Weyl's formula for a class of pseudodifferential operators with negative

WEYL'S FORMULA FOR A CLASS OF PSEUDODIFFERENTIAL OPERATORS WITH NEGATIVE ORDER ON L2(R n) Monique Dauge and Didier Robert U.A.C.N.R.S. 75g, D~partement de Math~matiques et d'lnformat~que 2, Rue de la Houssini~re F. 44072 NANTES C~dex, FRANCE

Introduction The starting point o* this work is a paper by Birman and Solomjak [BI-SO I] in which they study the eigenvaiues asymptotlcs of a class o[ integral compact operators on L~(Rn). These operators are pseudodifferential, with symbols that are homogeneous (or quasi-homogeneous) of negative order with respect to the phase variable ; there are very few regularity assumptions with respect to the space variable. If such an operator A is sel:[-adjoint, (X~ (A)) denoting the positive eigenvalues decreasing sequence of A, [BI-SO I] gives an equivalent to the X+ (A), by powers of /

j when j + + o~ , and the corresponding result for negative eigenvalues (cf § 2.A). The method of [BI-SO t ] is based on Courant's mini-max principle and consists in reducing to a model problem on the torus'[ 'n. In this work, we intend to extend and precise [BI-SO 1] results in two directions : (l) relaxing the homogeneity assumption (2) getting a remainder estimate. As the symbol may vanish and change of sign, there are some difficulties we will overcome by making

an hypoelliptic regularization of the operators and then using

a Mellin functionnal pseudodifferential calculus. Here is the structure of our paper §l : Assumptions and the main result. §2 : Special cases and examples. §3 : Further information about spectral theory of globally hypoeiliptic pseudodifferential operators on R n, §4 : Proof of the main result. Annex A : Weyl-Ky Fan inequalities. Annex B : Composition formula with precised remainder. Annex C : Computation for an example.

92

The main results of this paper have been announced in the authors'

note [ D A - R O ] .

§1 : Assumptions and the main r e s u l t

Our operators are described by the properties of their Weyl's symbols, according to L. H6rmander's

formalization

[HO

l]. For the sake of s i m p l i c i t y ,

we r e s t r i c t our-

selves to the diagonal metrics. L e t us recall some d e f i m t i o n s . On T -~ R n, i d e n t i f i e d to R 2n : Rnx x R ~n , let qb and cp be two w e i g h t f u n c t i o n s ,

with thezr values in ]0,+oo[. (1.1) D e f i n i t i o n L e t m be a w e i g h t f u n c t i o n , m : R 2n ÷ ] 0 , + m [ . (1)

If t h e r e exists C O , C 1 > 0 such t h a t : [y[ ¢(x,g) + Inl ~(x,g) , 0

on

R2n

such t h a t K' m Y' , ) ~

-

For the functions a decreasing function

dx 0' YI ' Y2 >0 such that : f

I is derivable on ]0,t 0]

(T) ~ ¥ 1 f(t) 0 : ql a 4

(ii2)

3 61,82 > 0 : .~

IVal

w i t h (it 2) and (2.3), t h a t yields c o n d i t i o n (T) a b o u t V+0~ ; a). Note t h a t

if a is quasi-homogeneous, with weights k t on x i and h i on

~j , we

may take : F(x,~) = -(k I x I , ... , kn xn ; h I ~I ' "'" ' hn ~n)

2.C : S c h r ~ d i n g e r o p e r a t o r s L e t us c o n s i d e r : A -- (-A+E) 0/2 V(-A+E) °/2 w i t h A t h e L a p i a c e ' s o p e r a t o r on R n, E >0, o

(2.#)

< 0 , and t h e p o t e n t i a l V such t h a t t h e r e is s < 0 :

!a~ V(x)[ .R i=l i i x i

In t h a t case, we have : N+& ; A) = V+(~ ; a 0) + 0(~ ~ fs,o(X)) where fs,o00 = ~n/2s if s < o and fs,a(~) = t n / 2 s L o g t i f

s = O.

