+n2

and this proves (8.6).

Go

0

(~2-iE)f (x)

- 51 -

L. Garding

9. The perturbed Fourier transform, the wave operators and the scattering operat or, us to express the

The results of the prec eding s ect ion allow

w~ve

opez-a'tor-s

and the s cat t er i ng operator in

terms of the Fourier transforms F 0 and F. The comjugat e A ,

of

a linear transformation is defined by Au = Au • In particular,

Ha

Ho

Hand

=

HO

, par t of the formula

so that, by

(9.1)

(3.1), w

=

W+ •

This explains

below.

Theorem 2, Under the hypothes es of Theorem 1, one has ::I W = ~ + 0 The scattering operator S =

W

(9.2)

(Su,v)

w: W_

i s unitary and

=JJ S(I;,I;')~o(d~o(I;')

is given by the distribu tion k ernel

where

J

v(x)'fo(x,d
of Theorem 1, F*F i s the project ion on L2 FiiF o ' the last equality by Lemma

4.

. ,H.

Hen ce

Thi s proves

~

=FHFW =

(9.1),

in par-

t icul ar that W L 2 = L2 e N. Sinc e N i s invari ant un Ler c onj ugatio~, 2

2

being the sp~n of eigenfuncti ons of H, we also have W L = WL = + 2 L tile N s o that the s cat t er i ng operator S ",ii \1 i s unitary. To 11+ _ prove

(9.2)

(Su,v)

note that (~

u,W v) = « W - w+ ) u, ~ +v) -

+ = (u,v) +

«',1_

~ ','l + )u , ",+v )

+

- 52 -

J (eitF-Ve-itHo~u,W+V)dt -rOO

- i

-

A

<Xl

J\

L. G8.rdiqg

.()O

when u 6 cO" and v e e vanish close to the origin. In view o 0 o 0 of (3.2) the integrand can be written as ) • ( Ve-itH 0 u, W+e i t H 0 v Inserting e-e!tl of

1\

J\

U

o

and v 0

as a damping factor, writing u and v in terms

and notin"g that

W+u(x) = ?F'ou(x) =::

Jf

(x,1;') v o(1;')d1;'

which follows from ' (8.1), we can perform the integration ~ith respect to t explicitly. Letting e~O this gives the factor

J

e- i t (1; 2 - 1;,2)dt = 210 0(1;2 _ 1;,2) •

Hence (9.2) is proved modulo some technical details.

- 53 -

L. G~rding

III Summation theory of eigenfuncti on expan ci ons conn ecte d wi t h elliptic dif fer ential operators.

Introduction,

Th e partial sums

s (x)

n

=

L ~e1kx Ik l
of the Fourier s er i es

2. aneinx

of a function f can be wri t t en as

where e(x,n) =

.

Y IkT
1 = s in (n+2- )x

e i kx

I

~

sin x/2

1s the Dirichlet kernel. The more general kernel s

tX

~

e (x,n) = L.

Ikl
correspond to the

s~ a tion

.

(1 - k/ ln l)

0(

e

ikx

>

,o{=O,

of the Fouri er series by Ri esz'

means

Alll this has a natural conn ection wi t h the s pectral t h eory of, e,s" the elliptic non-neg ative dif f eren t ial opera~or D2, D=d/idx. In fact, this operator i s selfadjoint on L 2(o, 2~) i f we take n2 in its distribution sense on func t ions of

p~od 2~ .

It i s dia-

gonalized by the Fourier transform

(~f)n from L

2

2

~n .~inxf(x)dx

.. L (o, 2n ) to the space].

the inverse

2

of sequences a

=

t an} with

- 54 -

L.

(d- 1a )(x )

= (2lt)-1

>

in such a way that

if

J" d~

~ D2:;-

-1

I. an

partial sum s (x) when n

E~

t'

n

=

f

"t <)

(1

e inx

is multiplication by n 2 • Hence;

is the spectral resolution of D 2

G~rding

2,

E~f

is the

== A and if eX

-'t/A ) dE'(" ,

the Riesz mean can be written as Sn(x)

0(

= E).. f(x)

, n

2

=/1 •

Henoe all the classical results on the convergence and Riesz summability of Fourier series can be stated as properties of the convergence 0(

E>.f(X)-.,.f(X) More generally, if x=(x 1 ' ••• ,xn) are n real .va~iables, is the imaginary gradient and

A= D12+••• +dn 2

is

Laplace's operator, the convergence of the spherical means

of a Fourier series

L

al;e~';x , (

1;=(';1' ••• ,I;n) \'lith integral com-

ponents) or 'the spherical means

-f

(1 _

II; 1<" of a Fourier integral

II;I/~)

S

a,;ei,;xdl;

a,;eL-cl;d'; , (1;6

z"),

amount to the con-

vergenc e (1) wh er e nov! the E). are the proj ec t Lons of the c pe ctral re s ol ut i on of a su i table selfadjoint version of the La pl ace op erator. For a nunber of y e ars there \'las a sustained effort by many writ ers to extend the basic results on the conv er ge nce and summability of Fouri er se r i e s and int egral s t o t h e eigenf un ct i on

- 55 -

L . G£rding expansions connected with elliptic differ 8ntial

op 8r~ tors.

Tha

crovming effort s o far is · 'CilO~ "in e d in two papers by Hormand er

(1966,1968) TIhi ch shall form the basis of the third part of t h es e lectures. To describe the general s ituat i on , l et

e'"

II

be a para-

compactVmanifold and P an elliptic differen tial operator in~ wi th

Cpt'

coefficients. Assume that

the scalar product

P

,f c>O on

C.,.,

o

functions in

•

(u,v) = [u(X)v(X)dX where dx is a fixed

C~ den sity and let ~ ~ c be a selfadjoint

extension of P with spectral r esolution

( J~dE)

EftpN

it follows easily that

are bounded operators for all

E'}. has a C

cD

N,

·N

• Since P

E~

and

hermitian k ernel e(x,y,f. ). c3Xis ealleg. the!lpectr~

function. It h as the property that )le(x,y,A) 12dY is a con tinuous function of x and f6.

C:

""'> E;>.f(x) =

By 11he abstract

f

e(x,y,/')f(y )dy =

theo~y , ;>..., CO => E~f~f in

e(x,f,~) L

2

, say.

2

= L (Jl.) when f is

2

and sincs C~ is in the domain of all powers of P o co
0( dE). f(x) = e (x,f,).)

where

0(

E~

=

j (1 - 7./;>-.)0( dE.r,

cl e (x,y,'A) =

J

0(

(1 -"II).) de(x,y,7).

"Z.<~ . -z: 0The object is now to study the kind and rate of convergence t< '\ . (2) ~f...,f and e (x,y,lI) -'?' &(x,y) when ~ ~ 00

where &(x,y) is the Dirac kernel defined by f(x) =

j &(x,y )f (y )dy.

The starting point will be to estimate the kernel of the op erator

- 56 -

e

-itA

f

=Je

L. Ghding

-itA 11m

dE~

. ' , A

• 11m

=P

, m

= deg f,

for small real t and t h en ap ply a suit a bl e Tau berian argum ent . In the first part of this proc ess we s hal l use the f act that

u(t)

= .-itAu

satisfies the hyp erbolic equ ation

and construct ap proxim ate s ol u t i ons of i t by me an s of t he methoq of geometrical optics. Thi s fit s into t he no w curr en t t he ory of Pourier int egral op erators ( see Hormand er (1971), Duis t ermat and Hormand er (197 2 )) whi ch i s a 'v er y s oph i s t i c at ed tool of analysis. We s hall s t ar t by go i ng t hr oug h wha t we n e ed f r om t hi s theory ~nd then pro c ee d t o i t s a pplicat ions to ( 2). The out come i s t ha t , r oughl y s peruci ng , the conve r gen ces (2) are the same for al l el liptic opera t ors with t he s ame number of vari ab les and the s ame degree. The precis e r esults ' are f ormu lat ed in Theor em 2 of Section 5, the f ormulas (6. 4) and (6.5) an d t he 'cl os i ng s en-. t enc e s of Se c t ion 6. 1. The eal culus of psedd6d1fferentialopperators. We s hal l cons i der

linear operat or s u (1.1)

Fu(x)

Fu defi ne d by s ymbol ic i nt egrals

=~~ f(x,y,~)eif<X, y'~)U(y)dYd~

,

i . e . Fu(x) =j K(x ,Y )U(Y)dY wi t h t he di s t r i bu t i on k er n el (1. 2) ~h e r e ~

K(x,y):::-

Jf(x,y,deicr(X'Y'~) dl';

r ang es ov er

of ~1 -an d ~ 2

~T

R~

an d x ,y ov er op en

n

8~ b s e t s ~ L 1

r esp e ctivel y. Thc f un c t i on

an d

52 2

has bounde d i maginary

part an d i s ca l l ed a phas e f unc t i sn ~hil e f (x ,y ,~ ) i s c o~ pl e x and c alled an ampli t u de . Opera tors Cu(x)

= ~c(x, ; )U ( Y )dY

- 57 -

L . G~rding

defined by C~kernels o(x,y) are said to be smoot h and in t h e calculus that we are going to develop, su ch op erators are disregarded. Example. When Jl

=.n 2

1

=11 and

n 1=2=N a typical phase functi on

is the soalar product (x-y)~. Then if p(x, d -l>oc(x)~ is a polynomiRl in

~

0<.

and

P(x,D):.I.P«.(X)Do(., D = D = ix

1

0x

='1/i dx ,

i s the corre sponding diff eren tial operator we have

55 P{x,~)ei(X-Y)~

where

u(y) dyde =

fp(x,~)eiX~~(~)d~ =( 2~)np(x,D)u(x)

-rn, the s econd integral int erprets the. first an d

uECo~'

~(~) = Je-iY~u(Y)dY

is the Fourier transform of u. Henc e t h e

diff erential operator P(x;D) i s expres sible in t h e fo rm (1.1) wi t h ph as e f un ct ion (x-y)~ and amplitu de ( 2~)-np( x,~) indep end en t of y. Di sregarding the f actor (2~)-n , operators F wi t h phase functi on

(x-Y)~ and ampiitud~ p(x,d .ar e wr i t t en as p(x,D) and

t ho se of the form p(x,D) + C wi t h a smoot h C are called pseudodi ff er enti al op erators. () }T) To classify the amplitud es we shall u se s pa c es Sm =Smro ~1~~~,R cD

l ab elled by a r eal paramet er m and consisting of C fun c ti ons f su ch that all product s

D~ D~ f(x, y, ~ )(1+1 ~/)lpl-m x,y <,

ar-e bounded on c onpac t su be e-bs Of..ll1)(~

po

• Any C

f un ct i on whi ch

i s homog en eou s of deg re e m in ~ for l arge Z; is in Sm but e . g . c overs c ases of s l o\'l oscilla t i ons for large ~. ITe have SmSpC:S m+p

and d iff e r ~n t iati on

- 58 -

L. G~rding

wi t h respect to I; maps Sm into Sm-1 • An amplitude f in Sm and the porresponding operator F are s ai d to be of ord er m. The space of operators of the form F+C with F of order m given by

(1.1)

C smooth

and

will be denoted by Lm(f). Amplitudes in S-PO=

() Sm , e.g. those whi ch .v an i s h f or large I; gi ve COOk ernel s and hence smoot h operators. For t his r eason, amplitude s n e ed only b e s,pecified modulo S

-cD

and we agree to let t h em vanish f or small 1;.

In oTder to define the right side of

formulas (1.3)

i

(1.4)

i

obtained by

SJ:r~yei

udydl; = - fS(fyU +

J5:r~ei udydl; ~~gration

(1.1)

we shall u se the

:ruy)e~dYdl;

=' - JJfl;~1IfudYde

by parts when l' vani sh es for l arg e 1;. Here

the indices y and e denote the respective g radients. Si n ce

L=_11;1-2(~y

(1.5)

has .Q( Ie /-1) coefficients and the prop erty that L'f = 1 when f = (x-y)l; , the first of these fo rmulas shows that we can exp r es s Fu as a sum of operators of one lower order operating ,on u an d

its first derivatives. A repetition of this proc ess define s Fu when :r~ Sm provided the integral i s interpreted as the limit lim e:~O

iJIO

JJI

('j..(el; )fe

i

udydl;

where X~ Coequals 1 close to the o\rigin. In fact, when m+h
,the integral of (1.1) is ab solut ely conv ergent.

Defined in this way, F gives linear maps

(1.6)

sm~cY(~)--,.c1(~

end b;r dual1 t;r

o

- 59 -

L. G~rding

when m+n+ f


sets of Sm ( which is a Frechet space). The dif fer ential operator

also has ' the property that L

=

1 and used in (1.2) shows t h at

the kernel K(x,y) is a ~function wh en x4y. Henc e the pseudodifferential operators P have the property that (1.9)

si~supp. Pu~sing supp u.

We shall now present the cal culus of psendod1fferenti21

operato~s

under five headings: sums, uniqueness and symbols , product s, adjoints, other phase functions. Sumsa Let m > m >••• -+ - co and let f j 6 Sm j be an amplitude of 1 o po F j • Let Xf: C have the prop erty that

. Then, if the numbers t

/1:1<1~ =>~(FJ=O,

f = .

j

11;/>2 =>'f(f.) = 1•

tend to infinity SUf f i cien t l y f ast, '

L"Iif./t j

)f j

e

Smo

and the corresponding operator F has the property that for all j. order(F-F _ ••• _F.) ~ m' +1 o J J The proof is by direct verification. In the sequel we shall write

~~!~j for the last displayed property and analogously f~~fj • When f j has order m-j and is homogeneous in I; of that order f or large I; we say that .! is an Sm amplitude of homogeneous type. Such amplitudes were the first to be considered in the calculus of pseudodifferential operators.

- 60 -

L. G8rding Uniqueness and symbols.

The diff erence of t wo pseudodiffe-

rential operators P1(x,D) and P2(x,D) is smooth only if the difference

P1(x'~)-P2(x,~) is in S-~ • This statement is made

plausible by a formal Fourier inversion of the f ormul a (1. 2~ wi t h f =. p(x,~),~=(x-y)~ , x being considered as a param et er. The pro of i s not difficult but too long to be given here. Wh en P is a pseudodiff erential op erator we shall say that p(x,~) i s a s ymbol of P if P - p(x,D) is smoot h . Products.

The product PQ of t wo differen tial operators P and

Q with symbols p(x,~) and q(x, ~) has the s ymbol

e-tx~p(x,D)q(x,D)eix~ = p(x,D+~)q(x,~) where we may expand the f irs t f a ctor on t he r i ght in powers of D. Thi s i s the background t o the fact t hat the produ ct PF wi t h F gi v en by (1.1) an d P = p(x,D) a ps eudodiff er en t ial operator wi t h c ompac t su pport h a s t he ampl i tude

p(x, D+df(x, y, ~) N

(1.10) wher-e

p~)

pO

{S~J1

=10(p

wh er e

~

~

Lp (ol) (x, ~ )Dx f( x, y,d/od 0{

•

an d the r i gh t s i de has an exp an s i on- by the s cal a = or der p' + or der f +1-k. Th e pr oof i s by dire ct

v erif ic ation. In ord er to handl e products PQ wh er e P an d Q ar e ps eudodiffe-

r en tial we introduc e t he conce pt of compa ctly su ppor ted op erators A from ~ 'c.5l) to ha~ inG

:it' (J)J

behavin:; lik e convo l utions "Ii th fun cti ons

su ppor t cl ose t o t h e origin. The g en eral c ondi t i on i s that

t o ever y compact K

t h er e i s anot he r c ompac t K'

such that

supp uc K => su pp AuC K' , u=O in K' => Au = 0 in K • Such op ere.to r e ex t end in an obv-iou s '::a y t o S) I (JL). Th e point of

- 61 -

L. G&rding

this i s now that every pseudodiff erential P h a s symbols p( x,;) •

flO

()

such that, p (x, D) is compactly suppor t ed. In fact, if9JE:C (Jl.)C.,JL.) equals 1 on the diagonal but vanishes suitably cl ose t o it , the operator Q wi t h kernel~(x,y)K(X,y) wh er e K i s the ke r n el of P,

Q.

is compactly supported and P.;.Q is smooth. But

extends to " .(S})

and putting 1\

the continuity properties 'of Q allow u s to multiply by u(e),

u~ C~ o

that Q

(jD,

and integrate under t he operator s i gn . Thi s shows

= q(x,n).

The formula (1.10) has an i mmed i ate applic ati on to ps eudodifferential P .... p(x, D) whi ch are elliptic in t he sens e that

iocally uniformly i n x wi t h equivalence in t he s en s e of bounded quotients. For such a P there i s a

r-m ej(x,e)

_10

e(x,d'" in S-m such that

In fact, this amounts to a r ecursive system of

L

P<1+;l=k

'/1l·)(x,dDo(ej(x,~).k~"'(2~)-n

~na~~ans

0k _m

'

(x,~)rJ ~~)-n , all of them s ol v a bl e -m ) in view of (1.11). Hence, to ev er y P satisfying (1.11) there is with first member p(x,e)e

a compactly supported E(x,D) of order -m such tha t

PE = I + C where I is the id en tity and C is smooth. Suc h an E i s called a

- 62 -

L. Garding

right parametrix of P and a left parametrix i s def ined analogous17. What we have no w seen combined wi t h the sec t i on on adjoints

(1.11)

~hat follows shows that any P satisfying

has parametrices

of both kinds. Adjointe. Let us pu t
=~JLU (~X )dX

is then defined by t h e prop erty that

, t

(F

• The adjoint F t of F

u,v ) =
u,v£ c()O (~. If the phase functi on is (x-y)/;, then F t exists o andi.t: bU;·the:iiJIlP1U'.tiia-. ft(x,y,d

= f(y,x,-d.

This i s obvious,

but a les s obvious fact i s that the adjoint p~ ot' a p$eudO~ifte~

s.

rential P..-p(x,D) is pseudodiff eren tial wi t h the s ymbol f"V

L

(-D

x

(ol) (x,-I;)/ol!

just as if P wer e a di f fe r en tial operator. This i s a s pecial case of a more general re sult, viz. that F as gi v en by


(1.1)

with

(x-y)1; i s in t a ct ps eu dodi f f eren tial wi t h the symbol

L a I;

)

~O(O«

I

Dy f x , y,1; 10<. , y=x • The fo rmal proof i s t o r ep re s ent f by its Taylor s er i es

(1.12)

(i (y_ x))o<. D: t

(x,~, d lot !

and then use (1. 4) t o conv ert (i(y-x)) Oth er phas e f unc t i ons . of operators

(1.1)

01.

0< into~1; •

An import an t fact i s t hat t h e s pa c e Lm(f)

of or der m modulo smoot h ones r emains the same

f or al l phase functi ons
wi t h Im~So

wh i ch cl os e to th~

diagonal have the form

and are su ch t hat 1'1;=0 =) x=y

~h en

I; i s l arge. We shal l make a

rough sk et ch of t he proof . It suff i ce s t o wor k di ag onal and t he r e

l oc al ~ y

at t h e

- 63 -

L. Gltrdlhg

-t -I. 't/j'k(X, y, I;)(Xj-Yj)(Xk-Yk ) wi th

'frj k tE. S1•

Acco rding to (1.4), multiplic e.tion by x-y Lower-s

the order by 1 and henc e the se r i es expa n s ion of valent to a SO amplitude. Putting

r

=

CPo

ei~

i s equ i -

wh en ~= O this woul d

show that Lm(~)C·Lm(~o).and t h e op posite inclusion i s pr oved in the same way if we expres s jOo as a sum of pr odu cts of two c omponents of

1'l;.'

2. Pseudodiffer ential operators om manifolds. Ths s pa ces H • s

Let.Q. be a c.,cJ parac ompact manifold . In view of (1.13 ) i t makes sense to s ay t hat a linear oparator F fr om ~ 'CJl.) to;b' ~

is

pseudodiff erential if it is so when r estricted to any co ordi nat e patch. In su ch a patch wi t h coordinates x and canonic al c oor di nates l;. of the ootangent s pa ce at x , F t hen has a ke r n el

~f(x,~)ei(X-Y)l;.d~

+ smooth

where f€Sm for ~ome m. The symbol f (x, ~) transforms i n a complicated way under coordinate , coanges, but the paramet er m is inTariant and homogeneous type goes into homogen eous type and then the principal part of the symbol repre s ented by t he funct i ons lim t-mf(x,t~)

~-'t GO

of homogeneity m in ~ is s i mpl y a f un ction on the cot an ge n t bundle T- (.st) of 5l.

•

A pseudodiff erential operator F of ord er 0 on ~ h as t he re2 markable property that it maps L conti nu ousl y into L 2 • The loc proof is simple. We have Pu(x)

=~f(X' ~) eix~ ~(~)d~

- 64 -

L. G£rding and, assuming f(x,~) to vanish for large x it has Fourier trans-

....

form t(~,~)

such that

-

:r(x,l;)

=

(2rr.)-i:(~,t.)"eix'fld~

Hence,if vc Co we have

~ru~X)V(X)dX

(2.1)

and the estimate

~(~-I;,d

11;-1) I )-N) integrals ~K(I;,~)d~ andjK(F.,~)dl; of

= .2( (1 +

for any N shows that the

"

~...

<

K = (2rr.)-nf(~_I;,I;) are bounded. Inserting the estimate 2/vul = " " clvl +0- 1 lui and v arying

c>O shows that the right side of (2.1)

i s majoriz ed by a constant times

f'

l lvl l

1\

Ilull

where the double

2

bars denote the L norm. Hence, by Parseval's formula, FueL 2 l oc 2 wh en ueL • Say no w that a di stribution UE.&'(.5l), 5l.. an open part n

of X i s in H if s

j (1+1F.1 2) S

(2.2)

1~(F.)l2dF. < 00

Then the re sult above and a little reflecti on shows . t ha t 2

<=> FuE:: L f or all F of order s • l oc Here the right s i de make s sense also on a mani fold wh en U6-!.' ({L) uEH

s

and we t ake t h is as a definition of H (Sl). CorrespondinG to s

we then have F ell i pt ic inj ective Qf order s => wh en 2

n. i s

GL CL

2

c ompa c t .

I IFul I

i s a norm on H s

To prove thi s not e t ha t by t he l oc al t h eory ,

<'

whe n order G = o. Als o by t he loc al th eory, F ha s an

elliptic parametrix E su ch that EF=I +C vri th a smooth C. Since F i s inj ec tive, 1+C is nls o inj e ctive s o that by Fr edholm t he or y , Hence if Fu€:L 2 [mel order ( 1+C)- 1 = 1+C with a snoo th C 1• 1

Q~s ,

- 65 -

then Qu

= Q(I+C)-1 EFu

L. G£rding

is in L2 • In fact, t h e order of the pro-

duct of t he f i r s t three factor s on t he r ight is ~ O. Hence uEH s

(=)

and t hi s t ogeth e r with the cl osed graph t h eo-

FIlE H

0

rem proves (2.3). When.Q.. i s not compac t , this re sult has tobe rephrased to t ake i n t o account the suppo rts, bu t we do not gi v e t h e details. Powers. I f P is pseudodifferenti al .an d ell i ptic i n the (1.11) on a manifol d as when

!L

n,

s ~ ns e

of

the cons truct ion of a paname t rdx goes

i s a part of ~ and thi s shows tha t P has pseudod if-

f eren t ial ell iptic integral pos it ive and nega tive powers pk such that pkpj~pk+j

f or all k and j. Following Se el ey ' ( 196S ) one

can als o con struct ps eu dod i f f er en tial compl ex powers pS at l east in the special cas e when wh en Re

A~

i s, compac t; , P - ~ I

i s i nvertible

0 an d , e. g.,

; l arg e

Re p(x, d

= )

) 0

whe r e p i s the s ymbol of P. This c on s t ruc t i on i s based on the fo llowing wel l - kn own formu l a AS

=

(2ni)-1j /lS(A_/)I)-1

J

dA

fo r the power s AS wi th Re s < 0 of a closed dens ely d efine d inv ertibl e ope r a t or

A

on a Banach spa c e whos e spectrum does not

c ontain an angular neighbourhood of the neg ative r e al axis . Here

If

i s a c ount er clockwi s e l oop around t h e oriGin i n t h is n eigh-

bour h ood. Put ting A = P we defin e pS by thi s formul a . I t then turns ou t t ha t p S i s p e eu dodLf'I'e r -en t La'l, wi t h ilynbol (2ni )- 1

l. AS ir

e (x, ;,/\) -m

v:he r e t he s yn boL ~E S

dA

i s comput ed r-ecu r-s Lve Ly fr om t he syst em

- 66 -

(p(x,D+d -

A)e(x,I;,). )..v(2T1:)-n

•

MultiPl~cations by P extend the se mi gr ou p

s ~ps

with Re s < 0

to a ,group s-+p s , se~, of pseudodifferential operato rs. 3. Fourier integral operators. An op er2,tor F of the type (1.1) with ke rnel (1.2) is sai d to be a Fourier ,i nt egr al operator ,i f S1 , 1m SO and (11f'xl +

(3.1)

Id

1'f1;1)!11;1

and (Iepyl + 11;1 If~I)!11;1

keep away from 0 for large I; on compact subsets of

St 1'i-J1 2

•

Replacing the op erator (1.5) by

,Jj::,"

L

(I(l~l
,t

q>;):a; )

which still has the property that L = 1, a little calculation shovis that (-1.1) gives bilinear maps Sm)<.

c: (Jl2).-. c~(.n1)

Sm", ftf'- ~2)-~t" (~) continuous on bounded subsets of Sm Example. Let N=n-1, 1;=(e

2,

when m+N+~< ~ •

••• ,e n) • Then if

A(e)

is real and

homogeneous of degree 1,'1'=X (x 1- Y1) + (x2-Y2)1;2+ ... (xn-Yn)en .bel ongs to S

-~~but

cr

1

,
vanishes when

'

(x 2-y 2"" )!(x 1-y 1) equals

Ifx' = 1(A'~2, ••• ,en)1 is as large as I~I and hence

haa the property (3.1).

Singular support.

As intthe preceding example,

ife

can vanish

for some x,y and arbitrarily large e and the differential operator

'% ,-2 ~~d~

which is the analogue of (1.8) can be used- only for such x,y that

Ife l

keeps away from 0 for large enough e. Since f~x

and

- 67 -

L. G£rding


complement Sf is the singular support of the kernel of F

and (1.9) is replaoed b~ sing supp FuC

S1 sing

supp u

with an obvious interpretation af the right s i de . Calculus.

Sums of Pouri er integral operators behave like sums

of pseudodifferential operators but products of Pourier integral operators is a

ve~

complicated affair. What we shall need is

a simple result about the prOduct of a pseudodifferential operator and a Fourier integral operator. Let Lm(f) denote the space of operators (1.1) with a

flO

OIl

C o~2)-+ C~) which can fixed f satiSfying (3.1).

be written in the form Then if a(x,D) is a com-

pactly supported pseudodifferential operator in order a =

r- '

F GLm(
Jl 1

'

=> aF6 Lm-tf(f)

and aF has the amplitude

intepreted as (3.3)

.....,

La (0<) (X''fx)D z

f(z,y,1; )e i f /

ol..~

lz=x

where r(x,y,z,l;) =~(z,y,l;) - f(x,y,l;) -
proof is again rather straightforward. One

v~i t e s

F = F

where F 1 is smooth and F 2 is' such that zero for large I; wh er e the amplitude of F

does not vanish. In 2 the latter case one can calculate as fOr products of pseudodif-

ferential operators. 4. Asymptotic prop erties of the pseudodiff er entialoperator.

function of an elliptic m Let a(I;)ES , m>O, be r eal, ellipspectr?~

- 68 -

L. Ggrding

tic and po s itive f or l arge a(D) on xn is

~.

