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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 =(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
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.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~
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
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
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 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) 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-.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
in such a way that
if
J" d~
~ D2:;-
-1
I. an
partial sum s (x) when n
E~
t'
n
=
f
"t 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 aF6 Lm-tf(f)
and aF has the amplitude
intepreted as (3.3)
.....,
La (0O, 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;).
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,d0 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
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,d0
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
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
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
,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
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~I2UD2 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~
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 ",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" ,.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 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
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)
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 abl
, 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
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 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
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 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
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
(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
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
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
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
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/ 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 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
1.
J
+ e 2: n ,
If
Ct
+ e < n, put
By Holder's inequality. Iq (x)
Ix~yl 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 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
ils,p
sc
/lp(D)rt> lls_m,p' cp E C;,
2\m
which comes from the fact that (1 + lsi) Now if u E HS'P, to u in HS'P.
/P(s) is an LP multiplier.
there is a sequence (~} of functions in
c; converging
Thus
Ip(D)cpk'v)
Is
~ C (/lcpk-ull s,p
la(cpk-u,v)
I + I (Au,v) I
+ IIAu/l s-m,p ) Ilvl lm-s,p ,.
Thlls
Letting k
~
=, we obtain /lu/ls,p ~ C IIJt is injective and has closed range.
To-show
that it is surjective, assume that w E Hm-s,P' and (Au,w)
S u E H ,p.
= 0,
In particular, we have (P(D)v,w)
= 0,
v E S.
Since P(s) is bounded away from 0 for SEEn, for each h E S there is a v E S such that P(D)v
= h.
Thus
(h,w) = 0, h E S.
This implies w
= O.
Thus (70) is proved.
more delicate and is omitted.
The proof of (71) is slightly
- 21 6 -
M . Schechter
VI
Operators Bounded From Below As a special case of Theorem 32 we have Iheorem 39 .
Let P(D) be an elliptic operator of order m, and let
q (x) be a function in Ma,l with a < m such that (69) holds. p
=2
Then for
the operator P Q has a closed extension B satisfying (70). o+
If
p(Po} is no t empty, then (71) ho Ids. We shall now show how one can improve this result slightly in some cases.
We illustrate the method by taking P(D)
6 is the Laplacian and r is a positive integer. (78)
GS,A(X)
= (21T)-~n F [(A+lsI 2)
= (1m) -~n
r(~s)
-1
S'"exp o
(-
-cn
l!.f 4t -
=
(A-6)r, where A> 0,
We set
] At} t
~s -~n
-1
dt ,
and for q(x) locally integrable we put (79)
Bs,A (q)
= ql~~~>
0
s~p ql~x) Slq(y) I G2S,A (x-Y)ql(y)dy.
A result one can state is Theorem 40.
Suppose q (x) and V(x) are real valued locally integra-
ble functions such that V(x) :5q(x) and Br,A(V) O
sup Y
if and only if
-t-> JJK(X.Y)K(x,z)~(z)dx ~ Y
dz
The value of Co is not affected if we restrict the infimum
2 2 to those cp > 0 which are in L • Moreover. lIT 11
= co •
- 219 -
M . Schechter Before we prove the theorem, let us show how it implies (83). K(x,y) .. G ,.(x-y) Iv(y) r,I' 0 be given.
and define the operator T by (88).
Put
By (79) there is
a function V(x) > 0 such that
I G2
,(y-z) Iv(z) 1 V(z)dz
r,~
Set cp(z) .. Iv(z)
If K(x,y)
t(z) when V(z) .. 0 and .. 1 otherwise.
I [I Gr,A (x,y)
K(x,z) cp(z)dz ..
1.\
Iv(z)
s
r\
:5 (Br,~,(V)+ e) Hy)·
I G2r,A
cp(z)dz ..
Gr,A (x-a) dxJ Iv(Y)
(y-z) lv(z)
I
V(z)dz Iv(y)
true for any e > 0, we have IIT I/2 in S, and set v .. (A-6) (IVlu,u) ... (IVI G
r/2
.:S B
:5 Dr, A(V).
1.\
,*v,
G
,* v) ,(V) ([A-llJ
r,~
This is precisely (83).
r
u,u).
We now give the
Proof of Theorem 44. [9J and Gagliardo [lOJ.
