Estimates on Complex Interactions in Phase Space

Math. Nachr. 167 (1994) 203-254

Estimates on Complex Interactions in Phase Space

By ANDRE MARTINEZ of Villetaneuse

(Received May 14, 1993)

Abstract. We propose a method to study the complex interactions (or microlocal tunneling) between electronic levels that do not intersect in the real domain. The method consists in using a special kind of Fourier-Bros-Iagolnitzer transformation, and in adjoining an exponential weight to the L2-type Hilbert spaces which are associated to a complex Lagrangian manifold and have been introduced by HELFFER and SJOSTRAND for the study of resonances.

0. Introduction

This paper is an attempt to understand better the complex interaction between wells in phase space, in rather general situations. A very simple example where the usual techniques don't work is the following one:

Consider the matrix operator P on L2(lR") @ L2(lR")

where R is a symmetric 2 x 2 matrix of differential operators of order less than two. This is typically the kind of operator one can expect to obtain (at least approximately) by the Feshbach reduction for a polyatomic molecule, in the Born-Oppenheimer approximation. (In this case, h > 0 will tend to zero as the masses of the nuclei tend to infinity: see e.g. [KMSW] and references there.)

When h tends to O,, the principal part of P is

diag(-h'A + Vl(x), -hzA + Vz(x))

V,(x) = -x, - 1 , with

In particular, if one wants to study the resonances of P near 0, one should in principle consider the two potential wells

Vz(x) = xz.

u1 = V;'((-Oo,O]) = lR"-'x[-l , +co),

uz = V;'((-oO,O]) = (0).

204 Math. Nachr. 167 (1994)

But here U , is included in U , and one cannot reach in this way the interaction between the two electronic levels V, and V,. Then, one needs to be more precise, and instead consider the two microlocal wells (or wells in the phase space):

w, = { ( x , t ) E J R 2 " ; 5 2 + V, (x ) = O} = {<2 = x, + 1 1 , w, = ((x, 5) E R2"; t2 + V,(x) = O } = ((0, O ) } .

Now W, and W, are disjoint, and one can hope to estimate their interaction in terms of some "distance" between them. The problem is then to find a framework in which this can be exploited (e.g., by constructing an exponential weight that will measure the interaction).

For this purpose, we have chosen here to use the theory of resonances made by HELFFER and SJOSTRAND in [HeSjl], and more precisely to use a special kind of Fourier-Bros- Iagolnitzer (F.B.I.) transformation.

Indeed, the initial remark that has made the things possible is the following one. For u E C$(lR"), and p > 0 fixed, consider the (modified) Bargman transformation T

defined by

(0.2)

Then one can verify in a standard way that for some explicit C = C(n) > 0 and m = m(n) E IR one has

(0.3)

and also

U ( Y ) dY. T ~ ( ~ , 5 ; h) = 1 e i (x -y )Uh-a (x -v )2 /h

II TullL2(RZ") = Ch" IIUIIL2(Rn)

(0.4)

(0.5)

T ( h D p ) = hD,Tu = (5 + 2iphD,) TU

T(yu) = (X - hD,) TU .

(See [Sj2] for generalities about this transformation.) If P = p ( x , hD,) is a differential operator with polynomial coefficients, one then have

(0.6)

But since

(0.7)

and hD, - 5 is formally selfadjoint on L2(IR2") while 2iphD, is formally anti-selfadjoint, one has for any u, u in C~(IR")

TPu = p(x - hD,, hD,) T u = p ( x - hD,, 5 + 2iphD,) Tu .

2iphDSTu = (hD, - 5) T u

(hDSTu, Tu)L2(Rzn) = 0 .

In fact, making several integrations by parts and using (0.4)-(OS), one can prove more generally that for any C"-function f; and any multi-index c1

(0.8) (f(x, 5) (5 + 2 i ~ h D $ T u , T u ) L z ( R z n ) = ([(5 + i ~ h a $ f ( x , 511 Tu, T u ) L 2 ( R 2 9 9

( f x , C)(X - hD$ Tu, Tu)Lz(Rzn) =

where we have denoted

Martinez, Complex Interactions in Phase Space 205

For any y in Ca,(R2'), we then get by ( O h ) , writing P = C ua(x) (hD,)*, la1 I d

(5 + 2ipLapy + iph8,)" uor 2

(0.10) (e'+'Ih TPu, evih T v ) =

x eVlh Tu,ewIh To . ) In fact, all this can also be understood when adopting the point of view of Toeplitz operators (see e.g. [Sj2]), where (0.7) is just a &condition, and (0.10) comes from a direct integration by parts.