The pseudodiff er ent ial operatqr

th~/selfadjoint in

L2 (xn)

~ith'

domain Hm• It is co

diagonalised by the Fourier transform and the C funct i ons (4.1)

e(x,y,).) =

( 2~)-n

J

a(I;)<)

are the ke r nels of the proj ect ions

J)

d,E>.

of a(D).

in the s pect r al r esolution

Vie shall get a similar formul a when A i s a

formally selfadjoint order

E~

ei(x-y)1; 'dl;

~lliptic

pseudodifferen tial operator of

on a compact manifoldJl su"tlh that the limits '

(4.2)

,.l im t- 1a(x,tl;) , t ..... t:tD, u

where a is the symbol of

If.:,

exist and are po.s i tive. Th e s pa c e

L2~ with norm Ilull is taken wi t h a fix ed positive C<X> density dx. The closure! of A in L2 is

th~/selfadjOint

and bounded from below, ! ~ c. Let

J) dE~

wi t h ' doma in H 1 be its s pe ct r al re-

solution. Following Hormander (1968) ~e shall i nv estigate the C,. kernel.e(x,y,)

of E)o. for large/- by s tudying t h e Fouri er

transform

. ~ (t )

=

5.-it~

dE'). = e- i t A

for small t. Since, for every re~ s,

'"

II (A-c+1)BU II

in Hs ' all E(t) are strongly continuous maps Hs ~Hs • E(t)U o with u o e H1 is the unique H1 solution of (4.3) Dtu + Au 0, t=O => u=u o •

is a norm and u(t)

!l!he main step is now to show that locally inJ2, for small t

'"

and modulo smooth operators, E(t) equals a certain Fourier inte. ~ral operator Q(t) which can be constructed explicitly. Let x,y ,

be coordinates close to the diagonal chosen s o that dx is Lebesgue measure, let a(x,l;) be the local symbol of 4 and wr i t e

- 69 -

(4.· 4')

Q(x,t,y )

r(

L. G~rding

)

=)q x,t,y,e e i (x,t,y,e)d ~e

for the kernel of Q(t)~ In view of (4.3) we shall try to achieve that, modulo smooth operators, (1)t + A)li(t)"" 0, Q(O)tvidentity• . The first condition leads to

(4.5)

(ft + Dt + a(x,Di'x))q(x,t,y,d '" 0 which should be interpreted like (3~2). Hence it is n aturalto which makes the fi~st term in

require that crt + a(x'fx)E SO

the expansion harmless. To be able to control Q(O) we also require that ~= (x-y)1; + 0(lx-yJ

211;1)

when t=O. These require-

ments 5til1 give some leeway 'and it turns

ou~

that one should

put

cp'

(4.6)

-y(x,y,d - ta'(y,d

where a'(y,l;) is a real S1 function congruent to a(y,e) mod SO ( e.g. a' = Re a(y,I;)) and require that a'(x,'tk) x = a'(y,l;), "f:t(x,y,l;) = (x-y)1; +.Q(!x-y/2IH). ~his is a non-lin~ar Cauchy problem for~ and some reasoning using

(4.2)

shows that it has a unique

real solution in S1.

We shall describe this situation by saying that a" and ~are

adapted to A. An appeal to

(4.5) and (1.13) and some calculations which we

will pass over show that there is a SO amplitude q(x,t,y,l;) vanishing exoept when x and y ~ close such that Q(t) has the required property, namely that

(4.7)

A

X(x, t,y) -

.

'

Q (x, t,y)€

where A E(x,t,y) =

J

C

pO

~ e -itAde(x,y,~)

- 70 -

L. G!J:"ding

'"

1s the distribution k ernel of E(t), t i s smal l and y in a coor4inate patch. Inserting (4.6) into (4.4) and r eplacing q by its main term (21t)-n when t =O and x=y, we get in some s o far

J

unspecified sense

(4.9)

Q(x,t,y)

where

~

e-it"df(X,y,'A)

f(x,y,~) = (21t)-n~

a' (y,d<)

ei~x,y,~) d~

•

In view of this it seems natural to compare e and f and we have

Theorem 1. It A satisfies (4.2) and the phase function and the _.Pl1tb~ a'(y,~) are adapted to A, t hen , as

>. ~,.o,

e(.x,y,).) _ r(x,y,)) = Q(An-1) n ear the diagonal and e (x , y ,}. ) = Q(~ n-1) away from the

d~agonal.

In particular,

e(x,x,'}.) _ f(x,x,~) = Q~n-1) uniformly. The pr oof de part s from (4.7) and we s hal l sketch it. Consider convolutions

(4.11)

Jr ("A -~)de

(x,y,/<)

It?".... "") ) we with f.C ,.r(o) =; and 'p(t =0 exc ept for small t. By ..J 1''1

then know that (4.11) differs from (4.1 2)

(21t)-1~}d(t)

eit>Q(x,t,y)dt

by a rapidly decre asing fun ction of A. Hence the differenoe

(4.1 2)

Jf(~ -f-)de(x,y,{'f) -jR(x,j,_a'(Y,d,y,~).I.+<x,y,dd1;

i s also rapidly dec reasing

~h e r e

R(x,~,y,~) = (21t)-~ji(t)q(x,t,y,1;)eitA

dt

- 71 -

L. GArding

is a SO func tion rapidly d e c r' e aad.ng in ~. Rough estimat es sh ow that the second t erm of ( 4.1) 2 is

f

f

f\n-1 ) and henc e, cho sing

Q~

0 we get

(4.13)

n- 1) e(x,X,ft+1) _ e(x,x,}.) =Q(A

and the s'ame estimate for .e(x,y;/J. A new a t tack on (4.12) using (4.13) then brings ~he desired r es ul t . 5. The Sase of diff erential operator§. The pre cedine t he or em carries over to .di f f er en t i al op erators. As in the introduction, let P be an elliptic differential operator of ord er m>O on a paracompact manifold.!l of. dimensi on n and suppose t ha t P is formally selfadjoint. In a coordinate patch t.J wher e the gi v en density dx onJlcoancides with Lebe sgue measure we choose a funct ion

~x,y,~) , homogeneous of degree 1 in. e such that

(5.1)

p(x,~) .. p(y,d ,'o/=(x-Y)I; +.Q(/x_y/2/ e l)

.

where p is the principal symbol of p. Lete(x,y,)) be the spectralfUnction of a selfadjoint ext ension P of P bounded from below. Then we have HOrmander's main result Theorem 2. The s pe ct r al fUnction is QCl\(n-1 )/m) locally uniform. ly outside the diagonal while close to the diagonal

(5.2)

e(x,y,~) -

(2lt)-n

J

a i (x,y,dde = O().(n-1)/m) p(x,~)<) -

locally uniformly.

!

When .Q is compact, this follo ws from Theorem 1 if we put ·1/m = P which is pseudodifferential and hab the s pect r al func-

tion

A~e(x,y,~1/m). In

fact, (5.1) shows that p(x,e)1/m

and

~are adapted to A. The non-c~mpact case follo ws fr om t h e com·

- 72 -

L. G§ rding - compact case f ol l ows f rom the compact c ase 'an d on e of the

compariso n an d summa bili t y ·the orems in H8rmander . (19 66) whi ch we will summarize in t he nex t se ction . We

end this sect ion

s h~ l

by showing that ' (5.2) as a general r esult is be s t pos sibl e . When

11

is compaot,

P'has

a compl ete orthonormal s et

[fk~

of eigen-

functions and we have

Hence N

OJ =

J

e (x,x,).)dx

is the numbe r of eigenval ues

(5.3)

N().) _ . ( 2n; )-n

<)

/.J

p (x ,1; )<)

and (5. 2) s ay s t hat dxd l; = Q(

:>. (n-1 )/m ).

. n+ 1 When P i s La pl ac e I s ope rator on t he uni t s ph ere in lr , t h i s result i s be s t pos s ible. In f ac t, then the eig envalues are A = k k (k+n-1) with multiplicities (n +k ) _ (n +k- 2) '" kn - \ Hen c e t h e n n jump of N('A ) a t A i s of t h e ord er of magnitUde ).~n- 1 )/ 2 and k henc e equal t o the error t erm of ( 5.3). 6. Riesz means an d s unma bi l i t y. The Ries z mean of

ord er~

of a

f un cti on f(t) v anishing f or t~O an d l ocally of bounded v aria tion i s def i n ed by the fo rmula let)

=J:;

0(

(1 - si t ) dfCs )

whe r e Re d.. >-1. When Re ol >0 t his c an be . .·cri tten aa f d.() . t =o(t'~- 1 In

particu~ar,

if f

fo{( t ) =

at least wh en

«.

ft f (s )r,1 0

= ~~it

01t0

- s I t )0£-1 ds •

, a chan g e of

J~ e i tX:J ( 1_x )ol. -1 dx

and ~ ar-e i nt eGers 3.-1'1d

eof

0(

vari~bles s

= Q(t~

= t x gi ves

- Re O( )

+1. In f act , t hen 0<

integrations by part s ~how t he int eGral t o be Q(t~) . T~is sinpl e

- 73 -

L. Garding

example indicates that, taking the Riesz mean of order 0( of a function should diminish it s growth at infinity by a factor -ReD( of t provided the f un ct i on i s not too large and oscill ates suffi ci ently. Thi s i s als o t h e general picture. Th e f ol l owi ng precise r esult due to Hormand er (1966) i s s pe ci al l y ada pted to the s tudy of spgctf'al

' flinnUoi!s dhf :', ~ ll i p,t ic :' o pe rat o rs ~ Of :c' o:l'der

m. One as sumes that f = Q(t N) for s ome N and introduces the Fouri er-Laplace transform

whos e bound?xy v alue : in th e sens e of di stributions i s t h e Fouri er transform Ofdf(t m). That f os cillates suff i c i ent ly for large t i s then expre ssed by the condition that F(~) be an alytic cl os e to the ori gin and that F (Ink) ( 0 )

=0

for k=O,1, ••

There is al so a Taub erian condition

giv ing a nne-sided bound on the amplitude of the os cill ations of f. The conclus i on i s then t ha t

provided Reo( >0 oro{=O. We shall no w s et the s tage f or the a ppl i cat ion of (6.3). Let

J2 1

and

51 2

be t wo mani f olds and W a char t of both , i dentifie d

wi t h an op en su bs et of

xn.

Let P1 and P 2 be formal l y po s itive

elliptic diff eren tial op era tors in5L andJl 2 which are equal 1 in t..J • Sinc e

onl~'

constants c omnu t c ':'i t h ell ip t i c op era tors,

t he den si tie s Of~1 and

Jl

2

a re pro)or t ional ~nd wc QSOQDe

th 2m to be Leb e r-gu o mer.su r e inW. Le t ='1 ' P2 be t ':iO :;>o r: itive

- 74 -

L. G£rding

selfadjoint ext ensions of P1,P2 and e 1,e2 the s pect r al f un ctions. Then both e and e ar e of the form 2 1 dl;; + (n-1 )/m) p(x,d
J

s«

when x=y and, if Re 0()0 ortX = 0, 0( ) «n-1- Ree( )/m) e~ X,1,t ) - e~( 2 x,y,t = Q ~ where all estimates are locally uniform in x and y. We shall

(6.5 )

sketch the proof of this basic result. In the first Rl~ce,

(6.4)

follows from Theorem 2 when the

corresponding manifold i s compact and hence we may u se for one of the spectral 'funct i on s when provin~ in passing that the restriction

t~

(6.5).

(6.4)

We note

positive pperators rather

than ones bounded from below can be removed by changing the origin in the Riesz means. The proof ~f

(6.5)

depends on classical properties of Green's •

function G(x,yjz), i.e. the kernel of the resolvent (p -zI)where P

= P l' P2.

It is a C"'" function except when x=y or z ~

some c)O and analytic in z and we have G(X,y,i z) =

J o.~z)-1de(x,y,).)

and - 1( - ( 2~i ) -l~

i;;

e

-f3z 11m (

j

) G x,y,iz dz =

e

in the sense of kernels. Here Im3 '< 0 and of the angle Ze:: larg zl <

E:<~/2

-itS ( m) de x,y,t

ie

is the boundary

• Now the difference H=G 1-G2 11m

of the two Green's functi ons turns out to be Q(e- C I Z 1

),

(c)o ), outside Ze and hence if f=e the corresponding 1-e 2 is

1

- 75 -

L. G~rfting difference of the spectral functions vre get .~ 1/m -e• -.l.,.)Z -~UJ m -(2~~) . e H(x,Y,iZ)dz = e df(x,y,t).

=} . .

~trE

It i s then clear that t he left side i s an alytic at the origin and that i tsj -derivatives of order mk wi t h k=O, 1,. •• vanish forJ =0. m) It i s immediate to see t hat f(x,x,t h as the p~op erty (6.2) ·i f e 1 h as t h e prop erty (6.4) an~ this proves (6.5) wh en x=y which is suff i ci en t . The s ame line of ar gu ing gives s tat ement s about the Ri esz means

ro f) ) dy ec< (x,f,~) = je (x,y,~ r(y of the eigenfunction expansion of a func tion f. Here we s hall only n ention that if f~ LP(SlJ , 1~p~2 and if f h as conp ac t support, when

P<2i t hen (6.6) converges to zero locally uniformly outside the support of r when Reo( ~

(n-1 )/p and to f(x) at t he Leb esgue

points of f ( in the LP sense) wh en 2e~>(n-1)/p. For the details t h e r ead er i s rerer~ed to Ho rnander (1966).

- 76 -

Referenc es. E. Balslev and J.M. Combes. Spe ctral pr operti es Schrodinger operators wi t h di latat i on

Ju. M. Berezanskii.

Exp~~ si on s

o ~ m <~y-bodJ

in t e r ~cti ons.

Co~n .

rJ~th .

in ei g cni'unction£ of £elf2.Lj oi n c

operators. AMS translations 17 (196 8). L.D. Faddeev. Mathematic al probl ems of the qu antum s ce,tt e r i nc

theory of the three body probl em. Trudi Stekl LXIX(1963)1-120 ( In Russian )

L. Hormander. On the Riesz means of s pe ct ral f un ct ions an d eig enfunction expasions for elliptic differ en tial

op cr~to rs.

Rec ent

Advances in the Basic Scienc es. Yeshiva Univ. Coni' . Nov . 1966. 155-202. - ' • The spectral function of an ellipt i c oper at or . Acta Ma t h . 121(1968)193-218. T. Ikebe. Eigenfunct i on expansions

ass o c i ~ted

wi t h Schr odinc er

operators and their applicat i ons t o Sc att erinc The ory. Arch . Rat. Mech. Anal. 5(1960)1-34. -.

On the Phase-shift formula for the Sca tt er i nC oper a t or .

Pac. J. Math. 15(1965)511-523.

5.1. Kuroda. Scattering theory for differential opera t ors I,II. Journ. Math. Soc. Japah 25,1 and 25.2 (1973)7 5-1 04 and 222-2 34. K.

Maurin~

General eigenfunction

expans~ons

an d unitary r ep-

resentations of topological groups. Mon, llat .48 ( 1 9 68 )~ arszawa . R.T. Seeley. Complex Powers of an Kath. 10 (1968) 288-307. (AMS).

Elli p t~c

Operator. Symp. Pure

- 77 -

L . Garding B. Simon. Quantum Mechanic s f or Hamiltonians defined as quad-

ratic fDrms. Princeton Series in Physics (1971) • • Resonances in n-body qu antum systems and the

founda~i ons

of time-dependent perturbat i on theory. Annals of Math. 247-271. ~.

Weyl. Uber gewohnliche Differentialgleichungen mit

Singulari~

taten und die zugehorige Entwicklung wi l l kUr l i cher Funk tionen.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E.)

VALEURS PROPRES DE PROBLEMES AUX LIMITES IRREGULIERS: APPLICATIONS

CHARLES GOULAOUIC

Cor sot e nut 0

a

V are n n a

da1 24

ago s t

0

al

2

set t e m b r e I 9 7 3

VALEURS PROPRES DE PROBLEMES AUX LIMITES IRREGULIERS par

APPLICATIONS.

C. GOULAOUIC (*)

UNIVERSITE DE PARIS XI 91- ORSAY

PRELIMINAIRES

1.- Introduction.-

- Ce cours comporte essentiellement deux parties. - D'abord on donne quelques methodes d' etude du comportement

asym~

totique des valeurs propres de prob.Lsmee aux limi tes plus ou moins irreguliers ; on demontre quelques resultats sans pretendre faire une etude systematique et exhautive ; les principaux exemples traites Ie sont en vue des applications; les idees de base sont dues A.N. KOLMOGOROV [25]

et ont conduit

a de

a H.WEYL[36]

et

nombreux resultats recents

([ 12] [13] [18] [20] [29] [31] [35] ... ). On s' est beaucoup inspire ici d'un travail de

G. METIVIER [29].

- Dans une deuxieme partie, on s'interesse surtout

a donnees

a des

analytiques ; on etudie certains espaces de vecteurs

problemes 00

C

et

analytiques" 1*~La-Cont;IbUtion de l'auteur aux resultats exposes a ete obtenue en

collaboration avec M.S. BAOUENDI (Purdue University).

- 82 -

C. C.olll aOllj c dont on montre l' isomorphisme ave c des es paces de su ites (grace aux r esultats precedents de theor i e spectrale) . On met en evidence des relations entre, d'une part la r egularite analytique et, d'autre part, la croissance des valeurs propres et des inegali t es (di t es "de BERNSTEIN" et "de MARKOV") sur les fonctions propres ~t leurs prolongements. -

~plan

suivi est:

1 .- Formule du min-max ; epai s s eurs suecessives et localisation. 11.- Comportement asymptotique precise des valeurs propres de problemes irreguliers. 111.- Espaces de vect eurs

ceo

et analytiques •

IV.- Fonctions propres de "bons" prob.Iemes aux limites

inegalites

de BERNSTEIN et de MARKOV. - Les methodes utili s ees reposent es s ent i e l l ement sur des notions s imples d'analyse fon ctionnelle (en par t iculier la proprie t e de BAIRE) et des proprietes de certains es pac es hi l ber t iens de distributions.

-On commence par que l ques rappels et r emarques tres el emen t air es afin, ess entiellement, de pr eci ser Ie cadre dans l equel on se place et de fix er les notations et le s definit ions. 2 . - Compacite et espaces de SOBOLEV._ Soit

H un es pace de HILBERT de dimension infinie (ce sera en

gene r al ' L2 (Q) ou t eur

Q

e st un ouvert de

(A,D(A)) lineaire n on borne dans

mn

) ; on considere un opera-

H e t f erme

ral la r ealisation d'un probleme aux limites associe f er enti el

ce s era en gene-

a un

operateur dif-

• ~ suppo s era t oujours ~ l'inj ection ~ D(A) ~ H

- 83 -

. C. Goulaouic etant muni d'une norme equivalente a celle

D(A)

compacte, l'espace

~

du graphe. - Avant de donner quelques exemples, precisons les notations et defi. nitions que nous dOnnons des espaces de SOBOLEV

I a .' -- a f'·

a

D

= D~1

h I · ~n ·

••••• D~

et

DEFINITION 0.1.

I\: = -i

.

n

Q un ouvert de R

Soient

o

.

o~

et. m € pour

IN

; on note

Iexl

smI

et on le munit de la norme hibertienne evidente. ~(Q)

l'adherence dans

Lorsgue l'eepace

0

~(Q)

a support

e~

tions

nf) a support

de FOURIER) aux cas

tion dans

a

a

Q

0

de classe

tout ouvert

m€

0

IR •

=0

~(C1)' On note aussi

support compact dans

Q d'ele-

peut se prolonger (par la transformation

l'espace des distributions sur

1

a

em)

s'identifie avec l'espace des elements de

~(lRn)

~oc(Q)

des fonc-

0

dans

La definition de

On note

compact dans

s'identifie avec l'espace des restrictions o

~(

o

de l'esp~ce ~(Q) = e~(Q) o

est assez regulier ( par exemple

~(Q)

gID(Q)

(i.e :

Q dont la restric-

01 compact contenu dans Q) est

~omp(Q)

l'espace des elements de

gID(Q)

Q

11 est facile d'etendre toutes ces definitions aux cas de varietes

- 84 -

C.

assez regulieres (au lieu de

0) ; nous aurons

a y revenir

Goulaouic et nous

donnerons a.mesure des besoins les resultats utilises sur ces espaces (cf.[27] ~ar exemple) ; signalons une forme de l'inegalite de POINCARE Si

Q

est borne, il existe

u (

~(o)

II u II ~2(0) et

C(O»O

tel que l'sn ait, pour tout

< C(O)

C(O) tendant vers 0

quand le diametre de

Q

tend vers 0

- Voici quelques resultats elementaires de compacite de l'injection

• Lorsque de

Q

~(O) dans • Soit

~ompacte,

m>

a une infinite de composantes connexes. l'injection

2 L (0)

n'est pas compacte •

• Pour que l 'injection de

s.oit

il est necessaire que lim

dist(x,6o) = 0

IX 1-400 x(Q

Cette condition est d'ailleurs suffisente lorsque

Q

est un ouvert de

~(Q)

se

m

n (mais pas de IR ) grace a l'inegalite de POINCARE. • Des resultats plus fins de compacite de

o

trouvent dans des travaux, par exemple, de R. ADAMS, C. CLARKS, P.J. ARANDA et E.P. CATTANEO [2] ••• et repo~ent essentiellement sur un raffine ~ent

de l'inegalite de POINCARE. • L'injection de

lorsque C1 et

Q

~(O) dens

2 L (Q)

est compacte pour

est un ouvert borne assez regulier (par exemple

cf . [1] pour des cas plus generaux); mais si

0

m~ 1

Q de classe

est borne et

- 85 -

trop

il se peut que cette

irregul~er,

C . Goulaouic ne soit plus compacte •

in~ction

• (Espaces de SOBOLEV avec poids). On peut considerer

de nombreux

types d'espaces de SOBOLEV avec poids ; signalons seulement pour Ie moment un exemple Soit

0:>0

{u €

et

:15' (Rn )

; D u € L2(IRn) i

pour

i=l, • •••• , n

et

2 2 (1 + Ix1 ) 0: u € L (JRn ) I

II est facile de voir que l'espace 1 I injection de

E

L2(~n) vn

dans

0:

E

etant muni de sa norme naturelle,

0:

est compacte •

.Par contre, on verifie aisement que l'injection de

2 2 [u € L (1R) ; xu' € L (1R ) } dans

3.- Operateurs

auto~adjoints

2 L (rn.)

positifs et problemes variationnels.

En plus de la compacite de l'injection de suppose que

D(A) dans

H

A est auto-adjoint strictement positif, c'est

D(A) est dense dans D(A*) = {u € H

H

; ~

(Au,v) = (u,Av) u € D(A)

n.'est pas compacte.

et egal

dire:

a

(Av,u) soit continu sur

pour tous

a

on

,

u,v

dans

D(A) pour la norme de H

D(A) et

(iu,u)

>0

pour

, uFO

On sait qu'alors Ie spectre de A € ( tels que

A (clest

a dire

A-AI ne soit pas un isomorphisme de

constitue par une suite

l' ensemble des D(A) ~ H)

est

(A) j€ IN de valeurs propres de multiplici te

~ (repetees avec leurs multiplicites) que l'on ordonne en une suite

non decroissante et qui t end vers

(lp j ) j c IN

orthonormee dans

associees aux

~

• On peut trouver une base

H consti tuee par des f onctions propres

(Aj) j E IN • On a

- 86 -

C. Goulaouic

= {u =.E J=O

D(A)

u

m

jTj

si

et • Lorsque

A est seulemerit auto-adjoint borne super-Leur-emen t ou

inferieurement, on se r amsne au cas ci-dessus en prenant A +A o

avec

A o

ou

A -A

o

convenanble .

A es t auto-adjoint, on peu t, pour les proprietes des valeurs

Lorsque

propres et fonctions propres, le remplacer par

2+I A

qui est strictement

positif. D'autres reductions sont possibles et des arguments de perturbation permettraient anssi d'etendre

a des

cas non auto-adjoints les resultats

obtenus dans le cadre precise ci-dessus. Formulation variationnelle.- Soient que

V,H deux espaces de HILBERT tels

V ~ H avec image dense (*) • Soit

continue et

coercitiv~

sur

V

a

une forme sesquilineaire

a dire

c'est

qu'il existe

M et

a>0

tels que l 'on ait a(u,v) ~ Mlul~ l/vl~ pour 2

la(u,u)1 ~ a~u~v En identifiant

H

a son

pour

u

et

v

dans

V

u (V

antidual, on a

V G H G V'

de LAX et MILGRAM di t que l' on definit un isomorphisme V'

A de

V sur

par (Au,v)

= V' xV = a(u,v)

On definit un operabeur- non borne

D(A)

= {u

pour tous

(A,D(A»

Ci

F

dans

V H par

( V ; Au ( HI

Pour deui espaces vectoriels topologiques

par E

u,v

correspond ant dans

-------- - - - - - (*)

et le lemme

l' inclusion continue de

E de

F

E et

F

,on designera

- 8·7 -

II est immediat que, si l'injection

V c+H

C. Goulaouic est compacte, l'injection

D(A) ~ H est compacte. De meme,OIl montre aisement que L'operateur

A est auto-adjoint lre,p. auto-adjoint strictement positif)si

et seulement si la forme

a

est hermitienne (resp. hermitienne positive) .

II suffit pour ce La de voir que I' on a aussi v ~ (Av,u) de

H

sur

D(A)

soit continue pour la topologie

I .

On se placera dans la suite dans une formulation variationnelle, d'abord parce que cela permet de considerer des problemes "aux limites" dans des cas tres irreguliers (~o aux traces)

trop irregulier pour donner un Bans

et aussi parce que ce n ',est pas une restriction pour I' etude

que nous avons en vue, etsnt donne qu'a un operateur on peut associer

f

V

1

= D(A)

a (u , v ) 1

V

1

(Au,Av)

~

H

pour

u

et

v

dans

(A,D(A))

dans

H

-ss -

CHAPITRE 1.- FORMULE DU MIN-MAX

C. Goulaouic EPAISSEURS SUCCESSlVE3

ET LOCALISATION •

On donne d'abord les proprietes plus ou moins

cl~ssiques

des epaisseurs

successives d'un sous-ensemble d'un espace norme, qui nous s erviront par la suite; puis on fait I e lien avec les valeurs propres d 'operateurs, et on montre comment on peut en deduire des encadrements de ces valeurs propres. I.- Epaisseurs successives (cf.[25]) DEFINITION 1.1. - Soient et

k E IN

E un espace norme,

; on appelle

B une partie de

k-d eme epaisseur de

B dans

E

E Le

nombre (eventuellement 00)

II

inf sup inf X-"j liE EkE lh x E B y E Ek . designe l'ensemole des sous-espaces de E de dimension

En notant

d(x,~)

la distance de

x

a

E k

,on peut encore

ecrire inf

sup

E k

x E B

On peut -obs er ver aisement que La suite· k~ 0k(B,E) • Pour en

0

k=O

, 0 (B,E) o

et cont enan t

• l' application

est non cr cissante . est Ie rayon de la pl us petite boule c entree

B

B 1-+ 0k(B,E)

est non decrof.ss ante .

- 89 -

C. • Pour tout

a) 0

et tout

Ok

aB,E) = aOk(B,E) •

B

• En designant par

disquee de

B

Goulaouic

k E IN

l'adherence de

on a pour

k E

B et par

r(B)

l'enveloppe

~

Sont egalement classiques mais un peu moins evidentes les proprietes suivantes PROPOSITION 1.1.- Soit

F un sous-espace dense de

~neede

F

I

, k E

E,

B

une

; on a :

~

°k(B,E) = °k(B,F)

Demonstration : Soit

£)0

montrons que l'on a pour tout

!