Thus
r,~
2 ,(V) I/v l/ ... B
r,~
Since this is
Now let u be any function
u •. Then u .. Gr,A* v ;
r,~
2
1.\
(Br, A(V) + e) cp(y).
2 'l11us by Theorem 44, T is bounded and Irrl/ .:SBr,A(V) + e.
.. I/T*vl/
Then
We use the methods of Aronszajn-Mulla-Szeptycka Suppose C < o
co,
and let e > 0 be given.
Let W
be the set of those x E En such that K(x,z) .. 0 for aImos t all z E En. Bet V(x)
I K(x,z)
cp(z) dz,
.. I,
x EW
x~W
Note that v(x) > 0, and there is a cp > 0 such that IK(X,y) V(x) dx.:S (C e) cp (y ) o+ For ~,v E S we have I (Tu,v) I
:5
II 1«x,y) lu(y)v(x) 1 dx dy
a.e.
_ 220 -
M Schechter
~
~(y)-l
(SSK(X,y) , (x)
~(y)
X (SSK(X,y)
s (Co+ f:)~ 2
~Co.
dx
dy)~
2 t (X) - l Iv(x) 1 dx
dy)~
lu(y)
1
lI u ll llvl l.
Thus T 18" bounded on L have 1 ~1J
2
2 and 1~ 112 ~Co+
c.
Since c > 0 was arbitrary, we
Conversely, suppose T is bounded.
Then so 18 T*'r.
Let
C be any number greater than I ~*T!I , and "l et h(x) be any positive function
2• in L
Set
~o(x) ·0, and define ~
Then
~ ~
-1
• h +C
T*T
0 by induction, and
Thus (~} converges in L ~
Thus Co
~ ~
h > 0 and
.=: I ~*T I I
'k-l' k .. 1,2, •••
s
1 ~ 1I2 .
2
.. h
to a nonnegative fun ct ion ~ sa tisfying
T*T~ ~ C ~.
0
-1
+C
T*T ~
•
Thus Co is finite , and
- 221 -
M. S.chechter
VII A stronger Result We now show how to obtain information
concerni~g
the essential
spectrum of the operator constructed in Theorem 40. Theorem 45.
Let g(x) be a function satisfying the hypotheses of
Assume also that B (q) < 2r• A t(x) > 0 such that
Theorem 40.
(90)
vex) -1
J Iq (y) I G4r , A(x-y)
Then P + Q has a o
~elfadjoint
m
and that there is s function
V(y) dy ., 0 as Ix I 7"
co.
extension B such that
a (B) = [A,m).
(91)
e
Note that the theorem implies that if B has spectrum in the interval [O,A), it consists of at most a denumerable number of eigenvalues (of finite multiplicity) having A as the only possible limit point.
We shall
give the proof of Theorem 45 as a series of Lemmas. Put 6 .. B ,(V), and r,l\ define b(u,v) by (87). (92)
(1-6) IIule
By (83)
oS b(u)
s
(1+6)
IIull~,
where
Let W be the completion of
emo with
let W' denote its dual space. (94)
respect to the norm given by (93), and
Then we have with continuous inclusions
H2r,2 cw c Hr,2 c L2 CH- r,2 cw' c H-24,2
Let B be the operator associated with b(u,v) and let ed operator associated with it. P(D) • (A_A)r.
the extend-
Let Po be the minimal operator of
It is the operator corresponding to the bilinear form
a(u,v) given by (84). with it.
Bdenote
Let
P denote
the extended operator associated
The following are trivial.
Lemma 46.
-1 2 24 2 P is a bounded operator from L onto H " o
and
.....1 ~
- 222 -
M . Schechter
-2r 2 2 is a bounded operator from H ' onto L • Lemma 47.
B-
1 is a bounded operator from L2 into W, and a-I is a
bounded from W' to W. 2
Lemma 48.
is a bounded operator on L
and
2
bounded from W' to L •
If ~ E c=, then multiplication by ~ is a compact operator
Lemma 49.
o
2
from W to L • r 2
Proof.
It is compact from H'
Coro llary 50.
If
~
2 to L (Theorem 31).
=
E Co' multiplication by
~
Apply (94).
is a compact operator
2 from L to W.