As an a p p l i c a t i o n , tet us c o n s i d e r t h e s t a t i o n n a r y p r o b l e m for S c h r 6 d i n g e r e q u a t i o n : (2.6)

(A+gV) ~= C~

where E > 0 is the energy and g e R is a coupling constant. For a fixed E, we search values of g for which (2.6) has a non-null solution ~F in D(A)

n D(V). For V v e r i f y i n g

(2.#) and g ~ 0, (2.6) is e q u i v a l e n t to (2.7) ~F ~ LZ(R n) \ {0} , (-D+E) -1/2 V(-A+E) -1/2 ~= _l ~F g

It is t h e a b o v e case,

with

a = - [ . Thus, (2.6) has n o n - t r i v m l solutions for t w o

s e q u e n c e s (gk+ (E)) and (gk (E)) w h e r e ( gk+ (E)) is positive) i n c r e a s i n g and (gk (E)) is n e g a tive,

decreasing.

If i n f i n i t e ,

each

of

those

sequences

a r e n o t bounded.

(2.5), w i t h :

N+(g,E) = # {k/g k (E) >. -+g} we have, when g + + oo :

(g >0)

I~ V v e r i f i e s

97

(2.g)

N+(g,E) - _

(21I)-nn l'n ;

(g V(x)-E)+n/2- dx + 0 ( g - % I ( g ( l + l x t 2 ) Q E ) n / 2 dx

where Y is the volume of the unit bali of R n. n Many works have given formulas of (2.8) type ([SI],[MA]). 2.D : Equation A~ = ~B¥.

L e t A = Op W a and B = Op W b be two s e l f - a d j o i n t operators on L2(Rn), possibly unbounded w i t h domains D(A) and D(B). With (Gq0) weight functions v e r i f y i n g ( H I ) ~ (H2) , we suppose that :

A is positive and invertibte,

(i)

a = a 0 + a t where a 0 is a t e m p e r a t e weight such t h a t a 0 ~ S(a 0 ; O,O and a 1 ~ S(ao(~q~) -~ ; Gq~) with E > 0

b e S(q ; c),qo) where q is a t e m p e r a t e weight such t h a t (ii)

q ~ S(q ; #,q~) ; -1 m :: a 0 q v e r i f i e s the hypotheses (W) and (N) • And we consider the s p e c t r a l problem :

(2.9)

A T = ~,B~Y ,

~

D(A) r i D ( B ) \ { 0 }

.

By mean of c o m p o s i t i o n by the o p e r a t o r C : A -I/2, (2.9) is e q u i v a l e n t to :

~] = I C B C ~

,

$ ~ L2(R n ) \ { O }

Thanks to (n), t h e o r e m (1.3) can be appiied to C B C, and if the volume functions satisfy the condition (T), we obtain, for a

¢ >0, when X + + ¢o •

(2.10) N+(I ; A,B) = (2K)-n ] ]

dx de + 0(~-¢ a(x,~)< + X b(x,~)

where N+~

jj

dx d¢)

a(x,~)< X q(x,~)

; A,B) (resp. N (~ ; A,B)) is the n u m b e r oi eigenvalues of (2.9) belonging

to ]0~l] (resp. l-l,0[). Many papers are devoted to Weyl's formulas of (2.10) type, w l t h various hypotheses (see [BI-SO 3] and [ F L - L A ] ,

as w e l l as their bibliographies). Most of them suppose that

A is an elliptic d i f f e r e n t i a l o p e r a t o r and B is the multiplication by a function O. in particular,

for Fleckinger and Lapidus in [FL-LA], A is of Schr6dinger's type and 0

may be discontinuous (for instance~ the c a r a c t e r i s t i c function of a c o m p a c t set). However, those works generally only give an equivalent. L e t us note t h a t we find t h e same result as the one announced by Boitmatov and Kostyuchenko in [BO-KO] under assumptions t h a t seem to us more general and more natural.

g8 §3 F u r t h e r i n f o r m a t i o n

about spectral theory of globally hypoelliptic pseudodifferential

o p e r a t o r s on R n. 3.A : Introduction

Let (~,~0) be weight functions satisfying (HI). Let p be a symbol satisfying the condition (W), i.e : p ~ S(p ; O~c~), and the further condition (S) which is inverse of the condition (N) : (S) 3 C , C' >0, ~ ~' >0 such that C p6< ((~c~) ~~y>0. 3.B : Functionnal calculus : parametrix. For z ~ C \ R+ , we have to study (P-z) -I pK with K positive integer (the reason of the introduction oi this parameter

K will appear

in section 3.C). Thus, we build

a parametrix for the equation : (3.1) V ~(K) z o (P-z) = pK ,

(3.2)

As a first approximation, we get the symbol : (K) = pK (p_z)-I qz ; 0 Then, by recurrence over j, we define the symbols :

(3.3)

