This article was downloaded by: [Univ Bordeaux 1 Bat A33]On: 29 August 2014, At: 02:40Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in Partial DifferentialEquationsPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/lpde20

Scattering amplitude for dirac operatorsR. Brummelhuis a & J. Nourrigat aa Université de Reims , France, 51687Published online: 14 May 2007.

To cite this article: R. Brummelhuis & J. Nourrigat (1999) Scattering amplitude for dirac operators,Communications in Partial Differential Equations, 24:1-2, 377-394, DOI: 10.1080/03605309908821427

To link to this article: http://dx.doi.org/10.1080/03605309908821427

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to the accuracy, completeness, orsuitability for any purpose of the Content. Any opinions and views expressed in this publication arethe opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use canbe found at http://www.tandfonline.com/page/terms-and-conditions

COMMUN. IN PARTIAL DIFFERENTIAL EQUATIONS, 24(1&2), 377-394 (1999)

SCATTERING AMPLITUDE FOR DIRAC OPERATORS

R. Brummelhuis and J. Nourrigat

Universitk de Reims

B.P.1039. 51687 Reims Cedex 2: France

1. Introduction.

hl.Taylor [17] has constructed an approximate diagonalization of a system of pseudodifferential operators under the assumption that the matrix of principal sym- bols is diagonalizable in a C" way. Helffer and Sjostrand [9] have given the analogue of Taylor's construction in the semiclassical case. For some systems, like t h system of h9axwell equations ( [13] ) , the hypothesis of global Cw diagonalizability cannot be met, for example for topological reasons. and it seems useful to have a variant of the above result which does not need this hypothesis, and gives approximate projectors rather than an approximate diagonalization. We present such a result in section 2 below, with a proof which differs from those in [17] and [9].

Another result, presented in section 3. is an extension of the classical Egorov theorem for systems, going beyond the one stated in Taylor [18]. We will give a description of the operator P(t . h ) = e 2 i A ( h ) ~ ( h ) e - 2 i A ( h ) for m x m systems A ( h ) and P ( h ) of h-pseudodifferential operators, A ( h ) being self-adjoint, such that only the principal symbol of A(h) is assumed to be scalar. The variant stated in Taylor [18] assumes that also P ( h ) has a scalar principal symbol, and our formula for the principal symbol of p(t , h ) is rather different from the classical case. and is related to the results of Dencker [7].

In the remainder of the paper, we will apply these two results to the Dirac operator, depending on h > 0

3

(1.1) H ( h ) = xa, ( h D j - A j ( z ) ) + P + V(Z) j=1

Copyright O 1999 by Marcel Dekker, Inc

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

378 BRUMMELHUIS AND NOURRIGAT

where cv3 and are Hermitian 4 x 4 matrices such that

and where the Aj and V are compactly supported C" functions in R3.

Our first aim is to study the scattering amplitude and to find the analogue, for the Dirac equation, of the result of Robert-Tamura [16] for Schrodinger operators (we refer to section 6 for the definition, and to section 7 for the semiclassical expansion of the scattering amplitude).

In Section 8, we decribe the evolution of coherent states by the Dirac equation, (cf. Wang [21] for the similar result for Schrodinger operators, and Ichinose [lo] for the Pauli operator).

2. Approximate projectors for pseudodifferential systems.

We shall say that a family A(h) (h €]O,l[) of operators in the Schwartz space S ( R n . Cm) is in Lp if A(h) = a(x, hD, h) is associated, by the semiclassical Weyl calculus, to a symbol a(. , ., h) in S p (i.e. such that the functions (1 + I(l)-pflfiild~afa(x, [, h)l are uniformly bounded in RZn) . We will always assume that a moreover has the following asymptotic expansion

with aJ in Sp-3. The function a0 will be called the principal symbol of the family and a1 its subprincipal symbol. We shall say that A(h) N 0 if h-NA(h) is in CN for all AT.