On note de fayon generaIe

Si

X est un sous-espace de

.Soi t

~

un eous-eapace

E

la boule unite de l'espace norme

de

F

X

on a

un sous-espace de dimension F k

k E IN

de dimension

d(B,Fk) ~ d(B, ~) +

k

k

de

E

on a Ii chercher

tel que

£

Grace Ii (1.2), il suffit de realiser

ce qui est immediat. PROPOSITION 1.2 .- Pour que la suite tende vers

0

quand

k-ooo

une partie precompacte de

(Ok(B,E»kEN

soit bornee et

il faut et i l suffit que E

B soit

- 90 -

C.G01!J.a01!ic Eem2~~~~~~~~ : (on rappelle que

a

est compact et equivaut

dire que : Pour tout

nombre fini de boules de rayon Soit

B precompact

£ recouvrant

et soit

(B(Xi'£»i=1, •• ,N qui recouvrent engendre par les

(Xi)

;

Inversement, soit

< £/2

E

£)0

B • Soit

E£ de

(puisque

, i l existe un

B)

B

E

l'espace vectoriel

a

E est surement Ok - 0

quand

<£

k-
E de dimension f inie tel que

. II est alors facile de recouvrir e

B

; il existe des boules

£>0 et supposons que

fini de boules de rayon -£

£)0

la distance de

il existe done un sous-espace d(B,E ) e

B precompact s ignifie que

B par un nombre et que

d(B,E ) = d(B,26 (B,E)E ) e 0-£

est compact ). PROPOSITION 1.3. - Soient

E et

F deux espaces normes et

T € ~(E,F)

(espace norme des applications lineaires continues

de

F) . Pour toute partie

E dans

B de

E et tout

k€m

on a 0k(TB,F) ~ ~TU 0k(B,E)

E~~~~~~io~

: On a

T 'k(E) C

tld

'/F)

< inf sup inf I/Tx-y/IF - Ek € lJk (E) x € B y € T~

<

II Til

COROLLAIRE ,., . - Soient soient et

E,

F, C F

et

F,

inf ~

sup x € B

inf z€E

Ilx - zl~ k

E, F des es pace s normes et des espac es de BANACH

tels que

; on suppo se que la restriction de

continue et sur j ective de

E,

dans

F,

T €~(E,F)

T

E,C E

a

E,

• Alors i l exists

est C>O

- 91 -

C.

Goulaouic

t el que pour tout

k €

m

eela r esulte i mmedi atement de l a propos ition '. 3 et du fai t que ~1

cont ient un homot he t i que de

!1

PROPOSITION 1. 4. - Soient sous-espace de

E un espace pr ehilbertien. B une partie d e

E.

F

F un

k€N

on a :

Demonstration:

On peu t suppos er que

et

F un sous-espace f erme de

de

E sur

F

puds que

"P

E

II

E es t un espac e de HI LBERT

so it et

=

P

la pro j ection orthogonale

PB = B

on en ded ui t de la

propositi on 1.3 .

L'inegalite inver se es t evi den t e . PROPOSITION 1.5 . - Soient

E.F.G

que ' E C FC G

trois esp ace s normes tels

pour tout coupl e d ' en t ier s

on a

(k , k ) 1 2

ok +k (~ .G ) ~ ok (~.F)' Ok (!,G) 1

Demonstration

cca.k (F)

et

l'

Ek

2

x € E

ok

1

c ~k

c

1

2

(~,F) < £1

(G)

et

ok

2

(~.G) < £2

, il existe

t els que

2

E

3y € E k

Vz € F

3t € E k

Yx

Done pour

2

: Si

on peut t rouver

1 2

y €

" x-y l~ 2 £, UxIIE

II z- t lb ~

~1

£21~IF

et , en pos an t

z=x.-y

- 92 -

c. on trouve

t (E

k

2

I x- y - tIl G ~ y+t(~

au plus

k

Goulaouic

eomme ei-dessus ; d'ou

E

Ilxl E

1E2 +~

1

sous espaee de

G de dimension

2

1+k 2

Ce qui implique la proposition. Donnons tout de suite un exemple fondamental pour la suite. PROPOSITION 1.6. Soit

(~j)j(N une suite non deeroissante de

reels strietement positifs. On note semble de

i

2

defini par ':

liors, pour tout Ok ( B,i

;Q~!!!~~~ati~: k

2)

Soit

B Ie sous-en-

k (IN

, on a

-1/2

= ~k

~ Ie

sous-eapece de

i

engendr-e par les

premiers veeteurs de la base hilbertienne eanonique (eonstitue par les

elements dont toutes les eomposantes

f .

pour

J

j~k

sont nul.Les )

on

a immediatement

d(B,~) = ~1/2 Soit

F ,un sous-espaee queleonque de k

trouver k

f

~=olfjI2~j

= (f = 1

0

i

•••••• ,fk,O • • • • ) ( ~+1

2

de dimension orthogonal

a

•

On a evidemment d(f ,F = Ilf k) done

d(B,F k)

~ ~1/2

~ ~ ~t/2

,ee qui termine la demonstration.

II en resulte immediatement Ie

k F k

on peut avec

- 93 -

C.

COROLLAIRE 1.2. - Soient

et

Goulaouic

v=(v .)

deux suites de

J

reels strictement positifs tels que la suite

(~j/Vj)

soit non decroissante. On note: E

!f=(f .) € (l

B

I (f J.) c «?

; IIflE ; /If.J

J

On a, pour

k € vk

11

If J.1

= I:

2

J

u, < 1 J -

2v

.
J

I

~

Ok ( B,E) = ( ------)

1/2

~k

11.- Formule du min-max

([15J[36J).

On considere la situation variationnelle

V dans

(V,H,a) ou l'injection de

H est compacte et d'image dense et ou la forme

hermitienne positive continue et coercitive sur sur

V ). On note

(Aj)j€~ et

a

est

V (en abrege

A l'operateur non borne dans

h.c.c.

H correspondant et

(~j)j€~ respectivement les valeurs propres de

base de fonctions propres correspondantes orthonormee dans

A et une H • On a

donc V = If = I: f.~ . . j J J

plou on deduit, en utilisant la proposition PROPOSITION 1.7. - Pour tout

k €

1.6. : ~

,on a

.I 0k(!,H) = ~1/2 On peut enoncer ce resultat sous d'autres formes equivalentes : En munissant

D(A) de la norme I f I D(A)= 1/ Af II H

on a, pour tout

k € IN

Ou encore une formulation classique, mais que l'on

rl'utiliser~

pas iC1

- 94 -

c. Pour tout

k € IN , on a

Goulaouic

(Af,f)

inf

Ilf'~

fEV

f.L~

On utilise la proposition 1.7. pour "estimer" Lea valeurs propres connaissant d'espaces

Vet, inversement, pour calculer les epaisseurs successives V associes

a

des operateurs dont les valeurs propres s'obtien·

nent ai semerrt , Signalons d'abord quelques criteres de comparaison (corollaires immediats de ~a proposition 1.7). a

PROPOSITION 1.8. - Soient

A

1

H associes

a

tout

,

k €

~

PROPOSITION 1.9.- Soit dans

1

et

A>O

a

-

2

deux formes h.c.c.

2(f,f)

pour tout

H ; on note

sur

f € V •

A

les operateurs non bornes dans

et

a

2

2

respectivement, on a pour

W un sous-espace ferme de

rateur defini par

Soi t

et

a (f,f) < a

V et verifiant En notant

1

(~k)

dense

V

les valeurs propres de l'ope-

(W,H, a) ; on a, pour tout

k € IN ,

on note : E

1

k)M~(~,H) ~

1

c'est Ie nombre de valeurs propres de

A inferieures ou egales

il sera aussi designe par

N(A).

N(A,A) ou

a

et

On a in~ediatement un resultat sur les sommes hilbertiennes (finies ou infinies) :

- 95 -

C . Goulaouic - Soit tvl ,Hl, al)tEL .une suite finie ou

PROPOSITION 1.1 0 inf ini e

de situat i ons variationnelles t elle s que :

Pour tout

i)

,

lE L

a

et d 'image dens e et

ii) I l exi ste M>O u/V

l

On definit

1Je =

(u=(u )

'lr={u=(u ) . u EV

t

'

et

t

iu l

V'c; :Ie et ,

pour t out

N(X.,

'lr , %)

t N(X. , Vt,H

Demons trati on :

2

V

t

l EL et tout

J,

L'in j ecti on continue de

2 Elu 1 t H

t

a (u,u)

On a

'=

t

Mal(ul , u t )

t

et

u/H l

t

t

1

IIH ~

est compacte

C+ H

est h . c . c . sur

1

es pac e s

~ es

V1

tel que pour tout

IU l

' on ait

l 'in j ect ion

< 001

t

Ea (u ., u )

1 t

t

t

< 00 l.

DO l)

tt dans

~ r e s ulte de ii ) ;

ensui t e il suff i t de r emarquer qu' on obt i ent l es .val eurs pr opres de l' ope-

('li, 1e .,

r ateur defiIli par

des operateura definia par

a)

en pi-enant toutes

(vl, Hl ,a

l)

pour

lea valeurs propres

t E L

Remarquons que dans I e cas de sommes ou pro duits fin i s non necessairement hilbertiens , on a encore aisement de s inegalites sur l e s val eurs proprea ou lea epaisseurs successives ; par exempl e , soi ent

(E ).

i J.=1, ••••• ,N

pour chaque

et

(F i)i=1, ••. • ,N des espaces normea t els que

i=1, •• ••• ,N ; on note

normes respectives

N Ilf liE

i~1 N

II f /IF

i~1

I f. J.

E =®E . et i J.

F =(±)F. i J.

, munis de s

P ) 1/1' IE. J.

I f . liP )1 / p J. F. J.

' avec

EiC:;Fi

~p~oo

- 96 -

(k.. ).

Soit

1 1=1, ••• , Ok

+ •••. • + k

1

C. Goulaouic on a :

N des entiers posi tifs ou nuls

<

n

On va appliquer ces quelques resultats

sup ok ~,F.) 1
a l'estimation

d'epaisseurs

successives dans Ie cas d'espaces de fonctions ou de distributions.

III.- Localisation et exemples.Localisation :

E

sur

tels que et

E

et

(&') ' -1 1

1- , • • • • ,

N des ouverts de

~n

1=1 1

Soient

de

Q

Q c.U &.

tels que

Q

Soient

N

et

F deux espaces de Banach de f onctions ou de distributions

Soient

E q.F

par restriction

F

a

Q

i

corollaire 1. 1. on de duit que, pour

E(Q.) 1

et

les espaces deduits

F(Q .) 1

' munis des normes quotients. Du i=1, .• •• ,N

, i1 existe

C. >O 1

tel

que l' on ait pour tout Par ailleurs N

2

t=1ei=1

dans

soit

sur

E(Q.) 1

Q

et

de

F

1

N

@

i=1

F(Q ;)

1

€ W~)

i

pour

i = 1, • ••• ,N et telles que

. On suppose que Ie produit par

continu de .e .E dans de

8

clans F

dans

E

F(Q .) 1

pour

k € IN

e

i

est continu de

E

et que Le prolongement par 0, ... , est i=1, ••. ,N

• Soit

P

l'application

definie par N

p((u.)) = I: 1 i=1

~ 1 1

On a fait l es hypotheses suf f i s ant es pour assurer que

Pest continu de

-97 -

@

i

F(O.) J.

@E(O .) 1

,1'

J.

dans' F et que sa restriction dans

a

C. Goulaouic est continue de

Ef> E(O. ) i

J.

E et surjective; il resulte encore du corollaire 1.1.

existence de

C>O

fel que lion ait. pour tqut

k€1N ,

0k~,F) ~ Cok(t E(O~), ~ F(OI)) Et en utilisant (1.6) on obtient, pour tous

<

( 1 .S)

C

sup l
k ••• • ,k dans N. 1

m

Oki (E(O.), F(O.)). __J._ J.

Les hypotheses ci-dessus sont verifiees en particulier dans le cas des espaces de SOBOLEV ou des espaces de SOBOLEV avec poids ; on pourra done obtenir, par une etude locale. une estimation dlepaisseurs successive$ dans de tels espaces. c'est

a dire

aussi une estimation des valeurs pro-

pres de problemes aux limites convenables sur un ouvert

0

de

n

R .•

Exemples d'estimation dlepaisseurs successives dans les eapaces de SOBOIEV

(cf. aussi (16]~7]) •

Pour deux suites (ou

~. ~k)

pour tout

a~(~)

~=(~k)

et

si et seulement si il exist$

k (IN

(ou du moins.

Exemple 1.1. Soit

Exemple 1.2. Soient

0

k

et

Cl~C2>O

~

tels que l'on ait

assez grand)

un ouvert borne de

m >1

a:

• on convient de noter

0

n

Il.

et

m>

un ouvart borne regulier

on a

Q de

- 98 :...

C.

c Laaae

c!')

Goulaouic

on a

On trouve ces resultats, par exemple, en utilisant la comparaison avec le cas de problemes aux limites pour lesquels on sait calculer explicitement les valeurs propres (comme le probleme de DIRICHLET ou de NEUMANN pour

II dans un pave).

Exemple 1.3. On note

I =]0,1 [

et

2 V={u € L ( I )

muni de la norme hilbertienne naturelle. On a

On utilise, pour cela, la connaissance des valeurs propres de l'operateur

de LEGENDRE

-

d · . 2 ~(l-X ) ex

d

---dx

sur

]-1,+1[

Exemple 1.4. (cf [13J[20J). Soient

(g

o

variete compacte de classe a la distance a 00 V

= {u

C2)

et

0

un ouvert regulier de

n R

~ une fonction equivalente sur

on note

2 € L (0)

norme hilbertienne naturelle. On a :

pour

n=l

pour

n=2

pour

n>2

Pour cela, on calcule les valeurs propres de l'operateur D.(l-!xj 2)D. J.

J.

sur la boule

{x

c IRn

; Ixj

2

=

n

2

i~l xi

<11

puis

- 99 -

C.

Goulaouic

on utilis e l es arguments ci-dessus de localisation. Exempl e 1.5. Soit

V

=

1(R lu € H )

hi l ber t ienne naturelle. On montre que

.. k

-1/2

, en utilisant

Ie calcul des valeurs propres de l'operat eur d' Hermite 2 d 2 +x sur dx2 Remarque 1.1. - Soient non borne dans

L

2

(Q)

Q un ouvert de

~n.

et

(A,D(A»

un operateur

auto-adjoint positif et tel que l'injectidh

2

D(A) ~ L (Q). soit compacte. Si on a

Ifcomp(Q) ~ D(A)

alors, il existe

C>O

tel que l'on ait pour

A. (A) J

< C j2m/n

-

j~

1,

•

Ceci resulte des criteres ci-dessus de comparaison et de l'exemple 1.1.

- 100 -

C . Goulaouic

CHAPI TRE 2.- COMPORTEMENT ASYMPTOTI QUE PRECISE DES VALEURS PROPRES DE PROBLEMES IRREGULIERS .

Pour les applications que l'on envi s age ici, il suffit d ' une conn aissance assez grossiere du comportement asymptotique de s valeur s pr opr e s (noMe

::I)

Cependant, on peut, dans des cas asse z generaux , donne r un

equi v alent de ce comport ement asymptoti que : On s e s ert d ' un r esultat precis de L. GARDING [19J dans le cas de problemes tres r eguliers, et d 'une localisation fine degagee par NORDIN [31 J pour I II cas d ' un operat eur elliptique degenere.

Rappelons leresultat classique ([ 19J)

~ IRn tel que

a

soit de classe

2m C ; soit

tiel formellement auto-adjoint d'ordre elliptigue sur

Q

une realisation de

2m

,de partie princ ipale

dt. auto-ad jointe

un ouvert borne

Soit

Q

dt un

operat eur differen-

,uniformement fortement

Je' (x,n )

;

soit

2 L (Q)

positive dans

(A,D(A))

,verifiant

On a .al or s (2.1)

N(~,A) _ ~( Q)~n/2m

la mesure de densite

~I (X)

= (2rr)- n

J

qU~d ~ _ ~

' d~

•

J\! (x , ~ ) < l

Les diverses methodes usuelles pour prouver un t e l r es ul t at ut i l i sent l'etude d'une famille

a un

parametre de problemes associ es

on obtient une information sur une transform ee de

N( ~)

a

A ou

ou de l a fonc-

- 101 -

C. Goulaouic

tion spectra1e

e(k,x,y) =

E lp o(x);lYT et on conc1ut par l'applicaL
.

tion de theoremes tauberiens. Voici 1e schema d'une methode ([5][14][19]) On peut toujours supposer Pour de GREEN

t )O, l'operateur

Gt(x,y)

2m)n

(quitte

(A+tI)-l

a prendre

2 L (0)

de

dans

un itere de

A).

D(A) a un noyau

aBsez regulier et verifiant

Jo

Gt(x,x)dX

1

= °ON r:+t J J

D'autre part, on a

If(x)1

l

(Af,f) + t(f,f) d'ou i1 resu1te par application du theoreme de SOBOLEV qu'i1 existe

C)O

tel que

t 1-n/2m Gt ( x,x ) Ensuite, une etude locale de

~

C

pour

t)O

et

Gt(x,x), en 1e comparant

x € 0

a 1a

n e1ementaire d'un operateur a coefficients cons t ant s dans m Rue l'on a pour .tout

•

solution , montre

x € C

lim (t1-n/2mGt(x,x» t-On en dSduit lim (t,-n/2m E 1 t) = t-j€fi h j +

J

t(x)dx

C

Un theoreme tauberien sur 1a transformee de STILTJES 1e comportement de

1e s i dees ) . Soit (c' est

a di re

• permet d'en deduire

N(h,A)

A 1a r eali sa t i on du pro bleme de DIRI CHLET pour

m V=H ( 0» u

; on a a10rs :

Jt

I

-102-

C. quand 11 suffit de remarquer que

l'o~

J..

-'>

Goulaouic

00

peut approcher

O· par des ouverts

reguliers interieurs et des ouverts reguliers exterieurs dont la differen-

;A: sur un voisinage

ee

des mesures tend vers 0 (on ad' abord pro Longe

de

"0 ). Comme N(J...,If(o) ,H) est une fonction croissance de

quand

o .

la forme

a

est fixee , le resultat est immediat.

Remarquons que l' on peut mame supprimer l' hypothese sur

00

mais la

demonstration est alors plus difficile (cf[1S][29]). L'adaptation de telles methodes mes irreguliers(cf[S][9])

evo~uees

ci-dessus aux cas de proble-

est tres compliquee, voire impraticable.

Dans le cas de .probl emes "reguliers presque partout" , on va cher¢her un equivalent de

en utilisant un resultat precis dans la partie

N(J...)

reguliere et en estimant (par les techniques du chapitre 1) la contribusi cette derniere contribution est negli-

tion des parties irregulieres

geable devant la precedente, on a obtenu l'equivalent de

em a ramene quand mame le pr-obIeme

a.

N(J...)

; sinon

une etude locale au voisinage des sin..

gularites et dans de nombreux cas on peut encore conclure. Nous allon's d'abord degager un resultat de "localisation" precis pour _e comportement asymptotique de resultats permettent

au~si

modeles tres simples

a.

N(J...)

. On pourra remarquer que de tels

de retrouver le cas tres regulier

a.

partir de

coefficients constants.

Pour fixer les idees, nous allons considerer une classe de problemes /lUX

limites sur un ouvert

\loisinage de

00

irregularite de

0

n de IR

ou les singularites se situent ~

(perte de l'ellipticite, singularites des coefficients ou

oQ... ).

11 sera aise de voir les extensions possibles.

- 103 -

c.

11.- Localisation.-

2 V Ci- H = L (0 )

Soit la situation variationnelle n ouvert de IR

Gou1aouic

oii

l'injection et ant compacte et d ' image dense

If!comp(0)

Ci- V Ci-,jD

'~oc

es t un

0

on suppose

( 0)

(ce qui signifie que les difficultes ne provi ennent que du voi s inage de 00

).

On conaddsz-e une forme a h. c.c. sur a(u,v) =

Jo

aa~(X)DaU(X)D~V(X)

E

Ial <m

a la

dx

1~I~m

oii on suppose r a les co efficients On associe

V ,

forme

a

aa~

une mesure

mes { ~ ;

COO(O)!) .

assez r eguliers (par exemple

'E

lal=m

~

de densite

a ~ (x )~a~ <1} a

1~1 =m

Pour t out ouv ert

Uc 0

on note

c' est un s ous-espace fer me de V(U) = !f eU) o

V(U) = {u€V ; u(x) =O p .p x€Q-U }

V et en parti culier pour 'U a= 0

la norme s era enc ore Ilul v (u)= a(u,u)1/2

Pour et udi er la cr oiss ance de

N(A,A) on considere de s f onctions

a ~(A)

de compara ison (on peut pense r

n/2m = A qui est la pl us importante

mai s on peut avoir besoin d 'aut re f onc ti ons t elles que s >O ••. etc) . Dans l a suite

~

~

avec

designer a donc une f oncti on de

dan s JO,oo [ ·cr oissante et tendan t v ers 1 '00 ave c

J O,~

A , et de plus te l le

que Pour tout nue au point 1.

y>O

, ~* ( y ) = l im

y -+oo

exi s t e et

~*

es t

con t i -

- 104 -

C. Goulaouic

On note

N~(U)

2) N(A..V(U).L

= lim sup

A.-oo


2) N(A., V(U). L

N;(U) = lim inf A.-oo


= inf N;(Q-Q1 ) 0 a:: 0 1 (1 n'est pas necessaire de preciser ci-dessus

2(0) L

ou

2

L (U)

puisque le resultat est 1e mame. La resultat essentie1 est 1e suivant

PROPOSITION 2.1. - La comportement asymptotique de 2(0)) N(A,V. L

verifie

.;(0) ~ [':(0) .(0) +

lB;(O) Ceci resulte immediatemen t du lemme suivant IEMME 2.1. - Soit

~+

I

Q

I

un ouvert regulier de

+1

+

N;(Q) = N;(Ol) + N;(0-Q1) •

Pour voir que le lemme 2.1. imp1ique 1a proposition 2.1 •• i l suffit de considerer une suite croissante d'ouverts que 0i

U Q.

(Qi)iEW reguliers et te1s

=Q

• d'app1iquer le resultat rappele pour les ouverts + et le fait que la fonction U~ N;(U) estcroissante. i

~

regulie~s

O =Q-Q 1 et 2 2 dans L (0 ) est

La demonstration du lemme est plus delicate ; on note on peut supposer que l'injection de compacte. On note

v ; on a

d I oil

~videmment

Vo= V(01)

@

V (ou de

V(Q2) et

V 1

V(02))

l'orthogona1 de

V dans o

- 105 -

C. Goulaouic

N;(O) ~ N;(01) + N;(02) N;(O) ~ N;(Q1) + N; (Q2) Pour avoir les inegalites inverses, on doit eonsiderer

V 1

on a

d' abord Ie LEMME 2.2. - Soient

< A < ~ ; on a

0

~'V'L2) ~ N( ~A~A : Soient

Dem£~~~~~~~

et

A

O

' vo,i}) +

A

1

N(~,V1,L2).

deux r eels >0

; pour

i=<J,1

on note

11 existe done

tel que, pour tout

~

u

1 ~i€~ :

1

€ Vi' il existe

verifiant

~

< et a(u,u)

= a(uo'u o)

+ a(u

lu-vl 2 2

<

1u-v 122

~

L

Done

; done il

1,u1)

2 . 2) + E 1 a(u,u). L 2 2 2 Ok +k (y, L) ~ . EO + E 1 o 1 (EO

AOA

et

2

2

1 N(--r--->,V,L ) < k +k = N(A ,V ,L ) "0 +"1 - 0 1) 0 0

Admettons provisoirement que l'on ait obt~nu : N( A,V

2) =

1,L

0

( An/2m)

quand

A_

an en deduit Ie lemme 2.1; en effet, du lemme 2 . 2 . avee QIldeduit:

00

•

~=YA

et

y>l

-106-

C.

a la

et on passe

limite quand

Y-

~

Goulaouic

.

On va, en, fait, demontrer mieux que (2.8).

LEMME 2.3.- On a : N~~,V1,L2(Q))

= O(~n-1/2m)

Demonstration: L'idee est de montrer que

a un

espace de trace sur

h _

00

Vest convenablement

---------

isomorphe

quand

1

r=oQ et d 'utiliser les epaisseurs 1

dans les espaces de SOBOIEV sur r (qui est une variSte de dimension m-1 Notons X = IT gm-j-1/2(r) j=O Y

m-1

= IT

n-j).

. / H- J - 1 2(r)

j=O

Ces espaces sont definis grace s €

~

a la

HSURn ) pour tout

definition de

par FOURIER; ils peuvent etre munis d'une structure d'espace de ~

HILBERT et on a aisement. pour tout

j=O••...• m-1.

m

0k(gm-j-1/2(r). H-j- 1/2(r)) : k

n-l

d'ou immediatement

On note y .u = J

oju o\lj

\I

a

ret, pour

Y = (Yo' • •• 'Ym1)

Y est precisement

lineaire continu de

V ; l'application o

mame on peut construire un relevement des traces X dans S:(O')

V tel que ou

j €

~

,

Ir • 11 es t connu (cf[27]) que l'on definit ainsi un opera-

teur "trace" noyau de

la normale exterieure

0'

YoR

= idX

V dans

X

le

Y est surjective et R lineaire cont i nu de

(en fait, on releve l es traces dans

est un ouvert regUlier tel que

0

1

cr= QI cr=Q ) •

On note

- 107 -

C. Goulaouic

lapro jection orthogonale de

F

se un isomorphisme de

sur

X

V

V sur

dont la restriction

a

L'operateur

lineai re continue de

S

0k~.L2( Q» ~

a

Y dan s

X soit PoR : On aura alors demontre,

grace au corollaire 1'.1., qu ' il existe une constante

et donc grace

PoR r eali-

1

On-va construire une appl ication 2(Q) L

V 1

C 0k(x'Y)

te l le que

C>O

pour t out

(2.9). on aura pr-euv e Le l emme 2 .3 .

k € IN

Pour con struire

S

on utilise une formule de GREEN : On note

A

I ' oper-ateur defini par

lu € Vo

Au

c

0

0

2 L (Q)

I

t out

u € D(A ) o

dans

D(A ) 0

€ V

00

Hj + 1/ 2 (r ) m- 1 E

o

-v

D(A ) a

=

t els que l'on ai t ,

P OUll

e t t out v € V _, a (u,v ) - (A u ,v )

En effe t , tout

et

0

j =O , .•.• m- 1, i l existe de s operateurs

• Pour

lineair es continus de

't j

2

(V . L (Q),a)

v € V

j =o

h .u , J

v + v o 1

peut s' ecrire

y .v) ~

L~(r )

J

avec

; et il suff i t d 'obtenir (2 . 10) pour tout

v

1

€ ~(Q I )

u € D(A )

et donc et t out

0

v € ~ ( O, ), e t cel k r esulte de la f or mule de GREEN classique (cf. [ 27]) en o r ai sonnan t success i vemen t sur

e t sur

01

On defini t une application S

O 2

lineaire c ont i nue de

Y dans

2 L (Q)

par in- 1

<'tJ.u ,gJ' > J. 1/ H + 2 ( r)

E

j=o g Maintenant, pour

g

=

(g , .••• , g o

~

1)

c

/ xH- j - 1 2(r)

Y ettout

u € D(A ) . 0

€ X , on a aus si

a (u ,(PoR) g ) - (A u , (PoR)g) o - (A u , (PoR)g) o

m-1 E

j=O

hJ.u

y . (PoR)g ) 2 J

m- 1

E ('t .u, g .) 2

j=o

J

J L (r )

L

(r)

- 108 -

C.

done

Goulaouic

Sg = (PoR)g Ce qui termine la

demonstration du lemme 2 .3 . et done aussi du

lemme 2 .1. et de la proposition 2.1. Remarque 2.1. - On a suppose ci-dessus que les coefficients de COO(o)

prendre en consideration Ie cas ou les coefficients

~oc(o) pour

sont

([10][12][29]) de

i l est possible par un argument de perturbation

lal=I~I=m et

a

sont

co (0)

pour

lal+I~I<2m.