(K)

qz ; j

=

(p_z)-I [p(K) - j~l J

where F(cc,g) = ((~ ~ BI) -1

k=0

Z

r(~,6)(a~

0~+8--j-k

2-1~1 (- 2)-1~1

Dg x

p)(a~ o ~x q z ;(K) k)]

and pj(K) is the jth term in the asymptotics of

the symbol of p K More precisely, pj(K) ~ s(pK(~bq) )-j ; ~b~c~)and is a polynomial expression of the ~

x p for I~.g[. i , we get :

J

= ~ (_])k d(K) k=0 jk (P-z)-k-I

(K beiongs where the djk) are universal polynomial functions of the 3( 36x p for [~+6[~~2 k~ Tr(Pk ~ Z Op W Pk ~) f(k)(0) ~:0 We have showed that the contribution for k:2 is zero and that the others are generally non zero.

104 We can also derive a trace formula for short range Schr6dinger operators, which may be compared with the results of [CVJ and [GU]. Let V ~ C~°(Rn) be the potential, such that there is s X ;

f~,0(p)= { oo

g~,,O ~ CO (30~,0)) and g)~,O --I on IOn,O) ; (3.23) ~/k e N, [(p 3p)k f~,0(l~)l+l(p ap)k gx,e(u)l < c k x ke . Then~ we have : (3.24) IN~ ; P) - Tr fk,0(P)[ .< Tr li0,,0) (P) (3.25) Tr li(k,O) (P) ~1, t h e r e is pj ~ S(p(~ q0)-j ; qb, £0) such t h a t for e a c h N >~1 : N Z pj) = Op w r N P0 - OpW(p + j=l and e a c h

semi-norm

with

ol a pj d e p e n d

r N ~ S(p((~q0) - N - I ; do, ¢p)

only on a f i n i t e n u m b e r of s e m i - n o r m s

otb

and

b I - it is t h e m e r e c o n s t r u c t i o n of a p a r a m e t r i x Ior B 0. Thus, we a r e in a slightly m o r e g e n e r a l is w r i t t e n

as an a s y m p t o t i c

expansion

(just

s i t u a t i o n t h a n in §3 : t h e s y m b o l of P0 like c l a s s i c a l

symbols).

Nevertheless,

the

whole r e s u l t s may be a d a p t e d , by s t a r t i n g f r o m ; (K)

: (p_z)-I [p(jK)

qz ;j

- j -EI

k:O

Ej ~=0

E

Io~+6]=j-k-.~

F(c~,6) (3~ D B p~) (~ x

D c~ q(K) k) x

z ;

i n s t e a d of (3.3) (with P0 = p) Then, i n s t e a d of (3.22), we a r r i v e to :

[R(x

;

p0 ) - v(X,p)[

4 ( 2 . C 1 X0-6)[V(~.),I-0 ~ p)-~(x-xl-e

w i t h C 1 and C 2 depending only on a f i n i t e

;

p)]+C 2

number of s e m i - n o r m s of p , p t , . . . , p N ,

where

N is a fixed integer. Now, l e t us r e m a r k that, for 0 4 00 , ),8-6 may be r e p l a c e d by )-0 , and that, for any f i x e d c o n s t a n t c > 0, V(k+k 1-0 ; p) - V(' X'1-0 ; p) may be replaced by ~(X+c2, i - e ; p) _ V(X-ck 1-0 ; p).

On the other hand, we have, for la = k -1 : N(~ ; a O) = R(X ; PO) ; V(p ; b) : V(X ; p) and V0, -+ cA 1-0 ; P) = V(p(l+-cp0) -I ; b) We c h o o s e c=20-1 , and we h a v e

(l+clag) -1 > 1-10 (1-Cla0) -1 < l+k 0.

F r o m all t h a t , we d e d u c e (4.9). a L a s t l y , h e r e is a r e s u l t looking like G a r d i n g ' s inequality, It c o n s i s t s of a r e l a t i o n between

symbols positivity

and o p e r a t o r s

positivity

(that relation

is n o t s y s t e m a t i c

as

109

i t is in a n t i - w i c k q u a n t i z a t i o n - see [TU-SU]). (#.I0) L e m m a L e t b • S(m ; (b,~0)be a positive symbol. L e t be e > O. Then, there is d e ~ S(m(~ cO)-1 ; ¢,fl0) such t h a t : 02.