We consider a m x m self-adjoint system A(h) of pseudodifferential operators in Lp, and we denote by ao(x, E) the matrix of the principal symbols. We assume that ao(x, E) has only two eigenvalues A+ (x, <) and A- (x, [) of constant multiplicities m+ and m- , that these eigenvalues are functions in S p , and that there is C > 0 such that

Let us denote by .rr*(x, E) the orthogonal projection to the eigenspace corresponding to the eigenvalue A+(x,[). We assume that .rr*(x, 6) are functions in So.

Theorem 1. Under these hypotheses, there exist two self-adjoint h-pseudodifferen- tial operators lI+(h) and K ( h ) in Lo such that

and such that the principal symbol of II*(h) is n+(x ,<) . Moreover, there are two self-adjoint h-pseudodzfferential operators A+(h) and A-(h) in Lo such that

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 379

and such that the principal symbol of A+(h) is Ai(z,<). If A(h) = ao(x , hD) , the subprzncipal symbol of A+(h) is

The operators A+(h) are not unique. The equation (2.4) gives one of the possible

choices for a y ' ( z , <).

Proof. SITe will construct, by induction, operators r I y ) ( h ) such that

(2.6)n; [A(h ) : n y ) ( h ) ] = 0 mod CI(hN+')

If .V = 0. we take I I y ) (h ) = 7i+(z, hDz) . Now suppose we have constructed IIY) sat- isfyiug (2.5)., and (2.6),v. We want to find PNT1(x ,<) in S - " - ~ ( R ~ ) such that I I y ' l ) (h ) = rI iN)(h) + hN+lPh7+l(z, hD) satisfies (2 .5)*~+~ and (2.6)iL-1. Let

R K ( x . f ) be the principal symbol of h-"-' ( r IY1(h )* - r I Y 1 ( h ) ) and zSN(x.<) be

the principal symbol of h-N- l [A(h ) . I I iN)(h)] . Now ( 2 . 6 ) ~ + ~ is equivalent to

and (2.5),v+l is equivalent to

lire see that R N ( x , <) is an Hermitian matrix, satisfying

For that. we remark that ( 2 . 5 ) ~ implies n y ) ( h ) ( ~ $ ~ ) ( h ) ~ -rIyr)(h)) ( I - I I Y ) ( h ) ) =

CI(IL~"+~) . If we look at the coefficient of h N f l in the full symbol of this operator, and of its adjoint, we see that (2.9) is valid. We have obviously

rI?)(h)[A(h). rI?)(h)]IIiN)(h) =

= I l r i ( h ) A (h ) ( n Y ) ( h ) * - I I y 1 ( h ) ) - ( I I Y i ( h ) ? -T IY1(h ) ) A (h ) n y ' ( h )

If we consider the coefficient of hl'+l in the full symbol we see that Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

BRUMMELHUIS AND NOURRIGAT

and therefore

Then the following hermitian matrix

satisfies (2.7) and (2.8). If n + ( h ) in Lo is such that h-n'- ' (n+(h) - n i N ) ( h ) ) is in L-"-' for all N , and if X ( h ) = I - n + ( h ) , these two operators satisfy (2.2).

Then we construct: by induction, self-adjoint operators A Y i ( h ) in Lp such that

For N = 0, we take AYi (h ) = X+(x, hD) . Suppose that we have constructed

~ i ~ ) ( h ) satisfying (2 .11 ) ,~ . Let T N ( x , E ) be the principal symbol of h - N - l ( A ( h ) -

ALN)(h)) n + ( h ) . If we look at the coefficient of hN+' in the full symbol of ( A ( h ) -

A Y i ( h ) ) n + ( h ) n - ( h ) . we see that T N ( x , < ) n - ( x . < ) = 0. Similarly, we see that r+TA,r+ is hermitian. The following matrix

is therefore hermitian, and satisfies a y f " ( x , < ) r+(z , 6) = T N ( x . 4 ) . The operator

~ Y + l ) ( h ) = ~ j ; Y ) ( h ) + hN+'a ( tN+"(x ,h~ ) satisfies (2.11)j~+'. If A-(h) in LP is

such that hKv-'(A+ ( h ) - A Y ) ( h ) ) is in Lp-N-l for all N , this operator satisfies

(2.3). For the expre~~sion (2.4) of a:). we have only to follow the construction. N'e see that To ( x . E ) satisfies

The matrix a?) defined in (2.4) is hermitian and satisfies ajt')r+ = To. The theorem is proved.