III .- Quelques exemples.-

On a une premiere classification des problemes variationnels 2(0),a) L

(V,R

1-1(0)

verifiant les hypotheses ci-dessus, suivant la valeur de

=Jo(2II)-n mes{t;

1°) - Cas ou 1-1(0)

a al=I~I=m a

r' I:

~(x)t;a+~

<11 dx

est infini.-

Comme consequence immediate de la proposition 2. 1, on a : COROLLA.IR.E.2.1. - Lorsque

1-1(0)

lim (A- n/2mN(A,V,L2 (Q)))

A-tOO

est infini, on a

=

00

ou encore Dans ce cas, si on veut une etude pre cise du comportement asymptotique des + valeurs propres, on calcule avec una fonction ~ telle que

B;

An/2m =

O(~(A))

par~iculier

et qui est en general suggeree par l'etude d'un cas

servant de modele ; on ne detaille pas cette etape ici

Exemple 2.1.

- Soit tel que

Q

II rencon tre

.It: = D~ o

un ouvert regulier de

+

'1 ' axe QX

2

~2

(Q de classe C1)

et soit

X~D; + 1 ; on lui associe

l'operateur A defini

- 109 -

C. Goulaouic .

2

2

2

V = {uEL (0) ; D1uH (0) ;~D2UH (0) I

par {

a(u,v) =

I

et

{D 1UD1V + X?>2UD2 V + uvl dx

=

On verifie immediatement que ~(o) =

Si la frontiere de

0

seule-

ment rencontre Ox , on se trouve dans les hypotheses de ce chapitre ; .

.

.

2

si

0 rencontre

OX

' il est facile de voir que la methode sladapte ·

2

encore ; donc on a : ~=o(k) trouver un equivalent de l'intersection de

C

quand

N·(;\.)

avec

Exemple 2.2. - Soit

OX

k -

quand

;\.

= .

II est possible de

-> cc

,qui ne depend que de

2

0 un ouvert borne de

equivalente sur C a. la distance au bord

00

mn

~ Une fonction

et

• On conai.der-e l' cperateur

differentiel

~ = - div ~ grad+1

J {~

a(u,v)=

o 2(0);

V = {uH On voit que

a donc pour

nx t ,

~(O)

grad u • graav + u vi dx 2

~DiuEL (0)

est~fini pour

~=o(k)

quand

, avec

k

pour

i=1, ••• ,nl

n~2

et fini pour

)

un equivalent de

-> co

on sait deja. qu'il ne depend que de.la contribution de .00 nu dans [31] ; la fonction 20 )

-

Cas ou

~(O)

~

a utiliser est suggeree par

n = 1 N(;\.)

• On I

dont

a et e obta... l'exem~le

1.4.

est fini.-

8i Ie terme de bord

+ est nul avec

B~

, la croissance

des valeurs propres est donnee par la mame expression que dans les cas tres reguliers. II se peut que ce terme de bord ne soit pas nul (cela peut arriver par exemple pour un probleme de NEUMANN pour

-6+1

sur un

ouvert tres irregulier (cf. [18][29])) ; son etude peut encore se faire localement.

- 110 -

Goulaouic

C.

Exemple 2.3.

in

on note

~operateur

singulier).- Soit

~(x) = dist(x, 00)

It tuquel on associe

= -t. + ~a

pour

x € 0

avec

a

0

un ouvert borne dans

et on considere l'operate~

<0

A defini par

v=

[u € H1(0) o

--aa(u,v) = llgrad u .gradv+~ uV}dx

" a : N( ) ...... 1I(n),n/2 un A,A 1\

•

Q

En effet, pour voir que

i l suffit de le

comparer au terme correspondant pour -t. + 1 que l' on

~ai t

~tre

nul.

Un autre exemple moins evident sera rencontre au chapitre 3.

3°)- Remarques.- La methode ci-dessus s'adapte aisement

a d'autres

particulier au cas, ou il y a des singularites dans

Q

situations. En

(irregularite des

coefficients ou perte d'ellipticite ••• ) ; on peut aussi, par exemple, traiter le cas d'operateurs d'ordre variable 30i t

0

un ouvert borne de IRn

probleme de DIRICHLET pour

tA

et

~ € j) (0)

= lI~lI-t.+1

,~~O; on cons Lder-e le

sur

on trouve que le

0

comportement asymptotique des valeurs propres verifie : ou

O = 0 - supp 2

~

- Le corollaire 2.1. est vrai dans une situation plus generale. Soit V

2 (V,H=L (O) ,a)

et correspond

'3i on a

~(o) =

une situation variationnelle ou

a un co

operateur differentiel

dt

a

de degre

est h.c.c. sur 2m

,alors lim (A-

n/2mN(A,V,H))

=

00

•

11.-000 ~n

effet, on considere une regularisation elliptique de

a,

a =a + £

~b

- 111 -

pour

E

Bur V

E

>0 = V()

Pour tout

best h.c.c.

,OU

If'(0) 0

E>0

E

lIullv =

(a(u, u)+Eb(u, u)

1/2

E

,on a

_

est la densite de mesure associee E

o

muni. de la norme

N(~,V E ,L2(0» . ~ (0)

C. Goulaouic sUt gm(O) ; la forme a est h.c.c.

tend vera l'infini

quand

a

a

E

A -

00

,ou

quand . E tend vers 0

; d'ou Ie resultat annonce.

I.l'

E

_ 112 -

C. CHAPITRE 3.-

ESP~CES

DE VECTEURS

00

C

Goulaouic

ET ANALYTIQUES •

On s'interesse ici a des problemes elliptiques (eventuellement dege-

neres) a donnees analytiques et on montre la relation entre Ie comportement dU spectre et des proprietes de caracterisation des fonctions analytiques par les iteres d'un operateur, ou encore une propriete

d'hypoelli~

tici te analytique. 1.- Definitions et GSneralites.-

Soit, comme precedemment, un espace de HILBERT D(A)

H

un operateur non borne dans

, auto-adjoint> 0

et tel que l'injection de

H soit compacte.

dans

• On note . D (~O )

=H

et

D(A1 ) = D(A) et pour

k) D(A lui

• On note

AO

= identite A1 = A

et

k D(A +1 )

k > 0

Les espaces

pour

(A,D(A»

= {u

k) € D(A

k)} ; Au € D(A

et

k k A +1= AA •

sont munis des normes hilbertiennes

k = IAkul H ' pour k € IN D(A )

D(l"') =

tJr D(l)

et on I' appelle espace de vecteurs

COO

A; c'est un espace de FRECHET pour sa topologie evidente.

• Par la suite nona ne conatder-er ona que des·.operateurs a des operateurs differentiels d'ordre 2

A aasocaea

(pour simplifier les

notations, mais cela n'est nullement essentiel) et on notera en consequence

[u

c D(AOO

)

;

31>0

,Yk € IN

~AkUI

sup -----k2k! ie L

< 001.

- 113 -

C. Goulaouic ~ =(~ ')'€~ J J III

• Si

notera, pour

est une suite de reels strictement positifs , on .

k€fi

l i

=

(~~) jEfi J

= {(fJ€e

VIi =

fi

E

J

~

pour

b>O

, b

~

= (b

~j

E(~) = lim.ind. l2

(Vii"j) j€fi

If .12~. < J

J

00\

)j €fi

•

b>1 b~ NoUB avons alors Le resul tat :

~

PROPOSITION 3.1. - Le 'developpement

en

ser~e

de FOURIER

i)

de

(lpj)j€fi est un isomorphisme k 2 D(A ) sur lA pour k € IN 2k

ii)

de

D(l" )

sur la base

sur

i

lim.proj. k€~

Demonstration.-

.ontrer

D(AW)

de

iii)

Les points

sur

ten) .

i)

et

ii)

sont evidents. Pour

iii) on ecrit

D(AW) = lim.ind E

L

L>O .

a.f

00

k

' avec 2

IIA ullH

00

EL = {u € D(A ); E 2k 2 < 00 k=O L. (2kl)

~L

sur

at il suffit de verifier que l'on peut trouver

b1~

1'operateur

realise un isomorphisme de 00

{(f J € ~

;

J

Cb

Aj

1 1

b

1

et

b

2

00

Elf .1

2

j=O J

tels que 1 'on ait, pour

at

2k A

j

\

Ek 2) < oo} L (2kI) A2k

(

k=O

b

2>1

et

€ IN

Af

<E----

- k=O

tendant vers

L2k(2k! )2 1 quand

L tend vers l'infini.

C 1>O

- 114 -

C. Goulaouic

11.- Isomorphismes d'espaces de suites (associes On

a

et' D(AW) . _

D(A~)

a immediatement Ie resultat suivant. ~ =(~j)jSN une suite de reels posi-

PROPOSITION 3.2. - Soit tifs 'non decroissante

les proprietes suivantes sont

j

equivalentes :

i)

I am, ' pro J, • l 2k = s k€rf

~

ii) II existe

C

l'on ait Pour tout

C1ja1 On

o'est

a designe par

j € IN

1

2

,j';' 0

< ~j _< C2 J.a2 s

l'espace des suites

a decroissance

rapide,

a dire: s

=

Yk

c IN

a l'aide

des normes sur les espaces

, supllf .1

l2

J

j

ii)

Demonstration.- II est immediat que

i)

, a >O , a >O tels que

C 2>O

1>O,

<~}

implique i). En expliquant

avec poids, on trouve immedia-

tement :1.i). Dans Ie

m~me

ordre d'idees, on obtient avec plus de travail, Ie

ltesultat :

suites strictement positives, croissantes et tendant vers l'infini. On suppose que sous-espace ferme de t(~) que l' on ait pour t out ~ ,

< Ca .

J -

Demonstration.- Soi t

e

, J

j

t(a)

est isomorphe

Alors il existe

C>O

€ IN ,

•

un isomorphisme de E. (a)

sur un sous-

a tel

- 115 -

C. eBpaCe ferm e de envoie

£2 dans bC%

your t out

~(~ ) £2

k Em,

c~

; pour tout

avec une norme

, i l exi s t e c )1

L<~ ; s oit c

1

Goulaoui c tel que

E ] 1,c[

e

; on a

,2 ,2) < L 0 (,2 6k ( e ~cx'''~ k~ ~

(3.1 )

"0

d' aprea la proposition JI,e

b>1

c

b

c

1

1.3 c ~ / L( ~ k ;;l • c 1 tel que l'appl ica tion e-

membre de droi te dans (3.1) vaut d I apre s la propo sitio n 1. 6, D'autre part, il existe

b

1

E ]1 ,b[

soit continue de

Soi t

e- 1 dans ce s espaces. On a donc pour b 1 CXk/2 ( -) = b k bcx bcx 1

M la norme de

k E fl"

,

l )

° <.i

l

< MOk( -b cx

,

< Mok ( el ' bcx

l~ n e~(cx» c

l

1 )

c~

1

~ / e k 2 < ML(_1) e

.

]J)one 1 b1 1 e1 2~LO g(--b-- ) ~ Log (ML ) + 2~kLog(-c--- )

~kLog( ~1 ") ~ ~Log (

:1 ) + 2 l og ML

est une suite bornee .

done

On en deduit evidemment :

"

COROLLAl RE 3.1 . - Scient

l

cx=(cx . J' clN J J<

et

~=(~ ,) ,c~ J J«'

deux

suites strietement positives, non dee roissantes et

- 116 -

C.

tendant vers l'infini

Goulaouic

les proprietes suivantes sont

equivalentes i)

&(u:)

ii) j

&(s)

et

Il existe

sont isomorphes •

c ~ c 1 >0 tels que l' on ait, pour 2

c IN c a. <

~.

1 J -

< c 2a.

J -

J

D(A~)

Ces resultats s'appliqueront en particulier aux espaces

n(AW)

correspondant

a

A •

divers operateurs

qu'on aura une description de ces espaces

et

1ls ~seront interessants lors-

D(A~)

ou

D(AW)

;

on va le

faire .maintenant dans quelques cas. 111.- Caracterisation d'espaces

10

)

-

et applications.-

Cas d 'une variete compacte sans bord.-

•

Soit $ion

D(A~)

n

j

r une variete analytique reelle compacte sans bord de dimen-

on considere un operateur

operateur elliptique d'ordre

2

A auto-adjoint positif, associe a un

a coefficients

·Il est facile de montrer que

r ,

analytiques sur

k

D(A ) = H2k (r )

pour tout

k € If

avec

et

On sait aussi que le comportement asymptotique des valeurs' propres Terifie A. (A) J ~u

-

C .2/n J

C est une constante calculable 11 en resulte que

c~(r)

quand

a partir

j --

du symbole de

est isomorphe

Par ailleurs on sait decrire l'espace

co

a

A •

s

D(AW) (cf.[3][26]).

- 117 -

C. Goulaouic PROPOSITION 3.4. -

L' e s p8ce n(AW) es t l'espace ().(r) des

---I fonctions analytiques s ur

r

Demonstration: I l est facile de voir que o'(r)

a la

on se ramene u

tantes a.

situation suivante : soient

,

A- un

k €

~

Q

a coefficients

opera teur differentiel d I ordre 2

on demontre par recurrence sur

C >O et 1

un ouver t

Q

une fonction analytique sur un voisinage de

(notee : u € (1(0» dans a(g)

est con tenu dans

qu'il exi ste k €

C >O telles que l'on ait pour tout 2

~

2 cons-

et tout

c ttf

On conclilt W

Inversement, pour montrer que . D(A utilise une idee de [27] qui consiste

est "contenu dan s a(r)

)

a se

ramener

a un

, on

probleme plus sim-

Ple de regularite analytique avec une variable ·supplementaire : soit on note k

A

v(t,x)

at cette fonction

u(x)

2k!

vest definie et

C~

pour

x € r

et

Itl<E

assez

petit; de plus, on a

(D~

+ A) v

=0

dans

]-E,E[ x r

ce qui implique, par le .t heareJe de PETROWSKY( *) , que dana

]-E,E[

X

r

; donc

u = v(O,.)

Les propositions 3.1, 3.3

; v

est analytique dans

es t analytique

r

et 3.4 impliquent:

------------ -

(*) On peut consulter [23J pour une demonstration s imple de ce theoreme

p:Lr une ''methode d I ouverts emboites" •

- 118 -

C. Goulaouic

l

3.2. - L'espace
CORO~LAIRE

&(~)

de suites

avec

COROLLAIRE 3.3. - Soient

~j=

r

est isomorphe a l'espace

. l/ n

J

r

et

1

2

deux varietes analyti-

ques reelles compactes sans bord de dimension respective n

et

1

n

2

. Les espaces

et @,(r )

o.{r I)

morphes si et seulement si

sont iso-

2

n l=2

ao) - Operateurs uniformement elliptiques sur un ouvert regulier Soit

~n

Q un ouvert de

analytique compacte; soit

tel que

Q

A un operateur auto-adjoint >0 dans

dliptique d' ordre 2 aux limites

~

a coefficients

a coefficients

dans 0.(0)

at

cmn •

a bord

soit une variete

~ealisation d'un probleme aux limites pour un operateur

Q

2 L (Q) ,

uniformement

avec une condition

2 analytiques aussi et D(A) c H (Q)

On

peut penser, par exemple, aux problemes de DIRICHLET ou de NEUMANN pour 1'operateur

-~+1 OO

On a encore

D(A

)

= {u € C<Xl(Q) ; Vk € IN

On sait que les valeurs propres de A. (A) J

Demonstration:

quand

j

-+ co

D(AW) (cf[27])

PROPOSITION 3.5. - L'espace ferme de

I.

A verifient :

(~(Q)r2/n}/n

On sait aussi decrire

, ~ Ic.ku = 0

D(AW)

est un sous-espace

0.('0), a savoir

II suffit de suivre la methode utilisee au 1°) en

rempla<;ant seulement Ie theoreme de PETROWSKY de "regularite analytique t 'interieur" par Le theoremede MORREY et NIRENBERG de regulari te analytique . jusqu'au bord pour de bons problemes aux limites.

a

- 119 -

~(I1)

Il en r esul t e un i s omorphisme de ferme de 0.('0) (1' espace

s ous -es pace

de l' espace 0.('0) amene

W

D(A

av ec

) )

;

I1

j=

C . Goulaouic 1 j In sur un

s i on veut un i somorphisme

lui-mElme sur un espace de su ites comme

a u tiliser

i( 11 )

, on est

Q

et degen eran t

un operateur differentiel elliptique dan s

au bord, de fa90n

a fa ire

"dispara1tre les conditi ons aux limites".

3°) - Operateurs elliptiques degeneres sur un ouvert r egulier.Soi ent t~ls

un ouvert borne de

Q

mn

et

,

une fo ncti on analytique

que

On cons idere l 'operat eur

1ft avec on lui a s socie a (u,v ) =

= - div , grad +1 + Ao

..£L D. - ...£!.... ox. J ox. J

0

~J

f' Q

r

1s i , jsn D i

~

gr ad u grad vdx +

Ai j' , Ai j

f

uv dx + Q

f

.r .s.. u A.loJov dx ~J

Q~,J

A u (L ij e t on not e

A l' operateur non borne dans

L

2

(Q)

2

(Q)

pour

1Si, j sn}.

a s s oci e a cet t e s itua t ion

varia tionnelle ; cet operateur est au t o-adjoi nt positif ; on peu t aussi voit que l 'inj ection de s ion de

V dan s

V

dans

1 2 H / (Q)

L

2

(Q)

est compac te , so i t en mon tran t l'inclu-

,soit en compar ant ave c l ' exemple 1. 4. ou on

a de j a l'in j ec t i on compacte . Notons que l 'operateur

dt

est degenere au bord mais pas dans l es

- di r ect ions tangentielles au bord , comme un operateur --D1,I.ge de

0

dan s

n-1 ] O,co[ x lR

DttD

t

+

~y

au voi s i-

- 120 -

C.

Goulaouic

Nous avons Ie resul tat :

oo

PROPOSITION 3.6. - 10/ L'espace D(A ) est COO(Q) .

l

2 0 / L' espace

D(AW )

COO a ete traite par H. TRIEBEL [33][34j;

Le cas de

Demontration :

est (1(Q)

nous allons utiliser ici une methode differente, qui convient aussi pour Ie r-aa analytique.

D'abord Ie seul probleme est au voisinage du bord et on a essen tiel-

Dans la localisation au voisinage d'uh point du bord, l'operateur:A: n (y,t) € R - 1 x ]0,00[

devient en les variables

"" itt

= D t D + t t

L 11-11 ~2

= nn + a (y, t )DI-I + L b (y, t ) t D DI-I 1-1

y

~ coefficients analytiques dans un voisinage ~u'il

existe

11-11 ~ 1

& de

0

L'espace correspondant

~ designe

dans R

n

et tel

C>O verifiant :

,TJ,tO •

pour

QU

t Y

1-1

a

Vest

2 n) 2 n) V= {u € L (R • D u € L UR pour i =l, ••• ,n- l + ' Yi + 2(Rn- 1 ) . supp u c &} VtDtU € L + / n- 1 (IR x [0 ,oo[) n ~

On effectue Ie changement de variables

On verifie que I' operateur

Ji:

se transforme en un oper'ateur

E111iptique dans un voisinage ~

. t un sous-espace .. dev~en V distributions

a support

f erme•

de l'origine dans

6)

fortement

Rn-l~2 • L'espace

2_{0})) de H1(lRn- 1 x (1D n

cronstitue par les

dans unvoisinage de l'origine (que l'on peut

- 121-

aupposer conrenu- dans

C.

0'1) .

Goulaouic

On verifie aisement, puisqu'une distribution portee par

n+1

dans IR

ne peut

u

{

~tre

eu c

L

- x

au voisinage de

fie

2

- 10}H

R

; supp u c;:

0;

2)

1(lRn 1

-

2).

x R

C~ ou analytique et l'etude des vecteurs analy-

Done la regularite

C~ pour

(1R

2(lRn 1

impliquent en fait: u € H

tiques ou

-1 (. n+1) H R , que les hypotheses

dans

c H1(Rn- 1 x

n 1 IR - xIO}

se ranenerrt au m~me probl eme pour ~ elliptique

n 1 dans IR + •

0

On termine done la demonstration comme au 1 0 )

,

enutilisant en plus Ie lemm~

suivant :

. LEMME 3.1. - Soit et

1°1 La u

est

la fonction sur IRn-1 x IR2

U

fonction

u

est

00

C

def'Lru e par

si et seulement si

C

u est analytique si et seulement si

est analytique.

La demonstration de ce lemme se ramene essentiellement que si une fonction

dans

00

2 0 1 La fonction

u

n 1 une fonction de IR - x [O,oo[

u

v , definie

par

, la fonction

v

v(z)=v(z2)

a montrer

est analytique (resp.C~)

est elle-m~me analytique (resp.C~)

au voisinage de

0

au voisinage de

0; ce qui est immediat •

Sur Ie comportement asymptotique des valeurs propres, on a Ie resultat :

- .122. -

C.

Goulaouic

PROPOSITION 3.7. - La croissance des valeurs pr opr e s de A verifie :

~(Qh n/2

N(>",A) ..

quand

f

n

11(0) = (2nr w

>" - "" , avec

\~

w

n

n o r ,\x)

de la boule unite de ~~str~~~

~(>")=>,,n/2·

E

de

~n.

: On peut appliquer la proposition 2.1. avec

, et i l s'agit de montrer que Le terme de bord B;

c'est a dire de montrer que, pour tout ,0

designant Ie volume

dans

00

E>O

est

nul,

, il ·existe un voisinage

Q tel que l' on ai t pour

>"

assez grand,

En utilisant une localisation selon un recouvrement fini d'un voisinage de

00

soient

soit

Q

dans

() >0 0

v(w

o)

et un diffeomorphisme, on se ramen~ a la situation suivante : et pour

, b>O

{u

€

pour

W

o

, on note

0<0< 0 -

0

= {(y,t) €~:

2 L (w )

o

j

.,'(t'D t u

et

Iyl
c L 2 (wo )

et

O
c L 2 (wo)

i = 1,. 00 ,n-1}

et il suffit de montrer que l'on a : 2 lim (lim sup N(>",V(w , L (w ) ) ) = 0 o) o (reo >,,-Pour .cela, on consd.dere l'operateur sur

1

:dt-1=Dt t (Oo- t )Dt

- /:;,y

, avec

v1

= {u €

~2(wo

pour on aait que asaociee

A ' defini par

)

effet ~1

2(w Vt(Oo-t)DtU € L ) oo

o i = 1'000 ,n-1}

N(>" ,.1 ) .. '11 (w )>"2/n 1 oo

a Jt1 (en

j

ou

1

11

est la densite de mesure

est la somme d'un operateur de LEGENDRE

d'un laplacien en des variables separees)

j

il en r esulte que pour

et A ' 1

- 123 -

C. Goulaouic

; on a done aussi (3.4) en comparant

on a

II resulte done des propositions 3.6

et

et 3.7. les consequences

suivantes

:r

PROPOSITION 3.8. - Le developpement

en serie de FOURIER

A r ealise

sur les foqctions propres de l'operateur un isomorphisme de cO!> ('0)

sur

s

et

de CI(O)

avec

On en deduit (d'apres la proposition 3.3) COROLLAIRE 3.2. - Soient de

~n1

et

des varietes

mn2

a bord

COROLLAIRE 3.3. "- Soit

2

associe

a un

deux ouverts bornes

2

01

et

'0

2

etant

analytiques ; alors les espaces

A

1

un operateur auto-adjoint >0

, realisation d'un probleme aux limites operateur differentiel

a coefficients

Jt1

d'ordre 2

0

, tel que

localement bornes sur

2

l' injection D(A ) c; L (0) 1 Si

0

sont isomorphes si et seulement si

2

L (0)

et

1

respectivement,

o.('O~) et a('0 )

dans

0

soi t compacte.

est un sous-espace ferme de 0.(0) , alors

D(A~)

les valeurs propres de • .2/n J

, quand

A

verifient

1 j

-+

00

•

En particulier, avec les notations ci-dessus, considerons l'operateur

A associe 1

a:

- 124. -

C.

ttl a

= - div cp grad + 1

1(u,v)

=

f

2

tp

o

grad u • grad v dz + Iuv dx ,

2

V = lu ( L (0) ; "Diu ( L (0)

pour

1

On sait que , pour

Goulaouic

n~2

i=l, ••• ,n} •

, la croissance des valeurs propres ne verifie

D(A~)j Cl(o)

, l'inclusion etant OO ~tricte.( II est possible, mais non fac ile, de montrer que D(A7)=c (0), il en resulte done que

cf.[5]) • Remarque 3.1.

En utilisant de nouveau l'idee d'ajouter une variable ([27]),

on montre ainsi qu'il existe

u(x,t) ( COO(O x ]-E,E[)

et non analytique

et telle que

On peut en deduire aussi des exemples non triviaux d'operateurs non hypoelliptiques analytiques, en utilisant des changements de variables de la forme (3.2) (par exemple, en partant de

mX] O, E[ n2 t

x ]-E,E[

avec

2+(z2 2 + D2 +D z2 ) D 1+ 2 x zl z

£ )0

qui n'est done pas hypoelliptique analytique dans

4 R ).

4°) - Generalisation.. On peut etendre les resultats precedents au cas d'ouverts irreguliers (localement diffeomorphes anal yt i quement un pro duit de droites ou de

t

vois inage de

Q

(cf[9]

dans

1x 1D1

devient , par I e change men t de

h

pour une e tude ~ c omple t e ) ;

on montre s eul ement l'idee sur "un coin" : L'operateur D ians ]0,00[ x ]0,00[

0

on con struit un''bon operateur

droites)

el l i pti que degenere sur un tel ouvert

a un

+ Dx 2 2D2

variables[Xl=Z~+Z~ x =z2+z2 2 3 4

- 125 -

C. Goulaouic

on montre encore que lIon peut le considerer dana COO

m4

et obtenir ainsi la description des vecteurs

et analytiques. Lletude de la croissance des valeurs propres se fait encore en

utilisant la localisation et en montrant que la contribution du bord est nulle. 11 enresulte done un isomorPhisme de sur Remarque ·3.2.

avec

.1/n

1lj"':J

Coo(O)

sur

s

et

de

•

On peut aussi obtenir de tels resultats (et m~me parfois

Plus generaux, cf[7][37]) en utilisant les developpements, par exemple, sur une base de polyn6mes : Ce ne sont plus les vecteurs propres d1un operateur differentiel avec une certaine croissance des valeurs propres, mais

dlaut~e-

part on a des inegalites (de MARKOV) reliant les polyn6mes et leurs deriveea- ; on verra des choses plus voisines au chapitre 4.

- 126 -

Goulaouic

C.

CHAPITRE 4.-' FONCTIONS PROPRES DE "EONS" PROBIEMES AUX LIMITES ; lNEGALITES DE BERNSTEIN ET DE MARKOV.

Nous montrons lci que la caracterisation des fonct ions analytiques du domaine d'un operateur iteres de A

a un

A elliptique ou elliptique degenere, par les

A , est equivalente au prolongement des fonctions propres de

domaine complexe, verifiant de plus des inegalites semblables

celles de BERNSTEIN

a

ou celles de MARKOV pour les polynomes.