= r 2 For ~ E C , ~ - ~~ is a compact operator from H' to
Lemma 51.
o
H-r,2. Proof.
c (x)D~, where each of the
[~P(D) - P(D)~Jv =
~
coefficients c (x) is in COO 0
~
Lemma 52. Proof .
~
E C=, o
I q l ~B- l~
2• is a compact operator on L
Apply Lemmas 48 and 50.
Lemma 53. Proof.
For
For
Put T
~ E C=, ~Iq I~B-l is a compact operatn . on L2• o
= Iql~.
We have
~TB -1 _ TB-lcp = TB-1 [Bcp-qi3 ]B-1 = TB-1 [Pcp-c6P]B-1.
Apply Lemmas 47, 48 and 51 as well as (94). We shall also need a few results concerning abstract operators. Lemma 54. in ~ -1' A
If 0 E p(A), then \
Moreover,
+0
is in ~A if and only if 1/\ is
- 22 3 -
M . Sche chter. Proof.
Note that
A-""A. .. _MA- I _ ""A.-I)A. The result now follows from the fsct thst A is bijective. Lemma 55.
Let A.B be closed, densely defined linear operators on a
Banach space X. (95)
(j
e
n
If 0 is in p(A)
(A) ..
(j
e
-1
pCB) and A
-1 - B is compact. then
(B). -1
-1
By Leuma 27, ~ -1 .. ~ -1 and i(A - Tj) .. i(B - T]) for A B each Tj. By Lemma 54, 'A .. 'B with i(A-""A.) .. i(B-""A.) for each ""A.. This Proof.
gives (95). We now give the Proof of Theorem 45.
o ~ ~ ~ I,
Let
~
CD
'
be a function in C ' s uch that o
~(x) .. 1 for Ixl < \, ~(x) .. 0 for Ixl > I, and set C/\(x)
.. ~(x/k).
Now -1 -1 -1 -1 Po - B .. ~ T(synq) [C/\'I'B ]
(96)
+ (P-1 T(I-C/\) ]
(sgnq) 'I'B
-1
,
where T"
\ Iq 1 • Now for each k,
(Leuma 53).
'Mor e ov e'r , sgn q and
46 and (83». each k ,
and (104)
J Ix-yl< 1
for each 11, v.
[s
IJ,V
(y)h
I1v
(y) Idy ... 0 as Iy I
Let V be the set of those
...
CO
00
E COO such that Q(x,D)cp E LP• o
If the set (10) is not the whole plane. then the operator POD) + Q(x,D) on V has an s-extension E such that
(105)
cr (E)" cr(P ). e 0
- 226 -
M. Schechter
Scattering Theory
IX
We now describe an important application. abstract
b~ckground.
plex Hilbert space H. W+ u • lim
(106)
-
First we give the
Let S, T be two self-adjoint operators on a comIf the strong limits ei tT
e
-res u,
u c H
t-t±'"
exist, we call them the wave operators for'S, T (the ordered pair).
If
their ranges coincide, we say that they -ar e complete. The importance of completeness stems from the fact that it is
necess~ry
* W_ to be unitary. for the scattering operator S • W+ Theorem 61. that 'P(IQ
and sufficient We can state
Let peS) be a polynomial with real coefficients such
I ... '" as IsI ... "'.
form (97) such that
Let Q(x.D) be a symmetric operator of the
c; (lIp(D)cpli + Ilcpll), cp EC;
(107)
II o ]
and subject to the Dirichlet boundary condition
The coefficient c(y) in (1.1) is the disc ontinuous function defined by c(y) = {
c for 0 < Y < h 1 for
y > h
where hand c are positive constants and
·
~
2'34 -
Calvin H. Wilcox
(1 ~ 5)
0 h ,
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Calvin H. Wilcox
and
y+(~,~)
(2.10)
=
i
(*
sin
~h/c
z :~
cos
~h/c) e±i~h
The factor a(p,A) in (2.9) may have any value. defined below so as to normalize
'0
•
Its value is
in a convenient way.
Equati.ons (2 .8), (2.9), (2.10) define a solution 'o(X,y,p,A) of (2.2), (2.3), (2.4) for any value of p and A. ' Howe v e r , , (X,y,p,A) is a bounded solution only for certain o
values of ,t he s e variables. all p
~ ~
n-1
~(A) = {A
I
•
The factor e
i p• x
is bounded for
Moreover, it may be assumed that A
A~
oj.