Using (0.10), one can for instance study the exponential decay of Tu when u is a normalized eigenfunction of P with energy E(h) close to some fixed E,. In this case, the decay will take place in the region d where p(x, 5) - E , is elliptic, and the rate of decay will be estimated by choosing properly y as large as possible, in such a way that p ( x - d,y, 5 + 2ipLa,y) - E , remains elliptic in 6.

Now a technical difficulty occurs when one wants to study resonances, since the transformation defined in (0.2) is not adapted. Indeed, following [HeSjl], p must be replaced by a C" function p ( x , 5) which, near infinity, must have some particular behaviour (for instance in the study of (O.l), this behaviour must be (x)-' ((x,) + 52)112, where we have denoted (x) = I-). The main reason for this is that one wants to have some uniformity at infinity, e.g., when one has to make stationary phase expansions.

Then the idea is to take p(x, 5) constant in a large domain Q where the main contribution of the interaction will a priori take place. In the region where p is not constant, the identities (0.4)- (0.5) must then be corrected by a term of the same type as Tu, but with a factor of order (x - Y ) ~ inside the summand. Using the fact that this region is far from the interacting one, we then show - introducing also a particular kind of cut-off function in the definition of Tu - that its contribution is smaller than the one coming from s2. In s2, moreover, the remarks above can be essentially generalized.

Finally, we find a (vectorial) F.B.I. transformation 'lr such that for any y in Crn@1'") satisfying Supp V y c Q, and any u in C;(lR"), one has (with some constant 6 > 0)

(0.1 1) llew'h TPu11;2(R2n) = 11 p(x - d,y, 5 + 2ipJ,y) ev/h TullE2

+ @ ( h (x)q evih Tull& + e-"" llu11$)

uniformly for h > 0 small enough, and where now P can be any differential operator with holomorphic coefficients (growing at most as ( x ) ~ at infinity) inside a domain

{ z E (En; IIm zI I C + 6 , (Re z ) ] ,

where 6, > 0 is arbitrary, but C must be taken large enough. More generally, fixing an escape function G and associating to it an I-Lagrangian

manifold AtG (t E IR small enough) and a weight e-2fH'h (see [HeSjl]), we get (denoting

L:G = L2(A,,; e-2rH/h dx d5))

(0.11) tIeV/' TPUII;:~ = IIpb - a,,,v, 5 + 2ipa,,,v) evih TuII::~

+ @(h ll<Od (x)' ev~hwlE:G)

+ @ ( l l U l l $ ~ ( R n ; e-6cx)/hdx))

206 Math. Nachr. 167 (1994)

uniformly for h > 0 small enough, and where s < 0 is arbitrary and a,,,t is a n x 1 matrix of derivations of the same type as (and close to) a,,.

Formula (0.11) must be considered as our main result, and we refer to Section 6 (Theorem 6.1) for a precise statement.

From a technical pint of view, the proof of (0.1 1) strongly depends on a method developped in [Sjl], and consisting in making changes of complex contour of integration in oscillatory integrals. Here we should emphasize that, except for a minor technical point in Section 2, our paper is essentially self-contained. In fact, the results we need concerning the F.B.I. transform are in general more precise than those of [HeSjl], and this obliges us to prove them completely and to re-introduce all the objects we work with.

As an application, we then study the exponential decay of the resonant states of P associated to resonances close to 0, when P is a matrix of the form

P = diag (PI (x , hD3, P ~ ( x , hDJ) + hR(x, hD,)

with p j (x , 5 ) = t2 + Q(x) (j = 1,2), Vl(0) < 0, R formally selfadjoint of degree at most two, and assuming that, at energy 0, V, admits a non degenerate point-well and p l ( x , 5 ) admits a global escape function in the sense of [HeSJl] (i.e., a function G such that H,,G . - a av, a field of pl). ax a x a t is globally elliptic on (pI = 0}, where H,, = 25 - - ~ ~ denotes the Hamiltonian

We show that these resonant states are microlocally exponentially small (i.e., when one applies T to them) outside {x = 5 = 0}, with a rate of decay y ( x , 5 ) related to the Agmon distance d , associated to V,.