Let us give now our main motivation to introduce the approximate projectors n + ( h ) and n- ( h ) .

Proposition 2. We can write

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 381

n/Ioreover, for each T > 0 and N > 0, there exzsts CNT > 0 such that (2.15)

l l R h ( t ) l I c ~ ~ ~ ) + I % ( ~ ) ~ I L ( L ~ ) + I I R ~ ( ~ ) I L ( L z ) 5 C,VT h" Vt E [O.T] Vh €10, 11

\Ve omit the proof. which is a standard consequence of the preceeding theorem. using Duhamel's principle.

3. The Egorov theorem for systems.

We consider a ( m x m) system of h-pseudo-differential operator A(h) in L1. We assume that the following hypotheses are satisfied

(HI) The principal symbol ao(x,E) is real-valued: ao(x,<) is the identitity matrix. multiplied by a Cm real-valued function, also denoted by ao(x, E).

(H2) The operator A(h) is essentially self-adjoint. We shall denote also by A(h) the unique self-adjoint extension of A(h).

We consider also another (m x m) matrix-valued h-pseudo-differential operator P ( h ) in Lo; whose principal symbol will be denoted by po(x,E).

We want to study the operator

In the scalar case (where m = 1). the answer is given by Egorov's theorem (cf. Taylor 1181). We know that in this case: F ( t : h) is an h-pseudo-differential operator: whose principal symbol is

Fo(t,x,E) = po(@t(x,S))

where cPt denotes the hamiltonian flow of the principal symbol ao(x,E)

If m 2 2. the subprincipal symbol a l ( x , J ) will play a more important role. For all (x,E) E R2n, let us denote by I'(t,x,E) the matrix-valued function , solution of the following differential system

(3.3) ~ ( o . x , < ) = I

Thus we define a matrix-valued function r ( t ,x ,E) . We shall see that r ( t .x .E) is unitary.

Theorem 3. With these notations, - the equality (3.1) defines a (m x m ) matrix-valued h-pseudo-differential operator P ( t , h) in Lo, whose principal symbol is

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

382 BRUMMELHUIS AND NOURRIGAT

Proof. First we find an operator Q ( t , h ) satisfying

where R ( t , h ) is a bounded operator symbol of Q ( t , h ) will be written

The principal symbol qo(t. x , 6 ) has to

+ R ( t , h ) Q ( 0 , h ) = P ( h )

in L 2 ( R n ) , whose norm is 0 ( h W ) . The full

satisfy

In the right hand side, the bracket denotes the commutator of two matrices. \Ire can see that the function F0(t. x . <) defined in (3.4) is a solution of this system. Then we can find. by an induction. the functions q J ( t , x , < ) such that the operator Q ( t , h ) whose full symbol has the asymptotic expansion (3.6) satisfies (3.5). Then we show that the norm of the difference F ( t , h ) - Q ( t , h ) in L ( L 2 ( R n . C m ) ) is O ( h m ) . In fact, the operator P ( t , h) defined in (3.1) satisfies

dP i - = - [ A ( h ) . P ( t , h ) ] d t h

and therefore

Since (Q - P) (o , h ) := 0 , we can conclude using the "Duhamel's principle".

Proposition 4. 1. The matrzx r ( t . x . < ) defined zn (3.2) and (3.3) zs unztary. 2. Denotzng by Q t ( y , 17) the Hamzltonzan flow-out of the symbol a0 from ( y , 7 ) . we have

Proof. 1. Let v be in C m . Let us put

f ( t . 2, E ) = Ilr(t, 5, Ovl12

The equality (3.2) shows that

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS

In fact. al is Hermitian, hloreover, (3 3) shows that f (0, x . <) = I v I 2 . Therefore. the function f ( t , x . <) is constant and equal to llv1/2 In other words, l T ( t , T , <)vI2 = v 2 . and r ( t . x . <) is unitary. 2. Let I' > 0 be a fixed number. Let us put