~
0 because
Note that if A > Ipl2 then both ~ and ~
are real, since c < 1, and hence the square roots may be chosen so that
Thus (2.8), (2.9) and (2.10) define a bounded solution of (2.2), (2.3) and (2.4) for all A > Ipl2 and all of these functions appear in the eigenfunction expansion for A.
I f c = 1, so tha t
c (y)
1 for all y > 0, then 1;
1)
and (2.9) reduces to (2.12)
'O(y,p,A)
= a(p~A)
sin
ts
for
y > 0 •
In this case the functions '(2.8) with A > Ipl2 define a complete set of generalized eigenfunctions for A easily verif ied by Fourier analysis.
= -6 D
' as is
Moreover, if c > 1 then
-
~'±u
-
Calvin H. Wilcox
the functions (2.8) will A > Ipl2 provide a complete set see [4.] for the details.
However, if 0 < c < 1 there is a
profound change in the spectral structure of A.
In thi s
case the functions (2.8) with A > Ipl2 span a proper subspace of & and there exists a whole family of generalized eigenfunctions of a new type which are needed for the complete spectral analysis of A.
To discover the new generalized eigenfunctions look for aaditional bounded solutions of (2.2), (2.3) and (2:4.).
To
this end assume that
and define the square roots so that
In t his case
and hence 'o(X,y,p,A) is bounded for all (x,y)
E:a:
if and
only if y+(E;;,T]) = 0 or, combining (2.10) and (2.14.), i f and only i f
~
ain T]h/c +
c~,
cos T]h/c = 0 •
An equivalent form of this relation is (2.17) where
-¥
+ tan -1 (etr) = k1t , k=1, 2,3, •••
Itan-1(n/cE;;'>1
Ipl2 are parameterized by the points of the region
oo The generalized eigenfunctions 'k(x,y,P) are parameterized by the points of the region
It is shown in [4] that if 0 < c < 1 then all of these functions appear in the eigenfunction expansion for A. they form a complete
set,
spanning M.
Moreover,
These functions are
precisely the bounded solutions of (2.2), (2.3) and (2.4-), a8 was mentioned above.
The definitions of the eigenfunctions
is complete by defining
and choosing ak(p) > 0 so that
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Calvin H . Wilco x
2 ° 'k(Y'P) J oo
c(y)
-2
' dy = 1
J
k=1,2, ••••
These choices are motivated in [4].
§ 3. The spectral family of A. the spectral family {n(ll)
1 of
An ~xplicit construction of A in terms of the generalized
eigenfunctions 'k • k=O.1 .2 ••••• was derived in [4] by the method outlined in § 1 above.
It makes use of the generali-
zed Fourier transforms of functions f E
~
with respect to
the eigenfunctions 'k I that is. the functions (3.1) fo(p.>..) ,.
J
ll.n +
t (x.y.P.>..)f(x.y)c(y)-2 dxdy • (P.>..) EO. 0
0
'
and
J
(3 • 2) f k ( p) = ll. n \ ( x •
1" p ) f ( x • y ) e (y ) - 2 dx dy • P E nk • k =1 • 2 • • ••
•
+
Of course. these integrals need not be finite because 'k
;il.
However. if f E • and has compact support in lll.n = {(x.y) +
x E ll.n-1
• y ~ O} then the integrals converge for all
(P.>..) E 00 and p E Ok • respectively. because the 'k are bounded.
In this case i t is easy to verify that
for ' k=0.1.2.....
fk
E C(Ok)
For this class of functions the oonstruc-
tion of n(ll) may be stated as follows.
Theorem }.1.
tk
Let f.g E •
have compact support.
and gk are square-integrable on compact subsets of
Then
Ok •
the
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Calvin H. Wilcox
olosure of Ok for k=0.1.2 •••• and
for all
~
E:m.
(3.4-)
where Ak(p) =
and H(~) is Heaviside'
II
III
k
( l p l ) 2 • k=1.2.3 ••••
function
H(~) = {:
(3.5)
for
~
.c omp o ne nt s
tk
Theorem
of
t.
4.7 confirms the completeness of the generalized
eigenfunctions that were defined in § 2.
It rema ins to show
that the representati on (4.23) diagonalizes A.