Moreover, on s2 we also have

TPU = PTU,

where P" is formally selfadjoint on Lz(IR2") 0 Lz(IR2"). Then, applying a kind of generalization of the Green formula for P", in a domain 52' c s2 satifying

(030) E sz' c {(x, 5 ) E R2"; P I ( & 5 ) < O}

we find that the imaginary part of the resonances e(h) of P close to 0 can be estimated by the (microlocal) values of their associated resonant states on RZ" \ 52' and satisfy for any E > O

(0.12) IIm e(h)l = O(ep2(So-E)/h ) >

with some So > 0 related to the geometry of the problem, and optimized relatively to the

constant value of p in s2 for example in the case of (0.1) one finds So = ~ - 6). In some

sense this is an improvement of the main result of [Ma2]. Of course, the optimality of the rate of decay we obtain is not proved, but we do not know any generic counter-example showing the contrary. Moreover, the interesting question concerning the link between microlocal tunneling effect and complex classical trajectories is not discussed here, and constitute certainly a very difficult question. In [Ma41 we show that the same ideas can also be applied in adiabatic theory to get exponential decay of transition probabilities, and we believe that they are sufficiently general to give results in any problem involving in some way microlocal tunneling effect.

( 4


1. The Fourier-Bros-Iagolnitzer transformation

In this section we use the technical tools first introduced by HELFFER and SJOSTRAND in [HeSj 11, but in a rather special way here. Actually, for our purpose of controlling the rate of decay of all the exponentially small error terms, our F.B.I. transformations depend on an extra parameter 1 2 1 that will be taken large enough later.

Let us first recall some notations of [HeSjl]. For (x, 5 ) E IR2" we set:

(1.1) R(x) = (x) = (1 + x 2 y 2 , r(x) = ( x ) k o ,

i ( x , t) = (?w2 + t2)ll2,

where k , E [0, + a[ is fixed. Let G E Cm(RZ"), satisfying for any y , y' in N" with IyI + 17'1 2 1

(1.2)

In this work, we also assume that there exists a positive constant Co such that

(1.3)

L@;'G(x, 5 ) = O(R(x)'- I y ' P(x , 5)' - "") .

SUPP G = {ltl I C o r ( 4 *

For t E IR small enough we then define as in [HeSjl]

aG

at a = (ax) a<) E C2"; Im a, = t - (Re a), Im cly = - t

which is an I-Lagrangian manifold, and R-symplectic if t is taken sufficiently small, equipped with the measure ( d q A da,)""lA,, that will be denoted by da for simplicity in the sequel.

Letting

tH(a) = -Re a,. Im a, + tG(Re a)

one then has t dH(cr) = -1m (a, daxlAtc), and one can define

(1.4) L:G = L2(AtG, e-2tH(a)'h da)

where h > 0 is a small parameter.

already said, d 2 1 will denote an extra parameter. Now let us introduce the F.B.I. transformations which we shall use in this paper. As

For any a E A t G and y E R" we set

208 Math. Nachr. 167 (1994)

C, 2 1 is large enough so that for any x, y in R, one has

7 Lko r(y) 9 ,Iko <-<-++. - - __ 1 C1 8 8r(x) r(x) 8 8r(x) I X - yl I - (A + R ( x ) + R(y)) (1.6)

We also fix, for j E (0, l}, pj = ( P ~ , ~ , ..., p j , J E (RT)", and set

(1.7)

A") = P O X 2 i(Re a) R(Re a,) + f(Re a)

= (Pl(t0, - - * 3 P " ( 4 ) E C"(C2"; @I*,)") 9

where k , = Max (1, ko), and xz E Cr(R) satisfies

(1.8) O I x z I 1 ,

x2(s ) = 1 if Is1 I 5 ,

SUPP x 2 = CIS1 I 8) >

and

(1.9) x2(s) 2 f if, and only if, Is1 < 6 .

Then for a E Cm(AtG x IR") we define the F.B.I. transformation T, by the formula

(1.10) a(@, Y) x&) 4 Y ) dy T , ~ ( ~ ; h) = S e i ( a , - y ) a y l h - q , ( a . a , - y ) l h

for u E C;(lR"), a E A,,, and where we have used the notation

Slightly extending this definition to vector-valued functions a, we also set

'IT = T-,

whereRA(a,) = 1 + ( 1 - x 3 ( ~ 'Jk:)) R(Re ax), x 3 E Cg(lR"), x3(s ) = 1 for Is1 5 5, so that

(1.13) t'(a,y) = (1,y - a,) if IRea,l 5 5 i k ' .