The two functions f and g are two solutions of the system

Since f (T) = g(T) . the uniqueness of the solution shows that f (0) = q(0). Since f (0 ) = I. we have

r ( T . y . ~ ) r ( - T . a T ( y . ~ ) ) = I

which proves the point 2

4. Case of the Dirac operator.

In all the following, the operator A(h) of Sections 2 and 3. now to bc lcnoted by H ( k ) , will be the Dirac operator of (1.1). Hence the integers m and 71 of sections 2 and 3 are m = 4 and n = 3. The eigenvalues of the principal symbol of the Dirac operator are A + ( x . <) and A- ( x . <) defined by

and the hypotheses of section 2 are satisfied. LfTe define self-adjoint matrices A,(q) by

and the spin matrices by

Theorem 5. T h e subpnnczpa l symbol of A+(h) constructed zn T h e o r e m 1 zs a Y 1 ( 2 . <) = &)(.c, < - A(2)) where

u ' h w e the S,k ure the spzn ma t r zces defined zn (4.3) and

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

3 84 BRUMMELHUIS AND NOURRIGAT

Proof. For each Cx function f on R6, let us put f"(x, 5) = f (z , 5 - A(x)). Then. we have

The eigenvalues of the principal symbol of H ( h ) are A+ = ,G+, where p*(x,q) =

V ( s ) i The corresponding projectors are 7;+ = &, where

Since { p + , p + } = 0, using (2.4), we can write 2iay) = F where

where. for j < k

and that rJk (x. q ) =,< 17 >-' cujak. The theorem follows easily.

The expression of a y ) ( x , < ) is somewhat similar to the usual spin-orbit coupling term, and perhaps could play. when h + 0, the same role that the spin-orbit coupling term plavs when c -. cx (see Thaller [19]. p. 189).

5. Transport equations and conserved quantities.

We denote by

the flow-out of (x. 5) by the Hamiltonian field of A+ defined in (4.1). The Hamilton equations of the symbol A+ are the equations of the classical relativistic mechanics of a particle in an electromagnetic field.

If f + (x( t) , ( ( t ) ) is a solution of these equations, let ~ ( t ) be a solution of the following system

(where ajt" is the subprincipal symbol of A+(h): defined in Theqrern 5)

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 385

SSk are interested to find some conserved quantities of this differential system. Onr of them (when the electric potential vanishes). will be the helzczty He(<) (cf [12]). defined for < # 0 by

We denote by & ( q ) the eigenspace of the matrix Ho(q) corresponding to the eigenvalue f < q >. and by T+(z .<) the orthogonal projection, in C4. on the space E+(( -- A ( x ) ) .

Proposition 6. If ( . c ( t ) .<( t ) ) zs an Hamzltonzan cume of A & , and v ( t ) a solutzon of (5.2). then 1 lVc have

d (5.5) - (.*(x(t). [ ( t ) ) v ( t ) . ~ ( t ) ) = 0 dt

2. If \ ' z 0 and ( ( 0 ) - A ( x ( 0 ) ) # 0 , then He(<(t) - A ( x ( t ) ) ) zs well defined and

(5.6) dt

Proof. If a matrix-valued function F ( q ) satisfies

then we have (using the Hamilton equations and the expression (4.5) of the Poisson bracket of F([ - A ( x ) ) and X+(x.E) = V ( x ) + < < - A ( x ) >)

By the expression (4.4) of p y ) , (5.7) is a consequence of

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

386 BRUMMELHUIS AND NOURRIGAT

These conditions are satisfied by F ( v ) = I. by F ( v ) = p+(q) and the condition (5.8)) is satisfied by F(7 ) == H e ( v ) .

Corollary 7. With the notations of Proposztion 6 , if u (0 ) is i n E+(E(O) - A ( x ( O ) ) , then u ( t ) is i n E + ( [ ( t ) - A ( x ( t ) ) for all t . If moreover the magnetic potential vanishes identically, then d ( t ) is i n E - ( ( ( t ) ) for all t .