1.- Generalites -. Soit

dont le b~rd est suppose au moins

C un ouvert borne de lRn

lipschitzien ; soit

A: un

dans 'l'espace acC)

et formellement auto-adjoint.

On note

(A,D(A))

operateur differentiel d'ordre 2

une J:8alisation auto-adjointe de

a coefficients

Je

dans

2(C) L

que nous supposons strictement positive; nous supposons que l'injection de

D(A) dans

par

(A)

et

L2 (Q) (epj)

NpuS supposons que

On nete

CO(Q)

est compacte ; nous designons comme precedemment

les valeurs propres et fonctions propres de ep , (<J.(O) J

pour

j

c IN

l'espace des fonctions continues sur

Nous supposons qu'll existe

D(i') 'G CO(Q) (on note

et que

D(!"O)

o.aD

Q

r ( IN tel que

(4.1)

a

A

M la norme de cette injection)

est ferme dans Q(Q) •

Ces .hypotheses ne sont, en fait, pas tellement restrictives : L'analyticite des fonctions propres est une hypothese bien plus faible que la

- 127 -

C.

regulari te analytique pour I' operateur

a pau D(A

)

It (l(e)

l 'inclusion (4.1) signifie

A possede un peu de regularite et enfin

pres que l ' oper at eur

oO

A

Goulaouic

est en general decrit comme Ie sous-espace de
fonctions verifiant des conditions aux limites sur

00

des

et est alors

evidemment un sous-espace ferme • • Donnons quelques exemples (plus ou moins rencontres deja, et auxquels nous refererons dans la suite). Exemple 4.1. dans un ouvert

0

Les pr ob.Iemea de DIRICHLET ou de NEUMANN

-6 + 1

pour

Q soit une variete analytique compacte.

tel que

Evidemment on peut remplacer

-6 + 1

et les conditions de traces par des

operateurs plus generaux. Exemple 4.2. -

Les problemes aux limites elliptiques degeneres ren-

contres au chapitre 3. Exemple 4,3. -

Soient

0

n un ouvert borne de R

tion analytique 'e qui v al en t e a la distance a

dt = -~6~

+ A*A

a coefficients D(A)

et

~

une fonc-

oQ • On prend

A designe un champ de vecteurs transversal

ou

dans (l(l5)

= [u

a

00

et

, avec

1 € H (Q)

o

; ~2u € H2 (Q) et A2u € L2 (Q) 1 (cf[4]).

• Nous utiliserons (4.1) aussi sous la forme suivante, qui resulte du theoreme de SOBOLEV : 11 existe

C>O

telle que l 'on ait pour tout

j €

~

sup 'I ~ .I < C"t-..~ Q

J

-

J

II.- Propriete des iteres et approximation.Designant par

~k l e sous-espace de

premieres fonctions propres de

A, on a

2(0) L

engendre par les

(k+1)

- 128 -

C. Goulaouic

les proprietes s uivantes sont equivalentes u € D(AW)

ii)

•

I I existe

d

~

2(u,wk)

On a note

d

et

2

c

JO,1[

tels que l'on ait

pour tout et

C>O

~

a

f.k

c

I I existe

dJu,W k)

et

C>O

a

c

JO, 1[

k€1i tels que I' on ait

pour tout

C jAk

k

e IN

d~ respectivement la distance dans L2 (Q) et

on pourrait d'ailleurs les remplacer par une distance dans n I importe quel

LP(O)

la suite des cas

p=2

avec et

~

j=a

lu . /

2

J

cc

,

mais nous avons besom

b

> 1 tel que

f3 <

co

•

Par ailleurs, compte tenu de (4.1), (4.3 ) equivaut (4.4)

{ I I existe

C et 1>0

IU j I ~ c1a~ Et comme

d

2(u,wk)

=

(E

j=k-+1

lu .1

a

pour

2)1/

2

1 j

a

€ JO,1[

t els que

c IN

, on a montre que (i) equi v aut

J

I I est evident que iii) implique

iO.

Montrons que i) et ii) implique iii) : on a k

d~ (u, wk ) < lu - E u .~ .1 j=o J J L~ (0 )

< MIAr(u

-

~

pour

et de ces cas seulement.

p~

si et seulement si il existe (4.3)


1

k

- E (A~u ·)~ ·1 2

j =o J J J L (Q) k MjIAru - E (A~u ·)~ ·11 2 j=o J J J L (Q)

a (i~).

- 129 -

C.

ce qui implique encore Pour la

iii) grace

a

Goulaouic

(4.1).

commodite d'expression dans la suite, donnons deux definitions: DEFINITION 4.1. - (Approximation analytique (par les (~)

une suite de reels> 0

paces (~k)

~k)) . Soit

. Nous dirons que les es-

ont la propriete d'approximation analytique

pour (~) si et seulement si : toute fonction dont la distance

doo

ce exponentielle de

aux espaces (~)

~k

CO(a),

u (

est a decroissan-

, est analytique dans

Q

Par ailleurs, nous avons deja rencontre la propriete suivante : DEFINITION 4.2. - On dit que

(A,D(A))

a la propriete des

iteres si et seulement si

On vient de voir que cette propriete est vraie dans de nombreuses situations (cf. chap .3) • On peu t exprimer le lemme 4.1. sous la forme suivante : PROPOSITION 4.1. - L'operateur

A ala propriete des iteres

si et seulement si les espaces

(~k)k(~

ont la propri-

ete d'approximation d'analytique pour la suite

(~) .

111.- Inegalite de BERNSTEIN.-

Rappelons l'inegalite cl as s i que de BERNSTEIN (cf [28] par exemple) pour les polynomes d'une variable: soient diametres

1 P+P

porte par..Rez

et

P

p>1 1

P

et

Q p

l'ellipse de

porte par ~lIlZ

•

Pour tout

- 130 -

c. polynome

P

de degre

~

sup z € Q

n

Goulaouic

, on a :

Ip(z) I ~ pn

p

I

su}> p(x)l. x € l-l ,+1 J

Les polinomes de LEGENDRE constituant une base de l'espace des polynomes et etant les fonctions propres de I' operateur de LEGENDRE d

·

- dX'( 1-x

2

d . >-ix-

sur J-1,+1[

, cette inegalite de BERNSTEIN apparaitra

comma un cas particulier (precise) des resultats ci-dessous . DEFINITION .-

(1\:) une suite dereela ,>0 ; on dit que la

Soit

suite

(~k'l\:)

verifie l'inegalite de BERNSTEIN si et

seulement si : Pour tout

o

en

dans

u € ~

b>l , il existe un ouvert tel que, pour tout

, u

ir voisinage

k € fi

se prolonge analytiquement

de

et tout

a 'lj et

verifie

sup lu(x) I < bilk suplu(x) I x€'" x€Q Nous avons alora Ie resultat suivant (susceptible d'etre generalise en ce Bans que les espaces

~k

ne sont nullement supposes dans la demons-

tration etre les espaces engendres par des fonctions propres d'un operateur) • PROPOSITION 4.2. - Soit

1l=(I\:)k€fi

une suite croissante de

reels positifs tels que l'on ait 00

pour tout La famille

a € JO,1[

E

k=O

a llk < 00

(~k'l\:)k€fi posaede la propriete d' approxi-

mation analytique si et seulement si elle posse de la propriete de BERNSTEIN.

te

Demonstration ------de BAIRE.

Elle est essentiellement un corollaire de la propriJ-

- 131 -

Goulaouic

C. -~~ab~~,

soit

supposons que la propriete de BERNSTEIN soit verifiee, et

f € CO(Q)

telle que pour

k €

m,

on ait

avec k €

Soit, pour

m

,f

k

€

W k

a € ]0,1[

telle que

d(f ,Wk) = If-fkll done

Loo(Q) < 2 d(f'~k 1) < 2 --

Ilfk-fk 1/1 - Loo(Q)

D'apres (4.6) la serie + (f

f est convergerrte dans

° CoCO)1-fo)et

On choisit

tel que

ssoeie

a

b>l

+ (f

+•••. •.•

sa somme est ab<1

et soit

f

V-

le voisinage de

H

k

_ f

_I

k-1 Loo('\)")

et done eette serie de terme general Co('/])

,ce qui montre que

mille

(W k,llk)

11

11

<

b k. 2Ca k-1

<

(ab)~ 2ca~-1-~

(f

k-fk_ 1)

est eonver~ente dans

fest anaJ.ytique dans 1)" , done que la fa-

a la propriete d'approximation anaJ.ytique.

In!~~~~~!,

supposons que la propriete d'approximation anaJ.ytique

est vraie • rour

Q

b dans l' inegali te de BERNSTEIN ; on a nf

-

2-f 1)

a € ]0,1[,

on considere E

a

= If

-~

€ CO(Q) ; sup (a k

d(f'~k»

<

00

I

c ' es t un espaee de BANACH pour la norme

L'espaee

E= lim ind E a O
s'injeete continuement dans

Remarquons que la propriete d'approximation signifie que :

c O(Q)

- 132 -

C. E c

Goulaouic

0.(0)

Par Ie theoreme du graphe ferme, on voit que cette inclusion est meme continue.

Q('Q) = lim.ind.(l('Ii) oii k€1N une base decrois.sante de voisinages de 'Q dans On peut aussi ecrire

les ('Irk)

constituent

tf

On peut alors appliquer un tneoreme classique sur la factorisation d'une ' appl i cat i on lineaire d'une limite croissante d'espace de FRECHET dans une autre limite croissante d'espaces de FRECHET (cf.[21]) que, pour tout

a € ]O,l[

, i l existe

k € IN

il en resulte

tel que l'espace

E a

soit contenu dans Ct(1!k) • II en resulte en particulier que tous les elements de V~k

se prolongent analVtiquement

a un

voisinage 'lj de

Q

~ff Nous allons maintenant supposer que les inegalites (4.5) ne sont pas verifiees, en supposant la propriete d'approximation. Soit

tf

dans

(l1 k )k €fi

un systeme fondamental de voisinages ouverts de

de croissants et contenus dans'li • On suppose done qu'il existe

tel que pour tout

b>1

0

(4.7)

il existe I-Ik· > b Jlf

j € IN

If~} LO:>(~k. )

k. J

et

I

satisfaisanJ;

f k . € ~k . J J

k j LO:>() Q

J Sans restreindre la generalite, on peut supposer que l'on a

Ilrk I

= 1 ,et que la suite k , LO:> (0) J effet il est aise de voir que si f k j ,

est strictement croissante; en reste dans un espace de dimension

j

finie, l'inegalite (4.5) est toujours vraie. Pour ' a € ]O,l[ F

,on designe par

= {u = (u.) a J

c f!1

I

I (u J. )I a = sup Ia j

-1-1 .

Ju.1 J

<

00

I

- 133 -

C . Goulaouic F = l im . ind F O
Done l 'application

(uj est c on t inue de

(}.(o)

F

~ ~ ,\ .fk .

)

J

J

J

= 'U((u j ) )

C'O) • Nous

dans

allons prouv er que son image est

,ce qui impliquera par Le t heoreme du graphe f erme que

nue de

1l

est con t i -

dans 0.(0)

F

cF

u = (u J J

Soit

est d an s <1.(0)

; pour prouver que ' U (u )

a

il

.uffit de monta-er (en utilisant la propri ete d'approximation.) qu ' il ex i ste

a

1

€ ]0,1[

et

tels que , pour t ou

C>O

d(1L (u)' ~k)

(4 .8 )

k € lN ,

s C a ~1

En fait, on a : 00

d( 'U. (u)'~k ) <

-

J

Four

E

.e=j+1

'\

.e

fk

II -< lui

.e

a

00

E

a~J,

a .e=j+1

11k · 1

J

avec

Par ailleurs, pour

k
d(U(U )' ~q )

' on a

s d (U(u )'~k .)

ce qui termine la preuve de (4.8)

J

< C a l11q

• Maintenant nous allons montrer que la c on tinui te de ~Q ) en traine une c on t r adi ct i on ave c ( 4. 7).

t

11. de

F

dan s

- 134 -

C. Goulaouic On a 0.(0)

= lim.ind g,{'lrk )

, ou g(V'k)

designe l'espace de BANACH

1tk

des fonctions snalytiques bornees sur

On utilise encore le theoreme de BAIRE qui entraine que pour tout a €

]0.1[ •

il existe

p €

N

et

1'U(u)II~1t) ~

On choisit

a

p tel que ab>l et j I-Ik' L < (ab) J et

Maintenant on prend dans (4.9)

0

L>

tels que

Llul a assez,grand pour que

'li

kj

ul=O

C

'lr

p

sauf pour

l=k .

et

J

u.K

'" 1

j

on obtient

Ifkjl~~p) < L a Ifk j

-I-Ik . J

at en utilisant -I-Ik ·

~ II fk} lJC'li ) < L a J < b p (4.7) et termina ·la demonstration

lQ.(17k)

ce qui contredit

(4.10) I-Ik· J

de la proposition

4.2.

Nous pouvons encore enoncer ces inegalites de BERNSTEIN de diverses ' fayons equivalentes en jouant sur les normes

(Cf1c)

ou

(~k)

2 L

et

00

L

et sur les

; on peut recapituler les diverses equivalences.

PROPOSITION 4.3. - Les proprietes suivsntes sont equivalentes i)

L'operateur

ii) La suite

A a la propriete des iteres.

(~k'

V\: )k€

IN a la propriete d I approxi-

mation analytique. iii) La suite

(~k'

f\:)

verifie les inegalites de

BERNSTEIN •

rv) Pour tout b>l dans

ff

• i l existe un voisinage

tel que. pour tout

analytiquement dans

'lY et

j € IN

verifie

, IJ'j

tr de 0

se prolonge

- 135 -

C.

Goulaouic

sup Icp.(x)l < bfij'sup Icp .(x)1 x(U' J x(O J

v)

Pour tout

i l existe un voisinage

b>1

C dans ff et C>O tels que, pour tout

tr

de

jEfi

se prolonge analytiquement dans 1)" et verifie

CPj

sup Icp.(x)l< C bV):j x(\t J ~~~~~~tratio~ : On a deja montre que ~ .= ~

(lemme 4.1) ; en prenant

J

J

i)

et

ii)

sont equivalents

, on a montre aussi que · iii)' impli-

que rv), D'autre part II r este Soit

iV) . implique

a montrer

i» 1

que

V)

; on choisit

b

V) grace a (4.2). implique

1

( ]1, b[

-VAk) (mes 0 )1/2

C = sup Vk+1 b ~ b 1 k(fi

(

associes li.

b

1

dans

iii): ; on note et on montre que ·s i 1iet

V), on a pour tout

sup Ir(x) I < C C b 1 'It -

r ( ~k

C sont

'

iXk sUPlr(x) I.

x(O et Ie reste de la demonstration resulte de la compaci te de la boule unite de

~.

J

pour

j

rixe.

IV. - rn egalites de MARKOV.On rappelle l'in6galite de MARKOV classique (cr[28]) polyndme

P

d Iune variable, de degre sup I pI (x) I xl~1

I

~

n

Pour tout

, on a

sup Ip(x) I Ixl ~1 Nous allons montrer que des majorations analogues sur les fonctions


2

-

pr opr es d'un operateur sont equi val ent es encore

a 1a

pr opr ie t e des iteres.

PRepOSITION 4 .4 . - Les proprietes suivante s s on t equ iv a1ente s

I

i)

A a 1a pr opr ie te des iteres (ou n' importe que 11e

- 136 -

C. Gou1aouic propriete equivalente donnee par la proposition 4.3.) ii) Pour tout lJlk

b>l

, i l existe

et tout indice

IDCllJlk IL~

C>O

uf ,

a E:

(0) ~ CI Cli Cl ., biAk

IlJlk IL~

(0)

Remarquons que lion peut encore remplacer la norme norme

2 L

en particulier et aussi

4.3, iii)) implique b>l

co

L

ii)

~k'

en utilisant la formule integrale de CAUCHY :

, i l lui correspond ~ par la proposition 4.3 iii) ; il suffit

ce qui est immediat d'apres la formule de CAUCHY, 0

a

17' ,

on a

C etant lie

a la

["li.

Inversement, supposons lJlk(Y)

=

ii)

; on ecrit :

E~

~a)(x)(y_x)Cl

Cl et cette serie est convergente pour on a psur

l~

par n'inporte quel element de

k

de montrer que pour une f'onc td.on '¥ analytique bornee sur

distance de

Ilar

On montre que l'inegalite de BERNSTEIN (proposition

Demonstration:

soit

lJl

tel que pour tout

b>l

et

Iy-xl

x E: Q et


<

E _1_, rICllcICl!Cl! b

Cl

<

Cl .

.

b'f'F:k SUpllJlk(X)I xE:O

y.E: "'"

On peut se ramener au cas

x

supllJlk(x) I

xE:Q

E (re)/al

Q

~

t el que

Lb

fAk supllJlk(x)I xE:O

1=1 en pas sant par un

en utilisant la compacite de

assez voisin de

0:

On a trouve un voisinage '\)" de sup IlJlk(Y)I

y

~l

pour

1

donne.

b

intermediaire et

- 137 -

C.

Goulaouic

BIBLIOGRAPHIE

S. AGMON - Elliptic boundary value problems, Van Nostrand 1965.P.J. ARANDA et E.P. H1 (Q) dans o

1295.N. ARONSZAJN -Sur les decompositions des fonctions analytiques ••• Acta Math. 65(1935),1-156.-

[4J

M.S BAOUENDI -Sur une classe d'operateurs elliptiques degeneres, Bull.Soc.Math.France 95(1967),45-87.M.S. BAOUENDI et C. GOULAOUIC - Regularite et theorie spectrale pour une classe d'operateurs elliptiques degenes, Arch. Rat.Mec.Anal.34(1969),361-379.-

[6J

M.S BAOUENDI et C. GOULAOUIC - Regularite analytique et iteres ••• J. of funct.Anal.9 (1972), 208-248.M.S. BAOUENDI et C. GOULAOUIC - Approximation of analytic

functions~ ..

Trans.Am~Math.Soc.I973.-

[8J

M.S. BAOUENDI et C. GOULAOUIC - Iteresd'operateurs elliptiques ••• Revue Roumaine de ~~thematiques pures et appliquees 10(1973)-

[9J

M.S.

[ 10]

R. BEALS - Classes of compact operators and eigenvalue distribution~ for elliptic operators. Amer J. Math.89(1967),10561072.-

C. GOULAOUIC et B. HANOUZET - Caracterisations de classes de fonctions COO et analytiques ••• J. Math. pures et appl , 52 (1973), 1.15-144.-

BAOUE1~I,

R. BEALS - On eigenvalue distributions •••• Bull Amer. Math.Soc.74 (1968),1022-1024.[ 12]

M.S. BIRMAN et M.Z. SOLOMJAK - Spectral asymptotics •••• Dokl.Akad. Nauk.S.S.S.R 205(1972) et Soviet Math.Dokl.13(1972),906-91a.-

[ 13J

L. BOUTET DE MO~nffiL et P. GRI SVARD -Le comportement asymptotique de s valeurs propres d'un operateur, C.R. Acad.Sc.Paris 272 (1971) ,23-25.-

[14 J

F.E. BROWDER -Asymptotic di stribution of eigenval~es and ei genfunctions •••• Amer.J.~~th.87(I965),175-195.-

- 138 -

C . Goulaouic

[ 15]

R. COURANT and D. HILBERT -Methods of Mathematical physics, vol.I, Interscience 1953.-

[ 16]

A.E] . KOLLI - n-ieme epaisseur dans les espaces de SOBOLEV,C.R. Acad.Sc.Paris 272(1971),537- 540.-

[17]

A. El .KOLLI - n-ieme epaisseur dans les espaces de SOBOLEV avec poids, C.R.Acad.Sc.Paris 273(1971),450- 453.-

[ 18]

J. FLECKINGER et G. METIVIER - Theorie spectrale •••• C.R. Acad.Sc.

[ 19]

L. GARDING - The asymptotic distribution of the eigenvalues ••• Math.Scand. (1953),237-

[20]

P. GRISVARD -Le comportement asymptotique des valeurs propres •••• Seminaire LERAY, Oollege de France (1972).-

[21]

A. GROTHENDIECK - Espaces vectoriels topologiques, sao Paulo 1958.-

[22]

M. GUILLEMOT-TEISSIER - neveloppements en serie •••• C.R. Acad.Sc. Paris 265(1967),419-421.- '

[23]

L.HORMANDER - Linear partial differential operators, Springer Verla~ 1963.

Paris (1972)

L.HORMANDER -HYpoelliptic second order differential equations, Acta Math. 119(1968),147-171.[25]

KOLMOGOROV -Uber die~este AnnMherung von Funktionen•••• Ann.of . Math.37(1936j,107-111.-

[26]

T.KOTAKE and S. NARASINHAN - Fractional powers of a linear elliptic operator,Bull.Soc.Math.France 90(1962),449-471.

[27]

J.L. LIONS et E. MAGENES - Problemes aux limites non homogenes t1 et 3, DUNOD 1968•..,.

[28]

G.G. LORENTZ - Approximation of functions, HOLT and ENEHART' and WINSTON 1966 -

[29]

G. METIVIER - Comportement asymptotique des valeurs propres de problemes irreguliers (a para1tre)-

[30]

B.S. MITIAGIN - ApProximate 'dimension and bases in nuclear spaces, Uspeki Math.Nauk 16( 1961) ,63-132.-

[31]

C. NORDIN - The &symptotic distribution of the eigenvalues ••• Arkiv ftlr Mat. 10(1972), 9-21.-

-139 . -

C.

Goulaouic

N. SHlMAKURA - Quel~ues exemple$ .de ~-fonctions d'EPSTEIN•••• Proc. Japan.Acad 45(1969),866-871 et 46(1970),1055-1069.-

[33]

H. TRIEBEL - Erzeugung des nukl'earen lokalkonvexen RMuines Math. Ann. 177(1968),247-264.-

C""(Q) • • •

H. TRIEBEL - Nukleare FunktionenrMume und singulMre el l i pt i sche Differentialoperatoren,Studia.Math.t.38(1970),285-311.[35]

I.L. VULIS and M.Z. SOLOMJAK - Spectral asymptotics of degenerate ell iptic operators, Soviet. Math.Dokl.13(1972 ),1484-1488.-

[36]

H.WEYL - Das asymptotische Verteilungsgesetz der Eigenwerte.~i~ea­ - - - rer parti~llerDifferentialgleichungen••• Math. Ann.71 (1911) 441-479).-

[37]

M. ZERNER - neveloppement en .ser i es ••• C.R.Acad.Sc. Paris 268(1 969), 218-220.-

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c .r.

M . E .)

ESSENTIAL SPECTRA OF ELLIPTIC SYSTEMS ON COMPACT MANIFOLDS

GERD GRUBB

Corso

tenuto

a

Varenna

dal

24

agosto

al

2

settembre 1973

ESSENTIAL SPECTRA OF ELLIPTIC SYSTEMS ON COMPACT M..J\NIFOLDS

by Gerd Grubb (University or Copenhagen, Denmark)

1. Introduction.

These lectures report on a joint work with G. Geymonat, Torino. Let

T be a closed, densely derined

in a Hilbert space A € G ror which

H. T - A

We denote by

~(T)

line~r

operator

the set or

is a Fredholm operator, i.e.

has closed range and rinite dimensional kernel and cokernel (the index =

dim ker T - dim coker T not necesaarily being

zero), and we derine the essential spectrum or ess sp T =

m\

~(T).

T by

- 144 -

Grubb

G.

Remark 1.

In Schechter's pre~erred de~initi~n

[17], the essential spectrum is the complement o~ the part o~

where the index is zero, so our essential

~(T)

spectrum is in general smaller than

his. The properties

listed below are proved in Wol~ [21]. The two notions coincide

~or sel~adjointoperators.

A point

"

belongs to

ess sp T

there exists a singular sequence

T* -~,

m,

and only

either

~or

SUn

-+

sUbsequence.

0

f'or-

n

u -+

ess sp T

n

E: H

00,

un

or S

is

for all

with

but

i~

T -"

here a singular sequence ~or an operator

a sequence of elements n E:

i~

having no convergent

is closed and . is invariant under

the addition of a compact operator to

T.

As is well known, elliptic boundary value problems on compact manifolds

rr

let

r

defi~e

be an n-dimensional

and interior

n =IT\r

r) 0

(r

COO

~ore

precisely,

mani~old with boundary

JOL

may be empty), let operator on

n

o~

(sCalar or matrix-~ormed) and let

B

be a

be a property elliptic order

Fredholm operators.

dif~erential

di~ferential operator (usually matrix-~ormed) defined near

r.

Then

Let

~

~

-"

be the realization o~

Jt

with domain

is a Fredholm operator for all

ess sp ~= ~, i~

"E:~,

i.e.

- 145 -

G. Grubb

(i)

rr is bounded,

(ii)

.It is uniformly elliptic on rr,

(iii)

B covers

(iv)rr,r

Jt,

and the coefficients in

B

and

Jt

are smooth. When

r = \2l,

(i), (a i ) and (i v ) alone imply that the maximal realization A in L2(rr) with domain

has no essential spectrum.

An essential spectrum may appear when one of the conditions (i) - (iv) is not satisfied. As for (i), we refer to Schechter's and Garding's lectures [17], [10], where operators on IRn are studied, having a large essential spectrum. Next, consider condition (ii). Here, it is well known that for

D(~) () Hr(rr)

~

to

(or A) to be a Fredholm operator from 2(rr) L i t is necessary that A be

uniformly elliptic. But we are studying operators in 2(rr); L for these, a mildly degenerated ellipticity may still give Fredholm operators, as in the works of Baouendi and Goulaouic, ct' , [ 8 esa sp ~ ~ ¢,

l-

A stronger degeneracy, for which

was studied by Wolf [21] and Poulsen [16].

Concerning (iii), we have similar phenomena. Non-covering boundary conditions, for which

ess sp

~

=

a,

were studied

by Egorov and Kondrat' ev [ 5 ], Hormander [11], Eskin [ 6 ] and others; on the other hand Tovmasyan [19] gave examples,

- 146 -

·G . Grubb

and Vainberg and

Gru~in

(20] presented a general class,

ess sp A = ¢ even though B does not cover Jt B - We shall not be concerned with (iv).

where

We shall now look at a rirth aspect or elliptic boundary value problems which may, perhaps surprisingly, impl y the presence or an essential spectrum even when (i) (iv) are all satisried. It appears when we consider systems elliptic in the sense or Douglis and Nirenberg. A rirst sketch or the theory was presented at the Goulaouic-Schwartz Seminar 1972-73 (expose No. 27). We are still working on the development or the theory, especially ror manirolds with boundary.

2. Mixed order systems. Bet

q be an integer

>

and let {m

1,m 2,···,mq] be a set or nonnegative integers.. Let .A:. = (Jt s t) s, t-1 () - ' •.. '-a be a q x q matrix or dirrerential operators )tst of orders

Jt

ms + m respectively. The principal symbol or t, is the matrix or principal symbols or the Jist

(2.1)

cro(A,) = (a-°(Jt.a t))S,t=1,oo.,q = (o-ms+mtCA::'st))s,t=1,-,q'

here

a-°(Jtst)(x,f) is a runction on the cotangent bundle n T*(ff) (~rr x R when rr eRn), homogeneous or degree rna + mt

in

f,

SQ that the determinant

det

a-0(Jt)

is

- 147 -

G.

homogeneous or degree be elliptic on

il,

.

2(m +.·.+m ) 1

q

Jt

f.

is said to

when

il,

and strongly elliptic on

when the matrix

~O(Jt)(x,f) + iO(Jt)*(x,f)

(2.3)

in

Grubb

is positive derinite on

T* (15:)\0.