Theorem 4.8.
The unitary operator
the - sense th at the operators t k
and
for a ll f E D(A) •
= JkPkt
i
diagonalizes A in
satisfy
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Calvin H. Wilcox
It follows from Theorem 4.8 that the project ionsP duce the operator A.
4.9.
Corollary
re-
This may be stated as follows.
IP
k
1; is
a complete family of orthogonal
projections which reduce A ; that is, PkP
t
(4.19)
k
= 0ktP!
,
P
k=
Pk '
holds and
Alternatively, the s u b s pa c e s
~k
=
Pk~
form a complete family
of reducing subspaces for A ; that is, 00
E
k =O
Ell ~k
Pk AP k
(an orthogonal d irect sum)
= Ak
' k =0,1 ,2 , • ••
,
and 00
§ 5.
Concluding Remarks.
The eigenfunction expansion of
Theorem 4.7 can be used to c onstruct functions of the ope rator A.
For example, c ons ider the wave equation
o
(5.1 )
The s olution of th is equation with init ial values
U
) E ( x ,y , 0) = f ( x ,y
~,
ou(x,y,O) ( ) c ~ 0t = g x ,y "' ..
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Calyin H. Wilcox
is
The coefficient operators in (5.3) are bounded operators on • by the spectral theorem.
It follows from Theorems 4.7 and
4.8 and Corollary 4.9 that, in the topology of Ie. 00
u(x.y.t) ::'
1:
k::O
where = lim M-..oo
J°hI M
p I.~{"A
uk(x.y.t)
.
, (x.Y.P,X) {(cos tA
1/2-
)f (p,A)
0
0
+ (1..-1/ 2 sin tA 1/ 2 )"g- (p,),,) o
1 dp d s,
and =lim M-..oo
JPk.~JpI.5.M 'k(x,y,P) + (A
for k=O.1.2 •••••
k(p)-1/2
sin tA ( p ) 1/ 2) gk ( P ) k
1 dp
Moreover, the "partial waves" uk(x,y,t)
remain orthogonal in Ie for all t dependently.
{(cos tA k ( p ) 1/ 2) f k ( P )
~ ~
and hence propagate in-
The completeness relation (4.19) guarantees
that every solution of (5.1) with initial values f.g ~ Ie is given by (5.4), (5.5). (5.6). These remarks are developed
more fully in [5J where (5.4).
(5.5), (5.6) are applied to the study of the propagation of electromagnetic waves along a dielectric-clad conducting plane.
- 252
-
Calvin H. Wil c oo
The technique develop.ed
in [4.] can be us ed to co ns-
truct eigenfunction expansions for other ope r at ors with p iece-wise c qnstant coeffic ients. 0=
It i s ev i dent that i f
c(y) has several d isc ontinu ities t he same t echnique c an
be used.
Another operator that can be tre ated in this way
is u
where
C(X ... .,x n) = 1
{:
for
x
2 2 2 h + ••• +x n < 1
for
x
2 2 2 > h + •• • +x n 1
In this case t he imp r o p e r e ig e nf u ncti o ns invo l v e Bessel f un ctio n s inste a d of e xp onential s .
Howev er, th e quali ta ti ve
f ac ts a bo u t
th e s pe ctrum a r e t he
s ame a s f o r t he case t rea -
ted a bove .
Det a ils wi ll b e giv e n el s e where.
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Calvin H . Wilcox
REFERENCES 1.
Kato, T.,
Perturbation Theory for Linear Operators,
New-York, Springer Verlag, 1966. 2.
Lions, J.L., and Magenes, E.,
Non-Homogeneous Boundary
Value Problems and Applications, V. 1, New-York, Springer Verlag, 1972. 3.
Stone, K.H.,
Linear Transformations in Hilbert spaces,
,P r o vi de n o e , A.M.S. Colloq. Publ. V. 15, 1932.
~.
Wilcox,
C.H.,
Spectral a~alysis of the Laplacian with a
discontinuous ooefficient in a half-space and the associated eigenfunction expansion, ONR Technical Summary aept. "" 21, Univ. of Utah (January 1973).
s.
Wiloox, C.H., Transient electromagnetic wave propagation along a dielectric-clad conducting plane, ONR Technical Summary aept. ~ 22, Univ. ~f Utah (May 1973).