Proof. The first claim is a consequence of (5 .5) . For the second one. we remark that

\Ye used p + ( ~ ) a Y ~ ( z . [ ) ~ + ( [ ) = 0. valid if the magnetic potential vanishes

Therefore. when the magnetic potential vanishes identically. and if u ( 0 ) is in E _ ( E o ) ? the transport equation (5.2) is equivalent to

and this system defines a parallel transport along the curve (rr ( t ) .E( t ) ) . by a suitable connection (cf also Lrnterberger [20]) .

6. Scattering amplitude : definition.

Let E > 1 be a real number and ~ u ' a vector in the unit sphere S2 of RJ (they are fixed In all t h ~ following). If u is in ~ + ( d d m ) , then the following plane wave

satisfies the free Dirac equation

We would like to add to this plane wave a function. behaving at infinity as a "spherical wave". to obtain a solution 11! of the complete equation (H(11) - E ) # = 0.

This will be possible under suitable hypotheses. We say that a real number E is non trapping for A+ if there exists a neighborhood W of E with the following property : for all R and R' > 0 , there exists T = T ( R . R ' ) > 0 such that we have the following implication (with the notation (5.1)

Proposition 8. Let E > 1 be an energy level, whzch zs non trappzng, nezther for A+, nor for A_ . Then. for each (u. E S 2 . 21 E E+(wv'-). and h > 0, there ns a functzon v E CS(R3. C4) such that 1. W e h a w H ( h ) u =: Ey zn R3

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 3 87

2. For each 0 in S 2 , there is a vector in the eigenspace E+(QJET), denoted by Th(w. O)u, such that, when 1x1 + oo

which defines a h e a r map Th(u. 8) of E+ ( u r n ) to E+ ( e m ) .

First, we recall the semi-classical limiting absorption principle for the Dirac op- erator, proved without magnetic field by Cerbah [6] and Jecko [ l l ] and, with an electro-magnetic field, by Bruneau-Robert [ 5 ] . For each s E R, let Lz be the follow- ing we~ghted L2 space :

Theorem 9. With these notations, let E be an energy level ulhich is non trapping neithe~ for A+ nor for A _ , and s > 112. Then, for each h > 0 sm,all enough, the resolvent ( H ( h ) - ( E + i~))-', defined z f E > 0, h,as a limit in C ( L z ( R 3 ) . L?,(R3)) when t: > 0 tends to 0 . Let Rh(E + iO) be this limit. There exists a constan,t C > 0 (independent on h ) such that, zf h is small enough :

Proof of Proposztzon 8. We can choose a function x such that

(6.5) x E c?(R3), x = 1 on supp ( A , V )

The) function $J defined by

a = (1 - X ) Phuu + h Rh(E + ,())((a . Dx) i^hru)

satisfies H(h)y? = E$. By the resolvent equation we can write

lii - phdu = -xphwu + h R ~ ) ( E + Z O ) ~

where R ~ ) ( E + 20) is the limit of the free resolvent, and

(6.6) g = (1 - ( ~ ( h ) - H O ( ~ ) ) R ~ ( E + 10)) ( ( a . D X ) a , u )

The distribution kernel of the free resolvent R ~ ) ( E + 20) is Kh(x - y), where Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

388 BRUMMELHUIS AND NOURRIGAT

This result is well-known (cf Thaller [19]). The support of g is contained in the support of the potential (A, V) . We remark that, when 1x1 tends to +co and 0 E S2, we can write, for all y in the support of g

Therefore we can write, when lzi tends to +m, setting 8"= 0 m ,

The Proposition follows easily.

This argument gives also a first expression of the scattering amplitude Th(w, 0). Tl'e have. for all u E E+(3) and .u E E+($), if w # 0 and if x satisfies (6.5) and g is defined in (6.6)

-E (Th(w,O)u, 21) = (9) inh~v)

7. Scattering amplitude : Semiclassical expansion.

This section is devoted to the asymptotic study of the linear map Th(w, 0). (the scattering amplitude) when h -+ 0. Recall that the similar result for the Schrodinger operator was given by Robert-Tamura [16].