Such systems are a symmetric case or the elliptic systems introduced by Douglis and Nirenberg [ 1 ], this case seems the most interesting for questions in spectral

theory~

in s decreasing order, and we assume furthermore that some of We may, ror our purposes, arrange the

them are

~:

m1 _> m2 ->

where

1

m

~

q'

••• -> mq I > mq , +1

< q.

m = 0, q

Then, even when conditions (i)-(iv)

are satisried, the essential spectrum will be nonempty, as the folloWing examples indicate: Example 1. b

E: JR,

n = JRn, q = 2, {m 1,m 2}

{1;O}.

Let

a

and

and consider it has

Clearly,

> 0.

b

A

'Y

~

is elliptic for

Assume that

b

~

°

b ~ 0,

strongly elliptic for

and consider the realization

def ined by the Dirichlet condition

-14!l -

G. Grubb

Ay

is a Fredholm operator. However, an easy computation

(exercise!)' shows that, f'or

b ~ 0,

A..y

the spectrum of'

,,+ converging n (counting mUltiplicities)

consist of' two sequences of' eigenvalues, to

+

and

00

Thus. ess sp

,,-

converging to

n

A..y

= b!

b

Certain dif'f'eren ti al opera tors f'rom

nuclear reactor theory are generalizations of' this, with b

replaced by a matrix of' f'unctions, and the

a

replaced

by suitable f'irst order dif'f'erential operators. Example 2.

'n

= 3,. !m1,m2,m 3 l = !1,1,ol.

E: ]R2, q

The

linearized Navier-Stokes operator is

(2.6)

.it =

a

:"7-

-/::;

0

~O

.:../::;,

"

a -ax -ay Here,

Jt ..It

~

"

ay ,

(TO (-A)

with

=

0

2

+T]

2

0

~

:I:

0

(~,T])

f'or

2

if iT]

+T]

0

-iT]

-i~

det (TO(Jt) == _(~2+T]2)2

0

=t: (0,0),

so

is elliptic. It is not strongly elliptic, but + cI

is, f'or

c

> 2.

Consider

A- "I,

its principal

symbol is ~

(2.7)

0

(T (Jt -,,) ==

2

+T] 0

-i~

with

2 ~

2

0

i~

2 +T] .

iT]

-"

-iT]

0 2 2 2 det (T (J+, -,,) = ( -,,-1)(~ +T] ) •

elliptic if and only 11' ?\ 1= -1 !

Thus

.it - "I

is

•

We shall see later that

- 149 -

G. Grubb

is in the essential spectrum or"any realization of this

-1

operator

Jt •

We observe that, since part or 1\

enters in the principal symbol or

interrere with the ellipticity! cussion or

(ii) in

is or order 0,

~ "

.A -

(Recal~

1\,

and thus may

however the dis-

th~ introduction.) Another ~ndication

that our operators have nonempty essential spectrum is that 2 the injection D(~) 4 L ( n ) <1 will be non-compact. We shall now try systematically to treat the rollowing problems, under the assumption that

(i)-(iv) are satisried:

1) to determine the essential spectrum or each when

r = ¢)i

~

(or A

2) to rind the asymptotic behavior or the

discrete spectrum, near inrinity and near the essential spectrum. Let us end this section with some notations. It will be userul to single out the zero order part or writing.A

..A.

by

in blocks, according to (2.4):

~

(2.8)

= [:

:]

here and and

Write

R =

(Jt s t) 8 ><1,

t<_<1 I' (they are all or positive order), • is the multiplication operator M = (~t) t> I · s s, <1 For certain purposes we also need a riner decompostion: I

- 150 -

G.

Grubb

{1 ' ••• ,1 ; 1 , ... 1 ; •... 1 ,···,1 I, 1. ~2 • 2, , 1 .. ., ~

r where 1

1

> 1 2 >••• >

I . ) r p = q - q.

1

entries

Ip = O.

r

2

~ote

entries p

that

...

.]

r. = q,

~.

According to this we write

(2.10)

r p entries

j=.1

J

.~

as

pp where each

Pjk

is an

operators or order

rjx r k ma~rix or dirrerential Ij + lk. The rollowing minors are given

special names:

... P j = [ P" P j1

Note that

Pp-1 = P

and

, P j

J

j = 1,

...

,p.

P jj Ppp = M

in the previous notation.

3. The case without boundary. 3.1

The theory is simplest in the case where

emp~y.

r

is

We present this case rirst, since it already shows

some typical f'e a tur-ea, So, assume in the rollowing that fij

on

is compact and without boundary, and let

n = n.

Then Jt.

is continuous

Jt

be elliptic

- 151 -

G.

m +a

q

n

H s

s=1 and Jt

H

s=1

s

(172) ,

all

has a continuous parametrix

site direction. The domain A

-m +a

q

(n) -. n

D(A)

Grubb

aElR,

in the o.ppothe maximal realizatio~

o~

('cf. (1.3)) is in general not a simple product o~ Sobolev

spaces, since none o~ the range spaces in (3.1) equals 2 q m L (n)q; we note, however, that D(A) CS~1H s(n), since -m n H s(n) c L 2 (n ) q . One shows that c;(n)q is dense in D(A)

by use of' .Jt(-1),

so .t he maximal realization

equals the minimal realization; extension o~

A

I

Proposition 1.

A -"

equals the Friedrichs

.it I.e"" when .:fc. is strongly elliptic, and

is sel~adjoint when

then

A

A

Jt

Eet

is ~ormally sel~adjoint.

"E:C.

I~

-:It -"

is a ·Fr edhol m operator in

is elliptic,

2

L (n ) q .

uses that when it - " is elliptic, it is q -m +a q m +a n H s (n) to S~1H s (n) ~or a Fredholm operator ~rom s=1 all aE E , the kernel consisting o~ COO ~unctions and The

proo~

being the same

~or

all

a, and the range being determined by

orthogonality to the same ~inite set o~ all

a.

Then the kernel

and the range

o~

A

COO

functions ~or

also equals that kernel,

is determined by the same set o~ 2(n)q orthogonality conditions (using that L contains the o~

A

product spaces in (3.1)

~op

large enough

tained therein ~or small enough

a,

and is con-

~or

the essential

a).

So we s ee that we only have to look

- 152

G.

~

spectrum among the It is not

a priori

ess sp A.

for which

Jt -

~

Grubb

is n o t elliptic.

evident that each such point is in

That this is indeed true will now be shown for

the case where

P

(cf.(2.8)) is elliptic; this gives a

particularly simple theory, whereas the general case requires further techniques that we

not go into here.

~hall

We shall use the following, easily verified observation

o

on matrices

and

I

denote zero resp. identity ma-

trices). TIemma 1.

Eet

H H G 1, 2, 1

and

G 2

be linear spaces,

and let H. G1 1 x. ..... x H2 ( J 2

with

and

T

bijective from

11

-1

T22 - T21T11T12

only if

T

H 1

to

G 1•

Then

is injective (r-e sp , surjective) if and

is injective (resp. surjective).

This is used to show

I

Lemma 2.

city) that

P

Assume that and

Jt

P

is elliptic and (for simpli-

are invertible. Let

- 15 3 -

G.

it is a pseudo-dirrerential operator or o rder o-°(S)(x,f) = so(X,f).

and

S

(3.6) o 0-

denote

Then

is elliptic, and invertible in

ror each

0;

Grubb

L2(n)~-~'. Moreover~

A E: Ie,

(Jt -AJ)

=

[ 0- 0 (p) 0

0-

~][~

(R)

and hence

ror all

(x,f) E: T*(rr)\O.

Now derine (3.8) w =

U

(x,f)E:T* (rr)\o

{A E:

n I

A

is an eigenvalue ror

We have as an immediate conse~uence or (3.7): W =

{AIJl - A

is not elliptic}.

The rollowing theorem completes the description or ess sp A:

- 154 -

G. Grubb

Theorem 1.

(3.10)

Under the hypotheses

o~

Lemma 2,

esB'sp A = ess sp 8 = w. It ~ollows ~om

Proo~:

(3.5)

that

Jt

has the inverse

-P

-1 -11

8

where the terms in the

is the sum A-

in

row and column are pseudo-

~irst

1_

that

[0 0

-

0 8-

1] comp~ct +

ess spvA = ess sp 8,

it

~ollows

S

are invertible. The ~irst identity in

using that

8

is in general

pseudo-di~~erBn~1al operator o~ order

~O(Q)

dif~erential

{oJ, ~rom A

which

and

(3.10) being established, we

shall now show the second one.

or

operator

ess sp A- 1 = ess sp 8- 1 u

~2(IT)q. Then

~O(R)

-1

operators o~ negative (mixed) order, thus

di~~erential

A- 1

QS

0

~

genuine

(only where

vanishes will the principal part be a

operator, i.e. mUltiplication with a matrix

~unction) •

net (in

~act

(o~ order

in

}.¢ w. II s 0

0)

L 2 ( fi ) q- q ' •

Then

- }.III

z.

sO(x,~) - }.I ~

c )0),

so

0

S - }.I

~or

alJL

(x,~)

is elliptic

which implies that i t is a Fredholm operator Thus

}. ¢ eSlll;

SD S.

Conversely, aSflUme that

- ' 155 -

G. Grubb A

is an eigenvalue for q q eigenvector 8.E c - ' .

so(xO,f

with a normalized o) As a singular sequence for S - A

we may then uSe the following sequence, often used in the literature on pseudo-differential operators (see e.g. Ho~mander

[12] p. 158-59 or Melin [15] p. 129): Consider

a local coordinate . system where with

IIvllo = 1,

x o = 0,

let

y

E ~(JP)

and set

(3.11)

w = kn/2Y(kx)ei<x,k2fo> 8. k

One finds that

Ilwklio =

for all

kE

]N,

and

supp w -+ . { 0 1, so w has no k k convergent subsequence in HO (IR n) q-q I • (Actually, wk in H- 1 ( JR n)q-q', [15].) Thus AE ess slJ S, which II(S - A)wk"o

-+

0,

but

-+

0

completes the proof of the theorem. Since

S

is a bounded operator, we have

corollary 1.

ess sp A is bounded.

(This will not in general be the case when

P

is

not elliptic.) Remark 2.

The singular sequence (3;11) for

be used to construct a singular sequence for E be a parametrix of

P,

S - A can

A - A:

Let

whose kernel (as an integral

operator) has its support close to the diagonal. Let and

- 156 -

G.

Grubb

is a singular sequence for A - A, (It is used is close to that .w -+ 0 in H- 1 and that supp EQw k k BUpp w ,) k then

uk

3.2. Having determined the essential spectrum of

A,

we

shall now discuss its discrete spectrum. Here, we restrict the attention to the case where

Jt

is strongly elliptic

and formally selfadjoint; in particular, the hypotheses of Lemma 2 are satisfied after addition of a sufficiently large constant to ~.

We need a more refined factorization,

using the notation (2.9) - (2.11): Proposition 2.

Assume that each

is invertible (holds e vg , when

Jt

P j (j

= 1""

,p)

is strongly elliptic \

and a SUfficiently large constant has been added). Then we have

where the

are invertible elliptic r x r j -matrices j j of pseUdo-differential operators . of orders 2l j, and Y1 and

Y 2

C

are triangular, of the form

with pseudo-differential operators of negative order outside

- 157 -

G . Grubb

Ithe diagonal. Proof: to

Apply Lemma 1 successively for

T = P j ',T 11 = P j-1

(the spaces

H1,H2 , CJ and 1 with the appropriate

C~(rr)k

being of the type

shows (3 .13) with bijective applied to ~o(~) order 21.).

j = p,p-1,· ·· ,2,

C j;

This

the analogous argument

showS that the

Cj

are elliptic (of

J

We note that in (3.13), Y1

=

Now,

and Y2 H2lj(rr) r. j ,

Cp = S

of Lemma 2.

are isomorphisms in

L2(rr)~,

we find as a special

conse~uence:

A is positive definite and unboUnded, and its

essential spectrUm is bounded; then it has a eigenvalues

(A~(A)).;cJN J

J- -:

ties) converging to b~havior

se~uence

~/ ~ . To determine the asymptotic

of these eigenvalues, we consider the decomposi-

!u 1 , .•• , u ~ I

as

w = {u~'+1'··· ,uq'l;

{

(3.15)

for large may set

of

(repeated according to multiplici-

tion (2.8) (where now P = p*, R = Q*, M = M*) u =

and

and write

u = {v,wl, v = {u1,·· ·,u~,I, we look for nontrivial solutions of (P-A)V + Qw = 0, .* Q v + (M-A)W = 0,

IAI > IIMII, M - A is invertible, so we -1 W = -(M - A) Q* v, reducing (3.15) to the nonA.

linear problem

When

- 158 -

G. Grubb

(p - A - Q(M - A)

(3.16)

-1 •

Here, it should be expected that the Q(M - A)-1 Q*

diminishes as

asymptotic behavior P.

~or

~irst

P,

which is usually P

increases, so that the

o~

the maximal realization

o~

o~

mixed order; the lowest order oc-

is 21p- 1·

Propesition

3.

The sequence

Aj(P)

o~ eigenvalues

(repeated according to multiplicity)behaves asympto-

tically like the sequence operator

C

p-1

-21

j

j -4~,

where

(3.18) . c(Cp_1)

-n/21

c(C

p-

the

~or

(3.13)),

(c~

A (p)j ~or

the term

the eigenvalues is similar to that

o~

determine the spectrum

P

A

e~~ect o~

This is. indeed what we shall show, but we must

curring in

o~

O.

Q)v =

i.e.,

p-1

1)

pseudo-di~~erential

/n -4

c(C

p-

1)'

is the constant determ ined by

p-1 = (2~)

-n( ) dx

J

0 tr ~ (C

n/21 p_1)

p-1 df •

n If l=1, This is proved by use

where

the

~act

that

- 1, and by using the p_1 for elliptic pseudo-differential operators given by

Y

~ormula

o~

is

o~

order

~

-21

Seeley [18] (we apologize for not trying to give a more

- 159 -

G. · Grubb reduced expression

c(Cp- 1)!)'

~or

We can now show Assume that A

Theorem 2.

>

IIMII

A greater ·t ha n

a

~ormallysel~adjoint. Let

eigenvalues

o~

A;(A) ~ A;(A) ~ •••• (3.19) Proo~.

~or

A;(A)j Let

values

a

A

-21

~or

and arrange the in an increasing sequence:

Then p-1

In

~ c(C _ p

T~ = P - Q(M-A) o~

is strongly elliptic and

-1 •

Q;

~or

1)

j ~ "".

we are then searching

which on e has the coincidence

(3.20 ) The spectrum

C(T)

o~

each

TA is a sequence o~ eigenvalues going to as ~ollows:

determined as in (3.18). For

a

< A' < A",

TAl - TAli = Q[(A'_M)-1 - (A"_M)-1]Q* ~ 0, thus f'o r' each f'Lxed

j, J.L/T A,) ~ J.Lj(TAII) (e.g. by the mini-~ax principle), so J.Lj(T A) is a decreasing ~unction o~

A.

It is moreover continuous

Now, when

~,

increases

through the interval j ' ~or which there exists a

J.Lj(P)

>

uniqu~

~rom

a

(c~.

to

Kato [13] p. 291 ) .

+ "",

J.Lj(T A)

decreases

]J.Lj(P), J.Lj(Ta)[. Considering only the a, we see that ~or each o~ t h os e j, Aj

such that

J.L

.(TA ) = A ,! j J

J

The

- 160 -

G.

numbers

Grubb

determined in this way satisfy (3.20) and

A.

J

constitute the sequence

Aj+(A)

(expect perhaps for some

of the fir~t terms and a shift in the enumeration). Finally, E ]~j(P), ~j(TA)]

since

A

~j(P)

> A,

j

for all

j

such that

> c(p) -

c(C p- 1)'

~ c(T

for each

we have

lim inf A. j j -+

-21

J

00

lif -+00 sup A . j J

-21

p-

/n

p-1

which im~lies (3.19) since

In

C(T~)

The discrete spectrum near

A)

-+

w

c(p)

A,

for

A

-+

~

•

is harder to pin down.

Let us just say this much (still for the selfadjoint case)~

w is the union of a finite number of compact intervals on the real axis. When

S

has a sequence of eigenvalues

converging to the end point of such an interval outside,

A

f~om

the

also has one (converging at the same or a

slower rate; this is seen by suitable use of the mini-max principle). This is the case in Example 1, when But even when

S

does not have such a sequence,

have one; this is the case for Example torus

~

1 = S 1 x S,

there

S = -I.

2

of

~,

and for which

may

defined on the

S

shall be sufficientlY

flat (Taylor coefficients vanish) at a point has an eigenvalue

A

A systematic criterion

seems to be that the full symbol of

Which " so(x,f)

a ~ o.

A

a-(Q)(x,f) ~ o.

(x,f),

for

that is an end point

- 161

-

G.

Grubb

4. The case or a manirold with boundary. 4.1.

We shall now study the aase where

n-dimensional is much more

manirold with boundary

C-

complica~ed

n

r ~

is a compac

¢.

The theor

than the previous theory, ror vario\

reasons that we shall explain below. Our results so rar are by no means complete, so we shall only describe them rather brierly. However, one result is easily obtained, thanks to Remark 2: Theorem 3.

Assume that

P

is elliptic on

n.

Derine

(4.1 ) ror

(x,f) E T*(n)\o,

and derine

- A is sp A ror

w by (3.8). Then

(i)

w = !AI~

no t elliptic}.

(ii)

w cess

any closed linear extension

Proor: (i) rollows rrom the rormula

o rt 0 det ~ (~-A) = det ~ (p) det (sO-A), which is still valid at each

(x,f) E T*(U)\O

(proved as

E n, and let E be o a ps eudo-dirrerential operator, which acts as a par ametrix or in Lemma 2 ) . To show (ii), l et

P

ror

C-

X

runct ions wi t h support in a nei ghborhood or

contained in n, and s ends such runc tions in to o When A is an e igenvalue or so( xo,f o) ro r some X

- 162 -

G.

the sequence

in (3.12) constructed ~rom

~

(Jt - ~)Id:'

serves as a singular sequence ~or

K

-~.

Since

ess sp

o~ the set of' such

A

w k

o

in (3.11) . and thus ~or

is closed, it contains the closure

(xo,fo)'

i.e.,

w.

Now we cannot expect to get equality in (ii), when is a realization

~

~

covers

is elliptic but

Example 2, continued.

J{ -~,

with

Jt

Jt ,

(-6-~)v

+ ow 0 ox ==

in

11,

(-6-~)v

ow + oy == 0

in

n,

-~w ==

0

in

11,

.

0

on

r,

.

2

-:~V1 _ov 2 oy ox

exclude the value

oju You == ul r , Yju == --., j == 1,2, •••• ) on J ~ .== -1 where Jt - ~ . is not elliptic.

Assume furthermore that

~

*

0,

then (4.2) may be solved ~or

and reduced to

(4.3)

l

~o~

as in Example 2, i.e., the problem

(He use the notation

~

~,

Consider the null Dirichlet problem

Y Ov1 == YOv2

~e

but ~or certain

does not cover . ..A - ~!

B

1

(4.2)

1

o~ a boundary condition (c~.(1.2))~ For

it may happen that ·B

-it -

Grubb

2 2 1 02 v 0 [ ( -1-,)"'"""2 == ~v1 - ~ =--:2. 2]v1 [ 13 oX oy oxoy 2 2 2 1 0 ( -1--)-]v == ~v2 , ~ oy2 2 oxoy + ox +

-t ~

[-~

YOv1 == YOv2 == 0

- 163 -

G. Grubb

the di ff e r enti al operat or on t he l eft is t h e

Fo r op era tor

L

which was s tudied b y Bi t sadz e [3], who showed that t he 0 i r i chle t p r oblem fo r

L

d efines a non-Fr edholm op erator; in

fac t t he Di r i c h le t c ondi tion does not cover Of

L

..A -

the b ounda ry . On t he o t h e r hand, i t c ove r s

any

A

~_12 ' -to

We s h a ll s e e l a t e r that

at any point A

for

ess

One of the d i ff iculti es i n a s yst ematic s tudy .of the r eal iza tions

is that the known Fredholm theory (Agmon-

~

Do ugli s - Nire nb e rg [ 1 ], Geymonat [ 7 ]) trea ts the operators onlly 2 In prop er sUbspaces of L (n )q. Th e methods Lions and .

or

[14] can b e ex t ended to yi eld statements in larger spa-

Ma g enes

ce s , b ut th is d o es n ot by the usual i n t e r pol a tio n te c hni que s l e a d to a s i tuat i o n where

~

maps a s ubs et o f . .L 2( n ) q

onto

-Instead of g o i ng fur ther into this and giving our partial results, l et us t u r n to an older method of defining re ali zations, nam ely v i a variational th eory. Assume f or simplici ty that ~'

Jt

= r 1,

q - q'

= r 2•

P

is of o rder 2 , so

. The sesquilinear f orms associa ted with

are the forms

(4.4)

a(u,u')

= p(v,v' )

+ (Rv,w') +(w,Q*v l

)

+ (Mw,w'),

- 164 -

G.

where

p

with

P.

runs through the sesquilinear forms associated This gives the "halfways" Greens formula

Llp

where

Grubb

and

~Q

are fixed

r,

matrices of functions on

I~1

and

x r

resp. r 1 x r 21~ runs through all

r

x r of first order differential operators on 1-matrices 1 (this is a special case of Theorem 3.3 in [9]; Yav and

tlp Y1v +

(i,Q"YOw

constitute the so-called reduced Cauchy

Av denote the operators associated, by the Lax-Milgram lemma, to the restriction of a(u,u') to 0 1 r 0 "z 1 r r 2 Vy = HO(n) 1 x H (n) , resp. Vv .= H (n) 1 x H (n) • Then data). Let

A

y

Ay

r

and

represents the Dirichlet condition '

(4.6) and

y v = 0

o

Av

represent a

-:

~umann

aP Y1 v

+

condition

aQy 0 w + ~y0 v

= O.

is invertible.) Example 2, continued.

If we to the Navier Stokes operator

associate the sesquilinear form

a(u,u') =

I) - (w,av 'lax) + (6v - (w,av21/ay) 2,6V 2') 1,6V1 1 -(av1/ax + av2/ay,w'), we find that (4.7) takes the form

(6V

= 0

- 165 -

G.

Grubb

rt

(nx,ny) is the normal to r. When A ~ -1, this condition covers Jt - A if'f' ,,~_J! Note the

where

2

dif'f'erence f'rom the Dirichlet condition. Our results are most satisf'actory f'or

Ay • We c an here

obtain a certain analogue of' Lemma 2: Proposition order 2, so

'1

1

4.

Assume f'or simplicity that

= r1,

= r2•

'1 - '1'

Let Jl

P

is of'

be strongly

n + ~ n * positive def'inite on elliptic, with ~ • . r Then P + P is posi ti ve def'ini te on c~(n) 1 ,

Py :

def'ines an isomorphism

H6(n)

r

1

r

~ H- 1(n) 1.

Def'ine

-1

(4.8)

Sy = M - RPy Q,

it is continuous on

r O H (n) 2.

(1) For all

(ii)

By

{v,w}

1 r E Ho(n) 1 x H0 (n) "z ,

is an isomorphism on

r

HS(n) 2 f'or all

s L 0,

(iii)

Proof': maps

(4.9) f'ollows using Lemma 1 and the f'act that r r r 1(n) H 1 x HO(n) 2 bijectivelyonto H-1(n) 1 x

°

this also shows (ii) f'or

s = 0.

Jt

r

HO(n) 2,

(iii) f'ollows f'rom the

- 166 -

G. Grubb

regularity theory s

2 1 by use

o~

Py ;

~or

and (ii) is shown

[1) and [7), for general

~or

s

integer

by interpola t t cn ,

This leads to Theorem the

Assumptions

Proposition

o~

4.

~orm

(4.10)

~or

4.

all

~t] ! ~ ,gl

[p-1 + p-1QS-1 RP-1 y y y y _S-1 RP-1 y

r

. ess sp

(4.11 ) in particular,

-1 Sy

y

E HO(~) 1 x HO(il) 2,

A.y

-'j[:)

-P-1 y QSy

-

r

has,

and

= ess sp Sy

is bounded.

ess sp Ay

In the proo~, (4.10) ~ollows by inverting each ~actor in (4.9), and (4.11) uses that, because

o~Proposition 4

(ii), the entries in the ~irst row and column o~ (4.10) map

°

H -spaces continuously into

H1 -spaces.

The theorem carries the problem essential spectrum

o~

This operator is not a

~

over into

equals

so(x,f)

determining the

~inding

pseudo-di~~erential

more complicated integral operator Boutet de Monvel in

o~

o~

that

o~

Sy

operator but a

the kind described by

[4); its (principal) interior symbol

but it also has a boundary symbol

O-~(Sy)

containing a term stemming from trace operators and "Poisson kernels". It ~ollows ~rom operator when

"

[4) that

Sy - "

is outside the set

is a Fredholm

wy = the closure of

- 167 -

G.

the union of' the

of'

. sp ect r-a

Grubb

We

and

remark that in f'act, by use of' the reductions in Proposition

4 on the symbol level, (4.12)

wI' =

W U

[AI A - A

is elliptic but

does not

l'

cover .A.

J.

We do not have a simpler description of' this set, except that we can ref'er to the calculus of'

:[4] .

We have

f'ound Icorollar y 4.

W

c esa, sp AI'

C

wI'.

As not ed earlier, the f'ailing of' the covering condition 2, does not necessarily give a non-Fredholm operator in L so a f'urther sharpening of' these inclusions requires special investigations. Remark 3.

In certain cases, the reduc tion analogous to .

(3.15)-(3.16) can be used to clear up the question completely: If'

W

contains all the eigenvalues of'

1

RP- Q - i s non-elliptic at each x the

A¢

W

M(x) (this holds e.g. if'

E.n), the investigation of'

can be carried out f'or the Dirichlet problem f'or

TA = P - Q(M-A) -1 R in stead of' f'or

Jt. -

A.

The Dirichlet problem f'or

well posed when it satisf'ies in Agranovi~ ['

2], in

i.e.

Then

r 1= 1.

t~e

T

A

is

condition described

p~rticular if'

T'A

(and

p)

is scalar,

- 168 -

G. Grubb

ess sp

A..;

= w.

This holds for Example 1 and for

[-~ -d/dX

A=

Example 3:

OX1] O/a

ilclR

n

, n > .1.

Here

1

w = [a-1,a] = eSB sp Ay We can also use

in

•

TA,

just as in Section 3, to investi-

gate the asymptotic behavior of the point spectrum at infini-

A

ty, when

Ay.

is formally selfadjoint, so that

is self-

adjoint. This gives Theorem

5.

Assumption of Proposition 4, with

selfadjoint. The to

+

se~uence

A;(Ay)

formally

of ei g envalues go i ng

sat isfies

~

The associated eigenfunctions . belong to 4.3

J{

C~(rr)~.

Let us finallw mention a few results concerning

A • v

One difficulty here is that the boundary condition links v

and

w

together, so that we cannot get as nice a descrip-

tion of

D(A) as in Proposition 4 (iii). But the ses~uiv linear formulation c an be us ed to show: When a(u,u') r r 1(n) (cf. (4.4 )) is coerciv e on V = H 1 x HO(n) 2, then v is bounded. Furthermore, in case a is symmeess sp A

v

tric, so that v alues g o i ng t o

A

v

is selfadjoint, the + ~

s e~uence

b e h aves like (4.1.4).

of ei gen-

- 169 -

G.