¢ >0, we denote, a e = a x(a/em) + emil - x(a/em)] •

We then have the obvious f o l l o w i n g properties :

(#.11) a e ~ S(m ; ¢@) (#.12)

ae

~

em

(#.13) a e co]'ncides w i t h a in the region {a >~2 em} (#.I#)

a

.0, V X < ~0 ' Vc~]0,1/2[ :

[V(;~ ; a¢) - V+(X ; a) I 4 C E~0 v(~ ; m)

Proof (l)

Majorization

ae > X

"a

of V(~ ; a ). We have :

+f

> Sup(X,2em)

~ ( a c > X) n (a < 2 e r a )

I1

12

Now :I 1 4V+0, ; a) and 12 ..< V(~ ; 2em) .,s 0

J =k=lZ xk+2 Therefore I T-K-SII

>xj

+

~x211ull 2j

•

(A.2) Corollary ~+

+

+

j+k+l (TI+T2) ~)'j+l (TI) + Xk+l (T2)

•

Proof

Let KI,K2,SI,S 2 be self-adjoint operators sucht that, rank(K 1) ~< j, rank(K 2) ~~ 3/2, o'

>1 3/2 depending only on (¢~¢9) such that for each N >~ 1

and q >~0, there is y : y(N~q,n) and j(n) such that for all (x,~) ~ R 2n : Z

I~l+lsl~q

]a~ a 13 r N (al,a 2) (x,C.)[ 4 x

y(q,N,n) (~sk(a I ; ml) s~(a2 ; m2)) ml(x,~) m2(x,~) (gb~)-N (x,~) N /~-

q02(x,~) + (~Z(x,~) >/2-

We deduce from that partition, a breaking up of b 2 into three parts, b~j) , the support of which being dose to Z. Ior j=I,2,3. At first we are going to study b

in l e m m a (g.g). We note that b ) is like b i )

At last, we will study b~3) in l e m m a (B.10). (B.g) L e m m a For all a i ~ S(m i ; ¢,~), for all t ~ [0,t], (x,~) ~ R 2n, we have : i

I t - 2 n J e~ ~ O ( Z , ( ; y,rl) bSl)(z,(,y,q ; x,~) dz dC dy drll ~
~l) and a(x,() .> X}

V 1 may be computed just like the volume function of a symbol with one predominant homogeneity, So : (C.2) Vl(a ; ~) = xn/°

f

b(x) -n/° c(~) -n/O dx d~

Bn x Sn-I + Xn/a

n

c(~) -n/° b(E) -n/O d( dE + 0(l)

.

~sn_ t xB n

We have denoted : ~,

x

_ ~

Bn

Let us compute V2r:V2(a ; X) =

I×1 n-i I~l n-i dlxl dl(]

J

{×l° b(E)l~l° c ~ ,~, l×{~ l, {~j{~i

= [ where :

v(x,~)

=

[

,#

Ix{ n-1 v(×,~) d{xl d~ d~ lxl ° b(~)c(~) >. ~, Ixl >- 1

~b-l(E) c- i C~ / G Ixl -I

rn-I dr

1

= I xn/o b(x~j-n/o c(~-n/o [x[-n _ _{ • n

Thus :

n

F V2(a ; ~) = J w(~,~) dE d~ - b (E) c ® ~ X

where :

(xb-l(x~ c-l(~))I/a

w(~,~) = [

,tl

We have : w = l_no Xn/° (bc)-n/° Log

1

--Inxn/a b-n/a (E) c -n/a (~ r -

xnla -~ (bc)-n/o + ~1

r n-!

n

dr .

dE

dE

121

Thus : V2(a ; )Q -

xnlc~ ( n2

n

>~,

(bc) -n/° [~- Log ), + Log(bc) - n / ° - 1] d'x d~' + 0(1)

NOW: ~C 1 ) . ~< (bc) -n/° Log(bc) -n/c~ dx" dE : O(~ - n / ° Log ~-

Therefore, we have : (C.3)

~,n/o V2(a,~.~) = no-

Log ~

(bc) - n / ° d'x' d'~

I xsn-I + xn/°

~ ~ -I x Sn-I

(bc) -n/° Log -(bc)-n/° d~ dE e

i

+ O(Log ~ ) So, (C.I)~ (C,2) and (C,3) give the asymptoUcs, when }, + 0 ; I 1 V(a ; ),) = }n/(~ (et Log ~+ B) + O(Log ~)

with (x given by (C.3) and 6 given by both (C.2) and (C.3),

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fRO]

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