In all the following, we shall set

Since E is non trapping for A+. and since the potentials are compactly supported. for each z E LU", there is a unique Hamilton curve of A+. denoted by

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 389

For the same reason, there is a vector Q,(z, w) of S2 and a point r,(z, w) of R3 such that, if t > 0 is large enough

We shall say that a point (w: 8) of S2 x S2 is regular if : 1. There is a unique point 20 E such that

2. The map z' + 0, (z', w ) is a local diffeoinorphism of w' on S2 in a neighborhood of 20).

We know that it would be more realistic to assume that there is a finite number of points 2, satisfying condition 1. but we shall however keep this formulation for sake of simplicity. We denote by J(zo) the following jacobian. where (21, 2 2 ) are arbitrary orthonormal coordinates in w'

80, 80, J(z0) = det (- , - 821

8.22 I om) ( 2 0 , ~ )

If (w. 0) is a regular point of S2 x S 2 , and zo the point of w' satisfying (7.2). then the action integral (7.4'1

is independent of t l < 0 et tz > 0 large enough in absolute value. hloreover, under similar conditions on t l and tz , the Maslov index of the following curve

in the following Lagrangian submanifold of Rs

is independent on t l and ts. We shall denote it by a

Let us explain now how a vector of C4 is transported along an Hamiltonian curve of A+. If u is a given vector of C4, let v(t) be the solution of the transport equation (5.2) (where x(t) = q,(t,zo,Z) and [(t) = poo(t ,zo,Z)) such that v(t) = u for t 5 t l (ti and tp chosen as before). Then v(t) is constant for t 2 tz : we shall denote it by M(w, Q)u. The norm of v(t), and therefore of M ( w , 6)u. is the norm of u and, if u is in E + ( Z ) , it follows from Corollary 7 that M ( w , 6)u is in E+(B").

\hTe can state now the main result of this section.

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

390 BRUMMELHUIS AND NOURRIGAT

Theorem 10. Let E: > 1 be a n energy level which is non trapping neither for A+ nor for A _ . Let (w , 8) be a regular point of S2 x S 2 , with w # 8. Let zo be the point of wi satzsfying (7.2).Then, we can write, for all u E E+(S)

The next proposition will be the first step in the proof of Theorem 10. In its statement, we mean by essentzal support of a family f h its support mod C?(hCD).

Proposition 11. For each ii: E S 2 , u E E+(W) and h > 0, we can find a function Fhdu in S ( R 3 . C4) such that 1. This function has the following asymptotic expansion when h -+ 0

2. Its essentral support zs contained i n the support of X . 3. Its L 2 norm is bounded independently of h: and we have, using the projectors of Section 2

4 . For all w and 8 i n S2 (u # O ) , for all u E E+(W) and v E E+($), we have

Proof. SVe can choose a function p(x .0 in C p ( R 6 ) , supported in a small neigh- borhood of the set I< = { ( x , Z ) , x E supp C X } , such that p = 1 on a smaller neighborhood of this set. Hence

Since (A-(x.E) - E ) # 0 on K. there is Q(h) in Lo such that

pw(r.hD) = ( A - ( h ) - E ) Q(h) mod C?(hCD)

and therefore

K ( h ) p" ( x . hD) = ( H ( h ) - E ) n- ( h ) Q(h) mod C?(hS)

It follows that. if u E E+(Z)

( a . Dx) S h w U = &U + ( H ( h ) - E ) Ghdu mod C?(hCD)

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 391

where

We have a similar equality for v E E+(B")). Therefore, if d f 8. we have, by (6.7)

By their construction, Fhdu and Ghdu are Lagrangian families of functions, with asymptotic expansions like in (7.8). Therefore. if d # 0, the scalar products like (FhLu . G h H r ) are ( ? ( h X ) , and the equality (7.10) follows. By the symbolic calculus, the first term in the asymptotic expansion (7.8) of Fh,.u is

f o ( x ) = p + ( 4 ( a . D x ~ " ) ) ZL

whew p+(Z) is defined in (4.6). SVe remark that. for all < and 71 in R3

Since (L E E + ( 5 ) . we have p - ( Z ) 21 = 0 , and (7.8) follows.