Grubb

REFERENCES [1]

S. Agmon, A. Douglis, L. Nirenberg: Estimates near the boundary ••. , II, Comm. Pure Appl . Math. 1l(1964),35-92

[2]

M.S. Agranovi~: Elliptic singular i n t eg r o- di r re ren t ial operators, Uspehi Mat. Nauk 20(1 965), 3-1 20.

[3]

A.V. Bitsadze: Uniqueness or solutions or the Dirichl et problem ror elliptic partial ,dirrerenti al e qu a t i ons , Uspehi Mat. Nauk l(1948) , 211-212.

[4]

L. Boutet de Monvel: Boundary problems ror pseudodirrerential operators, Acta Math. 126(~971), 11-51.

[5]

Ju. V. Egorov. V.A. Kondrat'ev: The oblique derivative problem, Mat. as• 78(1 20) (1969), 148-176.

[6]

G.I. Eskin: Degenerate elliptic equat ions or principal type, Mat. Sb. 82(124) (1970), 585-628.

[7]

G. Geymonat: Sui ·problemi ai limiti per i sistemi lineari ellittic i, Ann. Mat. pura ed appl. 69(1965), 207-284.

[8]

C. Goulaouic: Lectures at this CIME conrerence.

[9]

G. Grubb: Weakly semibounded boundary problems and sesquilinear rorms, Ann. Inst. Four. £2(1973), No.4.

[10]

L. Garding : Lectures at this CIME conrerence.

[11]

L. ·Hor ma nde r : Pseudo-dirrerential operators and nonelliptic boundary problems, Ann. or Math. 83(1966), 129-209.

[12]

L. Hormander: Pseudo-dirrerential operators and hypoelliptic equations, Proc. Symp . Pure Math. 10(1968), 138-183.

[13]

T. Kato: Perturbation Theory ror Linear Operators, Springer Verlag, Berlin 1966.

- 170 -

G.

Grubb

[14]

J .L. Lions, E. Magenes: Pr-ob Leme s aux limi tes non homogenes et applications, I, Ed. Dunod, Paris 1968.

[15]

A. Melin: Lower bounds for pseudo-differential Ark. f. Mat. 2(1971), 117-140.

[16]

E.T. Poulsen: On the local origin of the essential spectra of elliptic differential operators, J. Math. Mech. 11(1962), 728-748.

[17]

M. Schechter: Lectures at this CIME conference.

[18]

R. Seeley: Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10(1968), 288-307.

[19]

N.E. Tovmasyan: The Dirichlet problem for an elliptic system of two second-order elliptic equations, Dokl. Akad. Nauk SSSR 122(1963), 53-56.

[20]

B.R. Vainberg, V.V.Grusin: Uniformly nonelliptic problems, II, Mat. Sb. 73(115) (1967), 126~154.

[21]

F. Wolf: On the essential spectrum of partial differential boundary problems, Comma Pure Appl. Math. ~ ( 1959), 211-228.

operato~s~

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E .)

QUELQUES RESULTATS RECENTS EN SCATTERING

JEAN CLAUDE GUILLOT

C orso

t enut a

a

Va r enna

dal

24

agosto

al

2

settembre

1973

QuelqUes resultats recents en scattering par

J. C. GUILIDI' oeparterrent de Mathematiques Pures et Appliquees Universite de Dijon 2l<XXl .DIJCN

FRANCE

Introduction Dans ce senu.naire, j' ai l'intention de donner quelques resultats reeents concernant; la theorie spectrale de l' operareur de SCHRODlllGER -b.

+ q(x) dans ~ ou dans un ouvert borne

(00.

q(x) est une fonction mesu-

rable sur ~n verifiant certaines ronditions) dans le cadre du scattering. Quoique ce soft;

dej~

un sujet particuliererrent vaste, ce choix ex-

clut des questions tres inportantes cornre celles se rapportant au problem:

a

N corps en Mecanique Quantique et

~

la theorie quantique des chanps ,

Ne pouvant rralheureuserrent pas etre exhaustif, je signale neanrrodns :;ru'~

la suite de la COnference qui s'est tenue en Juin 73

~

Denver, un

livre sera publie en Janvier 74 qui rontiendra les derni.eres contrdbutdons en scattering et qui offrira un panorama assez c:x:>nplet sur la question. Je tiens

~

rerrercier particuliererrent les ProfICECCCNI et L. Gl\RDlllG qui

m'ont pennis de donner ce seminaire ainsi que le Prof. M. SCHECHI'ER pour m'avoir donne une ropie de ses derniers resultats e t le Prof. C. WILCOX pour de multiples et fructueuses discussions .

- 174 -

J.e.

Guillot

Comrencons par rappeler "c ertaines notations Notations . Si H ~st un operateur autoadjoint defini dans un espace de Hilbert, on notera (1 (H) Ie spectre de H, p (H) 1 'enserrble resolvant,(1ess (H) Le spec-

tre essentiel, (1c(H) Le spectre oontinu, (1ac(H) Ie spectre absolurrent continu, (1s (H) Ie spectre singulier et (1cs (H) Le spectre oontinu singulier.

en

notera enfin Hac la partie absolurrent oontinue de H. Pour toutes ces

definitions voir RATa

(1) .

C'est un fait quasinent experinental que les spectres essentiel et absolurrent oontinu d'un operareur autoadjoint sont stables lorsqu'on Le perturbe par une grande classe de perturbations alors que Le spectre continu ne 1 'est pas. La theorie du Scattering est concernee en partie par la recherche des conditions de stabilite du spectre absolurrent continu d 'un opezateur autoadjoint lorsqu 'on Ie perturbe. C 'est une propriete beaucoup plus forte que oel.Le du spectre essentiel et par suite plus diffieile

a derrontrer.

En fait, durant ces derni.eres annees les travaux se

sont ooncentires sur L'operat.eur' de SChrOdinger -6 + q traite oorme perturbation du Laplaeien -6 Plus preciserrent Scit q une fonction definie sur IRn,

1

a valeurs

reelles et rresurables

verifiant la oondition suivante, dite de STtJMMEL : sup

xe.1If

Ix-yl s

H

Iq(y)12 dy

1

<

ex>

pour

\i

>

n - 4

- 175 -

J. C . Guillot et.

lim

dy = 0

\xl~oo

~

1

Alors on peut rrontrer que (i) L'operateur H = -IJ. + q est un operateur autoadjoint dans

(ii) a (H) est borne inferieurerrent ; de plus aess(-IJ.)

= [0, + oo[ = aess(H) .

En particulier si q est une fonction suffisamrent reguliere qui verifie

en outre q(x) = O(....L)

S >O

Ixl S

et

Ixl ;:JR

Alors q verifie Lacondi.taon de Sturnrel. Mais quelle information precise sur le spectre de H peut-on deduire de l ' assertion : ae~ (H) = [0, +

00 [

? Essentiellerrent une information sur la

.

partie du spectre qui n' est pas . contenu dans [0, + a (H) f\ (-

00,0 rest

00] ;

il. savoir :

oonstitue uniquerrent de valeurs propres isolees (dans

a (H)) de multiplicite finie et dent le seul point d' accumulation possible

est {O}. Par cantre on ne peut rien dire de precis sur la structure du spectre cantenu dans [0, +

00[.

La theorie du SCattering perrret dans cer-

tains cas de precf.ser la nature du spectre en rrontrant que le spectre absolurrent cantinu est aussi stable . Plus precisenent les problerres que 1 'on cherche il. resoudre dans ce cas se resurrent dans le programre suivant: Programre de T. lKEBE :

- 176 -

J . C . Guillot

Trouver des condf,tions sur q telles que

- Que peut-on dire sur l'ensenble des valeurs propres plonqees

dans le spectre oontinu ? - Que

peut-on dire sur

0 cs

(H) ?

b) Montrer l'existenre at la oorrplHude des operateurs d 'onde W = S +

-

-

lim eitHeiti.

t ±'"

(II) Construire deux developpenents en fonctions propres generalises pour

Hac' enqendres par deux systEmeS de fonctions propres de L'operaeeur de SCHRODINGER

(Cf ± (x;k)) k

Rn tels que

(W+f) = L . i. m _

(IO+(X;k) ~nT _

f a (k)

aU fest la transformee de Fourier usuelle de f

dk

fE-L

2(Rn)

Eo L2 (Rn) •

En ce qui ooncerne la partie (I) des resultats corrplets ont ete obtenus par

S .AGMA~,

BIRMAN, T.IKEBE, T. KATe, S.T . KURODA, P.A. REJTO et M.

SCHECHI'ER lorsque q(x) =

°(Ixl -1-£ )

e : 0, et [x] ~ Ret par T.IKEBE,

P. ALSLOM et G. SCHMIDI', T. KATO et S. T. KUR)[)A en ce qui ooncerne la partie (II). Mais les resultats les plus rerents ant ete obtenus par

S.

A
[2.1 ,

S.T. KIRODA [3J et M.SCHECHI'ER [4]'et concernent les ope-

rateurs elliptiques. Plus preciserrent, consfderons 1 'operateur differential fonnel suivant .:

Hu =

L a. (i) p . a 1a.1,lsl ~m D (aa.S + aa.s(x))D u (Dj = -1 ;x)

au a. et S sont deux multi ·indices et on les oonstantes a (1) et les fanea. S tions aa.S(x) verifient les oonditions suivantes

(1)

-::uf a Go< et

i l existe une oonstante c > 0 telle 1


- 177 -

J. C. Guillot

a (1)

~

lal=181= m as ~

a+S

~ cl I~I

zm

(2) I.es fonctions aaS (.) cIefinies sur

sont bornees et rresurab.les

Ia I=18 I= In sont

lI'outes les fonetions aaS ( .) avec sur

IIf

unifomement continues

If. (3) aaS (x) = a

Sa

(x)

(4) Il existe une constante (:2 > 0 telle que

3s

(5)

c >

1 et c 3

>

3

avec

1aaS (x) I < : - - -

0

~ (l+lxJl o

n lal'ISllf In et 'VXER Soit PI (~) Le polynorre suivant

PI (~) =

~

~€ Rn

a~~) ~a+S

OJ;l

lal,ISllfIn n associe ii PI (.) l'operateur autoadjoint, rote HI' dans L2(lR ) de dcrnai-

ne

;rn(If)

et defini par Hlu = PI (D)u

De plus A E:

tel que PI (~)

,

u

E

n) ifrnOR

Rest une valeur critique du polynOrre PI s' il .existe ~

= A et

grad PI ( ~)

= O.

S. AGDN

E:

n IR

a IlOntre que 1 'enserrble e l

des valeurs critiques est un enserrble fini. Par suite, si on pose ~ inf

A

min .

~E

rrP

Pl(~) on a a (Hl )

= [ AminI

co) et Ie spectre de HI est absolu-

rnent continuo Maintenant ii 1 'oper'ateur fornel H on associe un operareur autoadjoint, note

~,

par la methode de Friedrichs. Plus precisement,

autoadjoint de dornaine D(H2) tel que

~

est l' operareur

- 178 -

J. C. Guillot

(~u,v)

D

=

u € D(H

.((a~~)

lal,lsJl;rn

+ aaB (x) )D(ju,Dav)

2)

On note E ( .) ).a famille spectra1e asaoci.ee 2

a

H 2•

a alors dem:>ntre 1e thOOrare suivant

S.T. KURCOA

THEOREME :

(ii) L'enserrb1e {An} de toutes 1es va1eurs propres de

~

dans

valeur propre An est de rnultiplicite finie. (iii) La restriction de H ausous-espace E ( (A 2

est unitairerrent equiva1ente

a

H

1

2

rnin

, oo)- (e {An }» L2(RP) 1

• En particu1ier

crac(~) - [Aminloo) De plus les operateurs d 'onde

) = s . W+ (H 2,H1 -

¥ID

t~ '

±oo

e i tH 2 e~itH1

existent et sent ccnp1ets. Recerment, M. SCHEX::HI'ER

[4]

S.T. KUroDA et S. ~

[2]

a amHiore certains des resultats oocenus par a obtenu des resultats analogues par une appro-

che differente.

Par ailleurs on sait rnaintenant que ce sont 1es neilleurs resultats que l'on peut obtenir dans une certaine direction. I1 n 'est pas possible en effet d'ameliorer 1a condition 5). Plus precisenent, consaderons l 'ope2 rateur autoadjoint H dans L (IR3) suivant o H o

=-/:,-

~

[x ]

x E 1R3

- 179

J. C. Guillot

cet exenple se distingue du precedent par le fait que le potentiel

~ ne verifie plus la oondition Iq(x) I~ ~+ e [x] [x] e:

q(x) = -

> 0

pour [x]

suffisarment grand. Il a ete etudie du point de vue du SCattering par J. DOLIARD [5] • Ce.lui,-ci a en effet rrontre que les operateurs d' onde ordi

naires it i t ll s - lim e Ho e t ±eo n 'existent pas. Par oontre i l a rrontre que les operateurs d' onde generali-

W±(HO,-A) =

ses suivants :

e

it H

0

E

ou

(t)

e

itll

i E(t)a

e

= {+l -1

2 (-lI) 1/2

Log '(- 4 It lll) .

0

si t > si t < 0

existent et sont a:nplets. J. DOLIARD a rrontre de plus dans sa these qu'on pouvait associer il. H

oac

deux developperrents en fonctions propres, enqendres par deux systen-es

('f~ (x;k»k(,- R3

(rJ? -

:>u

(H ,-lI) f) 0

de telle serte que l'on ait (x)

=

L. Lm,

j' (n:

3T-

2 3) f (; L (R

(x; k) f (k) dk o

fest la transfonree de Fouri:r ordinaire de f f L o

Aussi definit-on les potentiels Iq(x)

et les potentiels

a

I

~

a

2

3). (R

oourte poruee comre ceux verifiant

c

Ixll+€'

e: > 0 pour

x suffisarnment grand

longue portee, c:cmre ceux verifiant Iq(x) I ~

c [x] S

s

>

0 pour

x

suffisarnment grand.

APres le travail de J. DOLIARD, les recherches se sont orientees dans trois pj.rections principales.

- 180 -

J. C. 1. Obtenir les

tiels

a

~rateurs

Guillot

d 'onde generalises FOur taus les poten-

longue FOrtee et en derrontrer l'existence.

Pour l'operateur de SCHOODINGER, c'est un prablerre qui a ete resolu par

v.s.

W. AMRETh1, P. MMl'IN et B. MISRA [6J ,

P. AISHCM et T.

KA'ro

BUSIAE.V et V.B. MATVEEV [7

J,

[8] .

2. Obtenir directerrent des resultats concernant Le spectre de I' oparateur de SCHRDINGER directerrent c' esti-a-dire sans passer par I' exis

cenoe et la cx:rrpletude des operateurs d 'onde. Dans cette direction des resultats i.I\p)rtants ont ete cbtenus par J. AGUIIAR et J .M. CCMBES [9] , R. IAVINE [10] , T. IKEBE et Y. SArro

[ul.

J. AGUIIAR et J .M. CXMBES ont introduit des methodes analytiques qui

sont

r~elees tr~s

se

utiles FOur Le prcblare 11 N corps. Mais peut-etre les

resultats les plus a:xrplets et les plus generaux FOur L'operateur de SCHRODINGER ont ete obtenus par R. IAVINE :

n Soit q (x) une fonction definie sur R 11 valeurs reeUes teUEj que

avec l:iln ql(x)

, Ixl-. eo

~

-

C

(l+r)Y

=0

avec y

>

et 1

r=

[x]

et aU q2 (x) = _1_ (q2 (x) + q2 (x) ) (l+r) y , p , eo n avec q2,p(x) E rl(R ) FOur P. > max( ¥il) q2 ,eo (x)

c

L" (Rn)

On peut associer alors 11 l'operateur

-f).

+ q(x) un operateur autoadjoint H

- 181 -

J. C.

L2 (Rn) par la m!ithode de FRIEDRICHS et

1{ =

dans I' espace de HILBERI'

Guillot

R. IAVINE a nontre que les valeurs propres positives de H de rnultipl1cite finie et ne peuvent s'aCClmUller qu'il l'origine. De plus 'itcs(H) =iO}c'est

Ii dire que rous avons la

1t. = JtP (H)

d~sition

+

en satme directe suivante

*

'Itac (H)

T. lKEBE et Y. SAITO ont generalise certi1ns des resultats de R.IAVINE a

des

~rateurs

du type .

(f

n

E

aU b (x) j

e e'

aXaj

+

b),(X»2

+

q(x)

j=l n ) et q(x) sont des potentiels il longue portee ver1fiant

(R

certaines conditions. 3. Gereral1ser Le progranme conplet d' IKEBE.

Des resultats dans cette direction ont ete obtenus par V. GEORGESCU

et par J.e. GUILIm et K.

zrar

[1.3.1

[12] • V. GEORGESCU a considere Ie cas oil

Ie potentiel q(x) est il synetrie spMrique. En ce qui

nous concerne, rous avons considere des perturbations generales

du potentiel coulanbien. Plus preciserrent, nous avons consfdere principaLement; les deux cas suivants

n

a) Le problerre dans R

Cbnsiderons I' operateur H=

nI:

j=l

(I1 -a-a

+ b . (x»2 )

Xj

-

dans L 2(Rn)

a 1J(T + q () x IAI

n oil b (x) et q(x) sont deux fonctions definies sur R j

a

(n ~.. 3)

valeurs reelles ve-

dfiant cert.tins conditions. OOfinissons Q(x) =

~ (~~(X)+b.2(X))+q(X) ax ]

;=1

1

j

"Pour les definitions des sous-espaces

It p'

avec b],( .) Eel(Rn)

1t. ac

et

Itcs

voir T. Kl>oTO [1] "

- 182 -

J. C . Guillot

On suppose alor s que

(l + x

n

>2 ;

)a b .(x) J

L

P2 . n) ' J(R

1 ~ j < n

n

Max (2 , 2) < PI < 2n;

n < P2 . < 2n ,J 2 n) 2 alor-s H est un ope rateur autoadjoint L (R de domaine H (R pour

avec a

n)

1eque1 on peut deve loppe r une thor-ie complete en utilisant 1a theor-i e de factorisation de T KATO et S. T . KURODA

14 .

b) Le pr-obleme exter-Ieu r Soit

n

un domaine non bor-ne de R

deux var-iates

~

et

dont 1a f'ronti er-e est for-mae de 2, 2 de c1asse C disjointes et compactes . Supposons

que I' origine 0 appartient

l'interieur du compact determine Pitt' r 2.

~

$oit v 1a norma1e exter-i eur-e a1'exposant s (0 <

>

S

n

(x) une fonction ho lde r'Ierme n+l+ 2 (1 + x avec

et soit

~

< 1). Soit

=

(x)

0 fixe

$qpposons que Jr~elles

(x) q(x) soit une fonction dMinie sur

unifor-mernent holder-ienne d'exposant

(0 <

nu

~

valaur-s

< s) . Supposons

que

-

(x) q(x)

-

x

>0 >

00

Soit H l'operateur autoadjoint dans L

=-

2

(n) dMini par

g + qg et dont Ie do maine de dMinition est for-me de 2 tbute s 1es fonctions g H (n) telles que

Illg

g - - x

(x) g(x)

=0

=0

sur

sur

1 2

- 183 -

J . C.

Guillot

On peut alors en suivant la technique de N. SHENK et D THOE

15

construire des developpements en fonctions propres pour H et etudi er les ope rateur-s d'onde as socies

- 184 -

J. C.

Guillot

BIBLIOGRA PHIE 1

T KATO : Perturbation theory for linear operators, Springer (66)

2

S. AGMON

Spectral properties of Schrodinger operators , Actes du congr-es Intern Math . 1970 Tome 2 p 679-683 . Confer-ence donnee a Oberswolfach (Juin 1971)

3

S. T . KURODA : Scattering Theory for differential operators 1;11 Journal Math Soc Japan 25 . (1973)p 75-104;p . 222-,234

4

M . SCHECHTER : Scattering theory for elliptic operators of arbitrary order (Preprint)

5

J . DOLLARO : Asymptotic convergence and the Coulomb Inter-action J . Math. Phys

6

W. AMREIM . MARTIN

~

(1964) P 729 -738

el B. MISRA : On the asymptotic condition of

Scattering theory. Helv . Phys Acta 43 (1970) p . 7

313~n4

V S . BUSLAEV et V B MATVEEV : Wase operators for the Scattering Equation with a slowly decreasing potential Teoreticheskaya

Matematicheskaya 2 (1970) -

pag. 367-376 Traduction anglaise

Theo Math . Phys . ~J1970)

p . 266-274 8

P . ALSHOM et T . KATO : Scatter-ing with long range potentials Preprint , 1971.

9

J. AGUILAR et J . M. COMBES : A classe of a na l yt i c perturbations for Schrodinger- Hamiltonians I. The one body problem Comm . Math Phys. (1971) p. 269 -272.

- 185 -

J. C.

Guillot

BIBLIOGRAPHIE (suite) 10

R. LAVINE

; Absolute continuity of positive

spectrum for Schr0-

dinger operators with long range potentials. J. of Funct , Anal. 12 12

J. G. GUILLOT et K. ZIZI: Expos e (~uin 7~)

13

1973 p. 30.

sera

publi~

a

la confer-ence de Denver

en Janvier 74.

V. GEORGESCU : Expos e de W. AMREIM

a

1a Confer-ence de

Denver (Juin 73) . 14

T . KATO et S. T . KURODA : Theory of simple Scattering and eigenfunctions expansiOns. Functional Analysis and related topics. Springer (1970) .

15

N. SHENK et D. THOE : a} Outgoing solutions of ( - 6

+ q - k 2) u = f

in an exterior domain J.

Math . Anal. and App . 31 (1970) p.

81

b} Eigenfunction expansions and Scattering theory for perturbations of pl. 36 (1971) p.

313 .

- C1. J. Math Anal. and Ap-

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

THEORY OF PERTURBATIONS OF PARTIAL DIFFERENTIAL OPERATORS

MARTIN SCHECHTER

Corso

te n u t o

a

Vare n n a

dal

24

agos to

al

2

se ttembre

19 7 3

- 189 -

M Schechter

I.

Constant Coefficient Operators Let

(l)

1J.1 IJ.n P(Sl, ••• ,Sn} = L: a Sl ••• E: be a polynomial in n 1J. +••• +IJ. < m 1J.1,···,lJ.n tr 1

n variables with complex coefficients. troduce the following compact notation.

In the interest of ecology we inWrite

S=

(E:l, ••• ,E: n),

IIJ. I = 1J.1 +••• + IJ.n and IJ. 1J.1 .. IJ.n

~

= Sl

••• Sn

Then (l) becomes (2)

Corresponding to this polynomial we can form a constant coefficient partial differential operator in En given by (3)

P(D}

= L:

a

IlJ.bm

IJ.

where D .. (Dl •••••Dn) and D = -io!ox • 1 j

-i will be given later).

j

:s j :s n

(the reason for the

Co,versely. every constant coefficient partial

differential operator can be written in the form (3) and corresponds to a polynomial of the form (l). Once we have our partial differential operator. we must decide where to apply it.

Fortunately for this purpose mathematians have invented a

very convenient class of functions - the infinitely

differentiab~e

functions with compact supports (usually taken as complex valued). We Q) Q) n denote this class by C = C (E). Every partial differential operator o

0

with constant coefficients can safely be applied to this class. However, when one desires to talk about spectral theory (which I am obliged to do at the present moment). it is customary to deal with operators in either Hilbert or Banach spaces.

And since we do not wish to

- 190 -

M Schechter break with tradition, we shall have to find such a space for our operato~. For this purpose we have chosen the LP spaces because of their popularity. versatility and convenience.

In general we shall allow 1

~

p

but

~ ~,

in many cases we shall have to let go of the end points. Since C~C LP o

= LP(E n)

tor in LP'with domain C~. o

for any p, we can consider P(D) as an operaNow we can get down to the business of descri~·

ing the operator and its spectral properties. might ask is whether our operator is closed. 'vi ous l y negative, but don't go away.

(P(D)~,t)

All is not lost, for we have

ee

Let [~kJ be a sequence of functions in C

Proof. 0 and

The answer to this is ob-

P• The Operator .P(P) . ' on"C~ 0 is closable in L

Lemma 1.•

~ ~

The first question one

o

P(D)~ ~

= (~,

If

f

such that cio

V is any function in Co' then

P(D)V) by integration by parts (here P(D) is the con-

stant coefficient operator whose coefficients are the complex conjungates of those of P(D». that f

=0

Thus (f, t) = 0 for all V E C~. o

This implies

a.e. 0

= Pop

We let P

o

by the closure of our operator in LP•

This operato~

is called the minimal or strong extension depending on ones political convictions.

If you have guessed that there is a maximal or

~

exten-

sion, you are right. Now that we have our operator snugly lodged in a Banach space, we can now take the time tp review some concepts from spectral theory. A be a closed operator on a complex Banach space X.

Let

A complex number

A is said to be in the resolvent set p(A) of A if the operatoc.A- A has a bounded inverse defined on the whole of X. trum cr(A) of A. c Ics ed ,

Otherwise i t is in the spec-

I t i s a fact of life that p(A) i s open and cr(A) i s

- 191 -

m Schechter Another fact, which may not be so widely known, is that there are basically two types of points in the spectrum -- hard core and soft core , Some points of the spectrum are so completely ingrained that nothing short of a hydrogen bomb will dislodge them, while others move or disappear at the slightest whisper.

There are several definitions of hard

core spectrum, and most of them coincide for self-adjoint operators on Hilbert space (c f , r3,p.241]).

The definition I like the best (primarily

because I invented it) is

a (A) =

(4)

e

n

K compact

a (AofoK) •

We shall call this set the essential spectrum of A.

Clearly it is the

largest closed subset of a(A) which remains invariant under compact perturbations.

We shall see that it remains invariant under perturbations

which are even worse. [3,p.15].

If there is a seguence~ of elements in D(A) such that

lIukll = 1, (A-A)u

k

... 0 and {uk}

has no convergent subsequence, then

ae (A). Proof.

Suppose A ~ a (A). e

such that A E p(AofoK). lIull~c

(5)

e

For the moment we need only the following:

Lemma 2.

A E .

A complete characterization of a (A) is given in

II

Then there is a compact operator K

This implies

(AofoK-),,)ull, u ED(A).

Since K is compact, there is a subsequence {v of {~} such that j} {KV } converges.

j

verges itself.

A simple application of (5) then shows that {v conj} This contradicts the hypothesis that {uk} has no conver-

gent subsequence. 0 We can now return to our partial differential operator. s ta tement is

Our first

- 192 -

Lemma 3.

M Schechter If there is a ~ e En such that A = P(s), then A e a (P ). e

0

In proving the lemma we shall make use of a well known generalization of Leibnitz's formula: (6)

P(D) [uv] '" E p(lJo) (D)u DlJov/lJo!,

where

and IJoI

= ~ I ••• lJon'

(simple proofs may be found in [2,p.264] and [3, p.52]) Proof of Lemma 3.

We may take A

:0

O.

Let t(x) be a function in

which vanishes in a neighborhood of the origin and has norm 1 (all

ceo

o

~orms

will be of LP unless otherwise specified). Set (7)

~(x):o

t(x I k) e iE:x - Ik nIp

,

Then C/\ (8)

IIC/\II

e C0eo

for each k and

= 1.

But by (6) P(D)C/\(X) = EP(IJo) (sHIJo(x/R)eiSX/lJolkllJol where , (x) IJo

lit

IJo

= DIJo

v(x).

(x/k)/kn/Pil =

+~

Since

lit

IJo

II,

we have (9) " P(D)Cfl .... 0 in L P as k .... eo. k

Finally note that C/\(x) .... 0 pointwise as k .... '" (for P = "', we make use of the fact that

Vvanishes

in a neighborhood of the origin).

Now if [~}

had a convergent subsequence, the limit would have norm 1 by other hand

~hpre

(~).

would bp a s ubseQuence of this su bsequence wh ich

On the

-193-

1\1 Schechter

converges to the limit a ,e ,

Thus the limit vanishes a se , and cannot

have norm 1.

This contradiction shows that {~} cannot have a convergent

subsequence.

We now apply Lemma 2. 0

Since the essential spectrum is closed, we have Corollary 4.

The closure of the set

is contained in a

e

(P). 0

- 1 94 -

II.

L P Multipliers

We now search for other points of the spectrum. slight surprise. Lemma 5.

Here we are in for a

We have

If 1

s

p < "', then A E p (Po) iff l/[P(S)->.J is an LP

multiplier. In order to prove this lemma, it will be convenient to know what an LP

50 let me describe a few

multiplier is.

conc~pts

denote the set of functions v(x) E C'" such that Ix on Enfur each'k

2: 0 and~.

from analysis.

Ik ID~v(x) I

Let 5

is bounded

The Fourier transform of such a function is

defined as Fv(e:) .. (2n) -%0

J e -isx v(x)dx.

We recall that F maps 5 into itself and that F[D~vJ .. S~v (cf.,e.g., [4J). First we note 5 CD (P ) and P v .. P(D)v for v E 5. o 0

Lenma 6.

Proof.

Let V be a function in

V(x) .. 1

fo~

Ixl

c:

such that

5: 1. For ~/~ 5 set vk(x) .. Hx/k)v(x). Then vkE

and it is easily checked that v

k

~ v in LP•

formula (6) P(D)v .. E k .. E

p(~) (D)v D~v(x/k)/~!

P(~)(D)V t (x/k)/~!kl~l. ~

Thus lip(D) (vk-v)

+ ~

E

II 5: II [V(x/k)

~\O

05: V(x) 5: 1 in En and

- lJ p(D)vll

IIp(~) (mv] lit ""'/~! kl~1

Oask~"'.

~

c:'

Moreover, by Leibnitz's

- 195 -

M. Schechter Note that the proof of Lemma 6 requires only that p(~)(D)v be in LP for

Next we note

each~.

Lemma 7.

If

(12) n

Let S be any vector in E and let V be any function in Co

Proof.

Define ~ by (7).

such that IIvil .. 1.

IIp(D)~II ....

Ip(s) I

as k .... ~.

~

Then one checks easily that

Since CPk E

1 = !I~II ~ Co I Ip (D) ~ 1 1 .... Co Ip(E:)

c;. we have

I.

This gives (12). 0 A bounded measurable function m(S) on En is called an LP multiplier if

IIF [m( E:) Fv ] 11 ~ C IIvll , v E

s,

where F is the inverse Fourier transform.

We may assume A = O.

Proof of Lemma 5. holds.

Thus I/lp(E:) -

I

We can now give the

is hounded (Lemma 7).

-1

Then w = F [P Ff] is also in S.

If 0 E p(P ), then (11) o

Let f be any function in S.

Moreover P(D)w = f.

Since 0 E p(p), we o

have Ilwil ~ C I/f !/.

(l3) Thus pin LP•

l

is an LP multiplier.

Conversely, assume that IIp is a multiplier

Thus

Let f be any function in S and set w is in Sand P(D)w

= f.

Thus R(P ) o

~

= -F[P-1Ff]. S.

H.en.c.e 0

E

is bounded, w

Moreover, (14) implies (13).

Since S is dense in LP• this shows that for each f w E D(~ such that ~ = i ,

-1

Since P

~Po).

bI

E LP there is a unique

-196-

M . Schechter' Once we have Lemma 5, the probtem of determining the spectrum

reduces to one of determining LP multipliers.

Moreover, when p

= 2,

Parseval's theorem shows that the multipliers are precisely the bounded functions.

Thus ,XEp(Po) iff

l/[p(~)-X]

tained in the closure of the set (10). Theorem 8.

If p

= 2,

is bounded, i.e., X is not conThus we have

= ae (P0 )

a(p ) '0

consists of the closure of the

set (10). The situation is more complicated when p ,2.' The reason is that there is no simple nece.sary and sufficient condition for a function to be an LP multiplier. The criterian we found the most useful is Theorem 9.

Suppose 1 < P <

greater than n Il/p-1/21. (15)

IDlJ.m(s)I~C

and let t be the smallest integer

<Xl.

Let meg) be a function such that

le:rallJol-b,

11J.1~t, Then m is an LP mul tip lier.

for some real a < 1 and b > (1-a) n Il!P-1/21.

We'shall not. prove Theorem 9 here because it would take us to far afield.

For a proof we refer to [3,p.46J.

result of Littman [5J. The.orem 10. (16)

(17)

We apply Theorem 9 to obtain

Assume 1 < P <

<Xl

and that

p(lJ.) (P/p(s) - O(!srallJ.l), as lsi

IIp(s) • o(lsr

It is a consequence of a

b

) , as lsi ....

....

<Xl,

<Xl,

where t is the smallest integer> n Il/p -1/2 1, /.!! ~ 1 and b > (l-a)n

11/p-1/21. Then a(Po ) cons is ts of the set (10). . Proof.

It suffices to show that ll[p(s)-X is an LP multiplier i f X

is not in the set (10).

We may take X - O.

p(~)-l is bounded. Now for each .o f the form

Since a

~

1, b > O.

Thus

IJ., nlJ.(l/p) consists of a sum of terms

- 197 -

M . Schechter .

Constant p(~

(18) where

~

(1)

+•••+

~

(1)

(k)

) (E:) ...p(~

=~

(k)

) (E:)!p(s)k+l

(this can be verified by a simple

induction)~

Thus

D~(l!P) .$cle:ral~I.$.(..

(19)

An application of Theorem 9 shows that p-l is an LP multiplier.

Thus

AEp(P ) by Lemma S. 0 o

Assume that 1 < P <

Corollary 11. b > O.

co

and that (17) holds for some

If

Il!p-l!2! < "!n(m-h). then a(P ) consists of the set (10). o

Proof.

p(~)(s) is of degree at most m-I~I. Hence

p(~)(s)!P(s)

=

o(lslm-I~I-b).

For I~I ~ 1. we have m-I~I-b ~ (m-b-l) I~I.

Thus we may take a

= b+l-m

in Theorem 10. 0 Corollary 12. that Il!p-l!21 < Proof. holds.

If Ip(s>

n implies

I

~

co

as lsi ~

co,

then there ,i s an

n>

0 sucq

that a(P ) consists of the set (10). o

The hypothesis implies that there is a b > 0 such that (17)

Apply Corollary 11. 0

Coro llary 13.

If Ip(E;) I ~ co as

lsi ~

co

and pep ) is not empty, thert o

a(P ) consists of the set (10). o

Proof. prove.

If the set (10) is the whole plane. there is nothing to

Otherwise. there is a A not in this set.

There is a number b > 0 such that (17) holds. By Corollary 11. OEp(Q ). o

We may take A = O.

Take k > mn!b(n+2). and Since pep ) is not empty. o

P~ is a closed operator. Since it agrees with Qo on Sand Qo is its

- 198 -

M Schechter

k

see that Po is surjective.

Thus Po is surjective as well.

that it is injective also.

For suppose vEN(p).

sequence (v

j

Jc

Then there is a

o

S such that v - v and P(D)v - O. j

there are functions wjES such that P(D)

k-l

j

Now we show

-1

Since P

w =v j• j

is bounded,

Thus

Q~D)Wj" P(D)Vj- O. Since OEp(Qo)' we have wj- O. But P(D)k-l is closable. bijective.

Hence v

j

converges to 0 as well.

Hence v

= O.

Thus P

o

is

This means that OEp(P0). 0

Coro llary 13 is due to Iha-Schubert [6 J. To summaDize, we know that a(P ) is the closure of the set (10) when .

p = 2.

If Ip(s)

I-

for p close to 2. complex plane.

I~I

-

0

=,

then a(p

consists of the set (10)

For any p, a(p ) is either the set (10) on the whole o

o)

=

Iha-Schubert [6J

is the whole complex plane when

,.222 222 (sl-Sz-s3-S4-i) (Sl+S2+s3+~4+i), n

li/p - 1121 > 3/8. Theorem 10 applied a(P ) is the set (10) when

o

o)

The latter poSSibility does ·occ~r.

showed that a(p P(s )

=as

= 4,

and

to this operator shows that

Il/p-1/21 < 1/5.

- 199 -

M.. Schechter III.

Perturbation by a Potential We now discuss expressions of the form

(20)

P.(D)

+

q (x) ,

where q(x) is a measurable function.

Our first task will' be to define a

precise operator corresponding to this expression. ways of doing this. as an operator Q. those u

e L2

There are several

One simple way ·is to consider multiplication by q The domain of Q is easy to describe: it is the set of

2• such that qu E L

We can now define the operator correspono

ding to (20) as R=Po+Q,

(21)

where D (R) = D (Po)

n D (Q).

when will R be closed. Leunna 14. (22)

[su]

The firs t ques tion one may wish to ask is

One simple answer is given by

I f A is closed. i f D(A) C D(B) and

~ c I/Au!!

+ dllull, u

for some c < 1. then A

+

e D(A),

B is closed.

The proof of Lemma 14

is simple.

We leave it as an exercise.

To

apply the lemma we must find conditions on q so that an inequality of the form (23)

IIqull ~ c IIPoull \

+ dllul/, u8>(P o)'

We shall give one set of conditions. measurable function, set (24)

M

Q',p

(V) = sup y

.r

IV(x-y)

For 1

Ik

~

p < CD, Q' > 0 and V(x) a

Ix I o-n dx ,

Ixl< 1

Let MOfP be the set of those functions V such that MOfP (V) < CD. is a polynomial, we shall say that P (25) (26)

E O(a,b) if

p(~)(E:)/P(E:) = o(lsral~l) as 11::1 .... b l/P(E;) = o(ld- ) as lsi .... CD,

CD,

I~I ~n+l

If P (E;)

- 200 -

M. Schechter for i; E En.

We have Assume that P E O(atb) with a < 1 and b > a-k!-an.

Theorem 15. k

denote the smallest nonnegative integer such that k

o

assume that q EM

o <

(27)

o

Let

a > n-b. and

for some ex satisfying

a.p

ex < p(n-k ). .

0

If P (Po) is not empty. then D(Po)

C

D(Q) and (23) holds.

Moreover. we

can take c as small as desired. In proving Theorem . 15 we shall make use of Let Q', I3t yand p satisfy 1 < P < CIO. 13 < n < y. and

Theorem 16.

o <

ex < p(n-I3).

Let G be a function satisfying

IG(x) I s Kl I xl - I3.• Ixl < 1 . Ix l -> 1. ~K2 Ixl-Y

(28)

(29)

Then there is a constant C depending only on Q', 13. y. nand p such that (30)

IIq[G*fJ II Lenma 17.

~C

(Kl-fi{2) Ma,p(q)l/p II fll f E LP•

Suppose w E CCIO and DIJo weLl for

lul=

k t with

Then there is a constant C such that

The proof of Theorem 16 will be given a bit 'l a t e r . is a simple exercise.

We now show how they can be used to give the

Proof of Theorem 15.

o

e p (Po)'

b+allJol> n by (19).

13

= n+l.

=.kQ

Without loss of generality. we may assume

Let v be any function in S. and set f = P(D)v.

Fv = wFf. where w = lip.

IIJoI

That of Lemma 17

Note that w e CCIO and that D\.Lw ELl when

In particular. this is true when I IJoI = k

Thus by Lemma 17. G(x)

and Y

= n+l.

Then

Since v

=G

= Fw

o

ahd

satisfies (2 8) and (29) with

* f. Theorem 16 gives

- 201 -

Ilqv.ll

(33)

M . Schechter

IIp(D)vll, v E s.

~C

This implies (23).

To show that we may take c as small as desired, let

t

ee

be a function in Co satisfying 0 V(x) .. 0, [x

I>

~

V(x)

~

1 and

1

.. I, lx' < \. For Ii

>

0 put

GIi(x) .. V(x/ <5)G (x)

G (x) Let

.. G(x) - G/i(x).

13 satisfy k o < 13 13 n - (alp). Then

IG/i(x) I ~<5

(34)

e-k

0

K l

[x ,-13,

Ix I ~<5 Ixl > 6

.. 0, Thus by Theorem 16

Note next that (16 E C"'.

Moreover, since GIi(x) .. G(x) for Ixl > 6, we

have (36)

P(D) G<5(x) = 0,

Ixl > 6.

Moreover

By (36), P(D) G<5 E C:.

(~8)

IIq[Go*

Thus by (37) and Theorem 16

(p(D)v)JII~c<5MO',p(q)l/P

Ilvli.

The assertion now follows from (35) and (38) by taking <5 sufficiently small. 0 In proving Theorem 16 we shall make use of three simple lemmas. first is obvious.

The

- 2 02 -

M. Sc hechte r

Leuma 18.

For a~ 13 and 1 ~p < "', MS,p(q) ~Ma,p(q).

Leuma 19.

For any g there is a constant C depending only on nand

a such that. Mn,p (q) -< C Mg.p (q).

;

For a $ n, this follows from Lemma 18.

Proof.

let y be any point of En.

J

S2a-n

If Iz- yl

= 3/4,

Assume a> n, and

t hen

Iq(x-z)I P Ixla-n ~2a-n Ma,p (q).

\
0

Lemma 20.

If G(x) satisfies (29) with y > n, then there is a

constant C depending only on nand y such that (39)

J

Iq(x-y)IP IG(x)

I

dx ~C K_ M (q). - "2 n,p

Ixl> 1 Proof.

By

'"

(29) the left hand side of (39) is less than

J k< Ixl< k+l

There is a constant C depending only on n such that for each k the shell k < Iz I < k+l can be covered by C k

J

k
Iq(x-y) I P dx

n-l

s C k n- l

spheres of radius 1.

M

n,p

Thus the left hand side of (39) is less than C~ M

n,p

(q)

'"

1::

k-l

kn- y-l

(q)

Thus

- 203 -

M . Schechter 0

The series converges because y > n , Proof of Theorem 16. functions in S.

Then by Holder's inequality

l(qrG*u],v)! u(y) vex)

First aSSlDDe 1 < p < 00, and let u,v , be any

s If

I dx

dy

Iq(x)G(x-y)

= If

If

+

Ix-yl< 1

Ix-yl > 1 t p IIp Iq(x) P IG(x-y) p lu(y) dx dy)

s (If

I

I

I

Ix-yl < 1

(s.r

I

IG(x-y) I (l-t)p' Iv(x) P

,

dx dy)

IIp' .

Ix-yl
+

If

(

I

Iq (x) IP IG(x -y)

Ix-yl> 1

If

(

IG(x-y)

I

Iv (x ) IP

lu ( y) IP dx dy)

,

IIp

IIp' dx dy)

Ix-yl> 1 where t is any number satisfying 0

~

t

~

1.

The trick is to f ind a t

in this interval such that

S

(40)

[q (x) IP IG(x-y) Itp dx ~ K~P

Ma,p (q)

Ix- yl < l and

S

(41)

Iz

IG(z) I (i-t)p' dz

~

1< 1

C K (l-t)p'

l

For b y Lemmas 19 and 20,

S

(42)

Iq(x) IP IG(x- y) I dx < C K_ M (q) -L a,p

Ix-yl > 1 and by (29) (43)

S

IG(z ) I dz

s C K2

•

Izl > 1 Inequality (30) follows from (40) - (43).

To find a t such that (40)

- 2 04 -

M . S che chter and (41) hold, note that we may assume (44)

n • Bp ::: 01::: n

as well as 0 <

< p(n-a).

01

substitute n for

For if

n, then q EM, and we may n,p

On the ';lther hand, if

01.

stronger theorem if we replace :(n- ~ lap.

01 >

01 by

01

< n-llP, we are proving a

n-ap (Lenma 18).

We take t

=.

Then 0 ::: t ::: 1 by (44) and 1 - (n/ap') < t < n/ap

by hypothesis. have

01"

Thus (l-t)ap'< n ,

n - tap.

This implies (40) for the same reason.

theorem is proved for p ~ 1. IIq [G*uJ II:::

<

SS lx-yl< 1

This implies (41) by (28).

.r.f

When p

Iq (x)G(x-y)u(y)

JJ

+

Ix-yl> 1

by (28) and Lenma 20.

0

I

= 1,

we have

dx dy

< [K 1 Mn _ a, 1 (q)

We also

Thus the

- 205 -

M. Schechter IV Relative Compactness The next question one may wish to ask is how the spectrum of the operator (21) relates to that of P.

In general they are different.

o

However, one may ask if there are conditions on q which will guarantee that Q does not disturb the essential spectrum of Po' i.e., that a (R) • a (P ).

(45)

e

e

0

A simple but very, useful criterion can be given in the following way.

We shall say that an operator B is !- compact if D {A)c D (iii) and

implies that there is a subsequence of fBUk} which converges. Lemma 21.

If A is closed and B is A-compact, then

I shall give the proof of this lemma a bit later. it can be applied to our case.

the hypotheses of Theorem 15.

J

Ix-yl < 1

Let us see how

One result is

Assume that 1 < P < "" and that PO;:) and g{x) 's atiltf y

Theorem 22,

(48)

We have

Assume also that

Iq{x)I P dx -. 0 as Iyl -.

Then Q is Po-compact.

ee

,

Thus (45) holds.

The proof of Theorem 22 is not difficult, but takes a bit of care. We first prove some lemmas. Suppose w E Sand 0

Lemma 23. the operator Af Proof.

= p*f

is a bounded open set in En.

Then

is a compact operator from LP into L""
By Young's inequality

IIAft

+

n

E

j=l

IlnjAf ll",,:: C II f ll

p

is a sequence with LP norms bounded, then fAfk} is a uniformly k} bounded, equicontinuous sequence 00 ~ Thus it has a subsequence which If (f

- 2 06 -

M Sch e cht e r

converges uniformly. 0 Lemma 24.

Suppose q EM

S

(49)

%p

and that (48) holds.

Iq(x)I P !x_yla-n+€ dx ... 0 as

Iy l ...

Then for an y

(Xl.

Ix-yl< 1 Proof..

The lemma is obvious for

t - (n-~/e > 1.

J

+ e 2: n ,

If

Ct

+ e < n, put

By Holder's inequality. Iq (x)

Ix~yl
(f

Ct

IP

Ix-y I O'-n+e, dx

/q(x) IP !x_yIO'-n dx)

s

f

Ix- yl < 1

Iq(x)I P dx)

lit

lIt'

.

Ix-yl< 1 since (a-n+€)t' .. e-n, Suppose

Le1lllUl 25.

Suppose

(50)

t Ee;

This gives (49). 0 m(~;)

satisfies 0

t(I;)" I,

satisfies (15) for some a

:s H E;):S 1

and

11;1 < 1

ao,l sl>2. For r > 0 put

If I~I

:S.(, and al~1 + b > n, then

Proof.

Note that

ID~ 'r l :sc/rl~l. Put g (E:) .. D~[(l-'" )m] - (l-t ) D~ m. ~

-

'r

r

This function consists of a sum of terms of t he form (1) (2) (1) (1) (2) constant D~ t DlJo m, lJo ~ 0, u + lJo .. ~ • r

:s 1

and

.(,2: 1.

- 207 -

Thus g vanishes outside r S ~

Ie: I S

Ig~l SClral~l+b

(here we use the fact that a S 1). (53)

IIg)l

Moreover. (54)

M. Schechter 2r and satisfies

Thus

sC'/ral~I+b-n.

l

I-tr

Ill-'r)

Ie; I '< r , Thus S ds/'slal~l+b lsi> r

vanishes for

n~m"l SC

Since al~l+b > n. the conclusion follows from (53) and (54). Lemma 26. (55)

InJ.'w(s)I

Assume that

sC/lslal~I+b. I~I

where a S 1. koa > n-b. and k

o

Sn+l.

is an integer less than n.

1 S p < '" and that q (x) is a function in M

q,p

with

Assume

01 satisfying

(27).

If (48) holds, then the operator Tf = q [F(w)*f]. f E LP

(56)

is a compact operator on LP• Proof.

Let W E C'" satisfy (50). and let ~ be defined by (51). o

For

R > O. set (57)

qR (x) = q (x) •

Ixl SR !xl > R.

= 0

For each R > 0 and r > 0 write Tx = q [F(W w)*f] + q (F [(l-~ )w]*f} + (q-q ) R r R r R

Now T

l

is a compact operator on LP •

F(~ w) is in S. r

from LP to Loo(O).

Thus the operator Af

a>

For 'rw is in Co' and consequently

= F(~r w)*f

is a compact operator

where 0 is the set Ixl
Since q is

locally in LP • multiplication by qR is a bounded operator from Loo(O) to LP•

Thus T is compact on LP for each Rand r. 1

By Lemma 25 there is a

- 2 08 -

. M. Schecht er: function pr snch that

Moreover, Lemma 17 says that

Since M

a,p

Next let M

cr,p

<M

(qR)

a be

-

a,p

(q), we see by Theorem 16 that 1 ~211

.

any number such that a < cr < p (n-k ). o

... 0 as r ... "'.

By Lemma 24

(q-q) ... 0 as R ... "'. R

Another application of Theorem 16 gives IIT311 .s C Mcr;p (q-qR) IIp ... 0 as R ... "'. Thus I~-Tlil ... 0 as R ... '"

and r ... "'.

follows that the same is true of T.

Since T

l

is a compact operator, it

0

We now can give the Proof of Theorem satisfies (55). LP•

22.

Assume 0

E p(P ), and put w o

= IIp.

By Lemma 26, the operator defined by (56) is compact on

Let {uk} be a sequence of functions in D(Po) such that

"pouk" .s c , and set f

k

• Pou

k•

Then qU • Tf k k•

Since T is a compact

operator on LP, we see that {qU has a convergent subsequence. k} is P -compact. o

ground material.

Thus Q

0

We now turn to the proof of Lemma 21.

x to

Then w

Let X,Y be Banach spaces.

Y is called Fredholm if 1)

D(A) is dense in X

2)

A is closed

3)

dim N(A) is finite

4)

R(A) is closed in Y

(5) codim R(A) is fini te •

This requires a bit of backA linear operator A from

- 209 -

M , S ch e cht er

The (58)

~

of A is defined as

i(A) = dim N(A) - codim R(A).

We denote the set of Fredholm .oper at or s from X to Y as

The

~(X,Y).

following is well known (cL,e.g., Schechter (7,8J). Lemma 27. then A+K E If Y

~

~

(X.Y) and K is a compact operator from X to Y.

(X, y) and i (A+K)

= X,

=~(X,X).

tf A E

we let

~A

= i (A).

denote the set of

A such that A-A E

~(X)

The following characterization of cr (A) is useful. e

Lemma 28. Proof.

~ .~ 9"e1A) iff A E ~A and i(A-A) = O.

If A ~ cr (A), ' then there is a compact operator K such that e

A+K-A is bijective.

In particular it is Fredholm with index

Lemma 27, the same is true of A-A. A-A is Fredholm with index O.

Let ul, ••• ,u

be a basis for N(A ').

exist functionals uj and elements v u~ (u j) = v~ (v

o.

By

Conversely, assume that the operator

We may take A = O.

basis for N(A) and let vi' ••• ,v~

(59)

~hose

j

k

be a

Then there

such that

j) = &ij

Put Ku

=

Clearly K is of finite rank and hence compact. index O(Lemma 27).

Thus A-K is Fredholm with

One easily checks that N(A-ll:) = {O}.

index is 0, A-K must be bijective.

Thus 0 E P (A-K).

Since its

Hence 0 ~ cr (A).O e

Another fact we shall need is Lemma 28.

If A is a closed operator and B is A-compact, then

(a) IIAu l1 ~C(llu ll + II(A+B)ull), u E D(A), (b) B is (A+B)-compace, Proof.

If (a) did not hold, there would be a sequence {Uk} of

- 210 -

M . Schechter

elements in D(A) such that

In particular, (46) holds. by {uk} such that BU and uk

~

O.

k

Thus there is a subsequence (also denoted

converges to an element v.

Since A is closed, we see that v

possible, since "Auk" '" 1 for each k,

Then (a) implies that (46) holds. quence.

= O.

By (60), AUk~ -v But this is im-

To prove (b), suppose

Thus {BU

k}

has a convergent subse-

~

We can now give the Proof of Lemma 21.

If A is closed, we can turn DOA) into a Banach

space with norm

!/uliA

'"

[lull + /lAu!/ •

Now suppose A is not in (Lemma 28).

(j

e

(A).

Then A-A is in Hx) and i(A-A) = 0

It is obviously also in

~(D(A),X)

B is a compact operator from D(A) to X. has index 0 (Lemma 27). Thus A is not in (je(A+B).

with the same index.

Now

Thus A+B-A is in HD(A) ,X) and

It is also clearly in

~(X) wi~h

the same index.

The converse follows from (b) of Lemma 29. 0

- 211 -

M . Schechter

v.

Elliptic Operators The operator (3) is called elliptic if p(~) = E a e:~ 1~I=m ~ -

does not vanish for SEEn unless S an operator is in O(l,m).

= o.

It is easily checked that such

For such operations we have the following

improvement of TheorelllS 15 and 22.

Throughout we assume that 1 < P <

co •

Theorem 30.

If P(D) is elliptic of order m and q is im M with q.p a <mp. then (23) holds with c as small as desired. If (48) holds as well, then Q is Po-compact. The proof of this theorem is almost identical to that given above coupled with the following observations.

From the fact that Islm/p( s)

and its reciprocal are LP multipliers, one obtains the inequalities.

IIp(D)ull ~ C ( 1Itl~u ll

+

/lull) ~ c' (1Ip(D)u ll + lIui!).

From this it follows that one need only use one particular function G(x)

= F[(1+lsI2 )

-~

].

This function can be expressed in convenient

forms, e s g , , (61)

From this it follows that (62)

IG(x)

mn [x I - , m < n, Ixl

I ~C ~C,

m > n, Ixl

~ Ce-slxl , Ixl > for some s > O.

~

I,

s I,

1,

This allows for the slightly sharper result.

We can even extend these results to non integral values of m. s real, let HS'P be the set of those distributions u such that 2 1;s F[(l+lsl) FU] is in LP• Under the norm

For

- 212 -

M . Schechter

(63)

Ilulls,p

= IIF [(l+lsI

2

~s

)

HS'P becomes a Banach space. Theorem 31. IlquIJ < C M

(64)

-

Fu J1 1p' The proof of Theorem 30 gives

If s > 0 and q EM Q',P

(q) lip Ilull

s,p

g,p

for some

01

<

S

p, then

•

If (48) holds as well, then Q is a compact operat~r

from HS'P to LP .•

So far we have defined an operator corresponding to (20) for the case D(p )c D(Q) (i,e., when (23) holds).

The next question one might

o

ask is how to define an operator corresponding to (20) when (23) does not hold.

We now give a way of doing this.

Define

(65)

b(u,v) = (P(D)u,v) + (qu,v), u sv E C: '

and assume that there is an s such that 0 < s < m and (66)

!b(u,v)

I -< C

!Iu ll s,p IIv llm-s,p I s p

We can then extend b(u,v) to be a bounded bilinear form on H'

m-s p '

XH

'

We then define the operator corresponding to b(u,v) as follows: S u E D(B) i f u E H ,p and there is an f E LP such that

(67)

b(u,v) = (f,tT), v E Hm-s,pl

Clearly f is unique.

We define Bu to be f.

operator R given by (21).

B is an extension of the

We call it the s-extension for P(D)+q.

The

following theorem holds. Theorem 32. satisfy 0 <s <m.

Let P(D) be an elliptic operator of order m. and let s Assume that q(x) = ql(x) q2(x) , where ql EM •

q2 E Ma•p " with (68)

Q'