End of the proof of Theorem 10. IVe can prove also a limiting absorption principle for A + ( h ) : the resolvent Rh+(E + i ~ ) of this operator has a limit R i ( E + LO) in C ( L % ( R 3 . C 4 ) . L ? s ( R 3 , C 4 ) ) when E + 0, and the norm of this limit is bounded by Ch-I . Then we prove, using the results of section 2 , and (7.10). that we have, for all u E E~, (5) and 11 E E, (5)

Since the operator A+(h) has a scalar principal symbol. the end of the proof of Tht,orem 2 is the same as in Robert-Tamura [16] : by a partition of unity. we write the right hand side of (7.11) as a sum: some of the terms arr O ( P ) by Egorov theorem (as stated in section 3), and the last tern1 can be studied by the stationary phase formula.

8. Evolution of coherent states by the Dirac equation.

For each ( y . 7) E R6, and for each h > 0 , we define a function yh,, in S ( R 3 ) by

Z\'e consider now a point ( y , q ) , and a normalized eigenvector z. E C 4 in E+(71- A ( y ) ) . \Ye consider also a (4 x 4) matrix-valued &pseudo-differentia1 operator P ( h )

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

392 BRUMMELHUIS AND NOURRIGAT

in L o , with principal symbol po(x , 6). We are interested in the asymptotic behaviour, when h 0 and for constant t E R, of the following integral

(8.2) ~ ~ ( t ) = < e - 6 H ( h ) ( g h y r l u ) , P ( h ) e - ' i H ( " ) ($hzl+~) >

If @;(y , q ) = ( x ( t ) , [ ( t ) ) is the usual flow-out, we denote by v + ( t , y , q ) the solu- tion of the system ( 5 . 2 ) such that v+(O. y . 7 ) = v .

LVe main result of this section is the following theorem, (cf Wang [21] for the similar result for Schrodinger operators).

Theorem 12. W i t h these notations, if ZJ zs i n E+ ( q - A ( y ) ) , the integral I h ( t ) defined zn (8.2) satisfies, when h + 0

Lemma 13. For r a t h ( y , q ) zn R6 and v zn E+(q - A ( y ) ) , the functzons r ( t . ? / . ~ ) * 1 )

and v + ( t , y. q ) , defined respectzvely zn (3.2), (3.3) und above, are equal.

(8 .4) r ( t . v . ~ ) * v = r ( t , y . ~ ) - l = v + ( ~ . Y , v )

Proof. Lire kno~v that r( t , y . ~ ) is unitary. Put

(Lye uwd Proposition 4). The definition (3.2) of r shows that f ( t ) is a solution of the system (5 .2) . Since f ( 0 ) = g(O), the uniqueness of the solution of the Cauchy problem for (5 .2) shows that f ( t ) = g ( t ) .

Lye shall use also the following classical result (cf [21])

Theorem 14. Let A ( h ) be a h-pseudo-differential operator, wzth principal symbol o o ( r . [). and v E C4. T h e n , for each ( y , q ) E R ~ , We haue the following limzt when 11 -+ 0

< A(h)(7L'hyqu), ('$hy+J) > +< a o ( ~ , V ) v , v >

End of the proof of lheorem 12. First we prove the analogue of theorem 12, where H ( h ) is replaced b y >4+(h) . In other words, we have

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

SCATTERING AMPLITUDE FOR DIRAC OPERATORS 393

(8.5) = < u + + t t ~ , , 1 ) . P O ( @ : ( Y . ~ ) ) v + ( t , y , q ) >

This result is only a combination of theorems 3 and 14 and of lemma 13. Then we Put

U h ( t ) = e - Z i H ( h ) I I+(h)(+hyqv) X h ( t ) = e - Z i A + ( h ) I '+(h) (Qhyqu)

& ( t ) = II+(h) e-Z+A+(h) (Qhyqv)

We can write

Since the majorization by 0 ( h c X - ) is uniform on each compact set of R, and since X h ( 0 ) - Uh(0) = 0 , we can conclude, using Duhamel's principle that

(8.8) I/Uh(t) - Xh(t)ll = o ( h " )

In the other hand, by theorem 14, when h + 0

since 11 is supposed to be in E+ ( q - A ( y ) ) . Theorem 12 follows from (8.5). (8.8) and (8.9).

ACKNOWLEDGMENTS

We are very grateful to X.P. Wang for helpful discussions.

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014

BRUMMELHUIS AND NOURRIGAT

REFERENCES

[I] E. BALSLEV, B. HELFFER. Limiting absorption principle and resonances for the Dirac operator. Advances to Applied Mathematics, 13, (1992), 2, 186-215. [2]J.D. BJORKEN, S.D. DRELL: Relatzvistic Quantum Mechanics. Mac Graw-Hill. 1964. [3] A. BOUTET de IblONVEL-BERTHIER: D. hlANDA, R. PURICE. Limiting ab- sorption principle for the Dirac operator. Ann. Inst. H. Poinxare' 58. 4 (1993), 413-431. [4] V. BRUNEAU. Pvoprie'te's asymptotiques du spectre continu d'ope'rateurs de Dirac. Thkse. Uni~ersit6 de Nantes (1995). [5]V. BRUNEAU: D. ROBERT: Asymptotics of the scattering phase for the Dirac operator : high eneFTy, semi-classical and non-relativistic limits. Preprint, Nantes. 1997. [6]S. CERBAH. Principe d'absorption limite sexniclassique pour l'opkrateur de Dirac. Pr4p1~blication 95-6. Reims 1995. [7]N. DENCKER, On the propagation of polarization sets for systems of real principal type. J. Funct. Anal. 46, (1982). (3), 351-372. [8] C. GERARD. A. LIARTINEZ. Principe d'absorption limite pour des operatelm de Schriklinger a longue port6e. C. R. Acad. Sc. Paris. 306 (1989). 121-123. [9]B. HITLFFER. J . SJOSTRAND. Analyse semiclassique pour I'equation de Harper. 11. MCn~ozres de la S.M.F. 40 (1990). [10]\5'. ICHINOSE. On the semi-classical approximation of the solution of the Heisen- berg equation with spin. Ann. I. H. P, Phys. Th, 67. 1. (1997). 59-76. [11]T. JECKO. Approxin~ation de Born-Oppenheimer de certaines sections oficaces totales (i'une mol6cule diatomique. Rapport de recherche 97/01-1. Nantes 1997. [12]L. LANDAU. E . LIFCHITZ. Electrodynamique quantique. LIir (hloscow) . 1973. [13]A, KINET. SVork in progress. [Id] B. PARISSE. R6sonances pour l'operateur de Dirac. Heluetica Ph,ysica Acta. @ (1991). 557-591. [15] D. ROBERT, H . TALIGRA. Semiclassical estimates for resolvents and asymp- totics for total scattering cross-sections. Ann. Inst. H. Poincare'. Physique Th. a (1987). 415-442. [ l G ] D. ROBERT. H. TAhIURA. Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits. Ann. Inst. Fourier. a (1) (1989); 155-192. [17]LI. T.4YLOR. Reflexions of singularities of solutions to systems of differential equations. C.P.A.M. 3 (1975): 457-478. [18]hI. TAYLOR: Psr:udodifferential operators. Princeton University Press. 1981. [19]B. THALLER: The Dirac equation. Springer, 1992 [20]A. UNTERRERGER, Quantization: symmetries and Relativity, in Perspectives on Quantization, (L. Coburn and hl. Rieffel, eds.), Contemp. Math 214. Birkhauser. (1998). p. 169-187. [21]X. P. WANG. Et.ude semi-classique d'observables quantiques, Ann. Fac. Sc. Toulouse (1985). 101-135.

Received: March 1998 Revised: J u l y 1998

Dow

nloa

ded

by [

Uni

v B

orde

aux

1 B

at A

33]

at 0

2:40

29

Aug

ust 2

014