From [HeSjl] we know that T maps Cz in LtZG. More generally, we shall see in Section 3 that if a depends holomorphically on y in a sector of C" containing R" and is bounded along with all its derivatives, then T, maps C; in L:G.

2. Approximate resolution of the identity

in the next section to construct an approximate left-inverse of %. In this section we specify in our context a result of [HeSjl] Section 4. This will be used


(where C, is an explicit constant), one can prove as in [HeSjl] Section 4, by a complex change of variables, that (at least formally)

(v, Y', 4 da, s ( ~ - y') = ~ ~ h - 3 n l . 2 J ei (y -y ' )us lh-qLL(u.u , -y ) /h-qLL(u.ux-y ' ) /h J

ntc (2.2)

where J has the following form

with the J,'s in Cm(AtG), satisfying for any y, y' in IN",

(2.4) d u x Y a f J us 0 = O(R(Re m , ) - " / Z - IYI f(Re , )"I2- '''I),

agxd&Jk = O(R(Re cxx)-"/z- lyl- l f(Re a)"/'- "'I) for k 2 1.

Indeed, one can pass from (2.1) to (2.2) by setting for 1 2 j I n

and

Now let us define

(2.6) Y , Y', co L ( Y ) XAY') 3

y') = cnh-3n/2 j e i ( Y - Y ' ) ~ ~ / h - q l l ( ~ , u X - Y ) / h - q L L ( u , I r x - y ' ) / h J (

'46

where the right-hand side must be considered as an oscillatory integral. We then prove:

Proposition 2.1. Zt(y, y') - S ( y - y') is in Cm(IRZn), andsatisfies for any E > 0 and2 2 1 : There exists So = So(&, A) > 0 such that i f f is small enough, then for any y E IN2" one has

) ~ Y ( Z , ( ~ , y') - s ( ~ - y')) = O ( e ( & - S r ) / h - d o ( r ( y ) R ( y ) + r ( y ' ) R ( y ' ) ) l h

uniformly with respect to h > 0 small enough, where (denoting p,: = Min {pj, 1, . . . ,

Proof. We prove it by a deformation argument, as in [HeSjl] Proposition 4.1. To make the reading easier, we also denote sometimes R(a,) = R(Re a,) and f (a) = f(Re a).

14 Math. Nachr., Bd. 167

210 Math. Nachr. 167 (1994)

Consider first the case where r is constant and G = 0. Then Formula (2.2) can be justified, and it shows that we have in this case

zf(y, y’) = ~ ( y - y’) + cflh-3n12 j” e’q(Y*Y’,a)lhJ (Y , Y’, a) ( x ~ ( Y ) Xa(y’) - 1) da 7

R3n (2.7)

where we have denoted

d Y , Y’, 4 = (Y - Y’) ay + iq,(a, a, - y ) + iq,(a, a, - y‘) . By (lS), we also have

so that on this set

1) = {la, -

1 V(Y> Y‘, a) 2 ___ p - (a) (A + R(a,)I2

16C: (2.8)

with p-(a) = Min {pl (a), ..., pn(m)}.

Inserting (2.8) in (2.7), we obtain for any 6 > 0 small enough

ZJY, y’) = S(y - y’) + J O(e-e(avYqY’)/h ) dm R3n

(2.9)

with 1

16C: &a, Y7 Y’) = __ (1 - 4 P L ( 4 (2 + R(mJ2 + 6q,(a, a, - Y ) + 6q,(a, a, - y’)

Now we claim that: If R(a,) + i (a ) I 6ik1, then

PLg 2

(2.10) p P ( a ) ( i + R(a,))2 2 - A 2 *

If R(a,) + ?(a) 2 6Ak1, then

CL; (2.11) 2

Indeed, due to (1.7) - (1.9), if we set

pP (a) (2 + R(CX,))~ 2 - [3(1 - 6) Lk1 + GF(a) R(a,)]

(2.12) @(a) = R(a,) + one has on @(a) I 6Ak1

so that (2.10) is immediate. Moreover, on @(a) 2 6;lk1, we get

Martinez, Complex Interactions in Phase Space 21 1

and also, since R(aJ and F(a) 2 1,

so that (2.1 1) follows easily. Inserting (2.10)- (2.11) in (2.9), we obtain

1 da I t ( y , y’) = S ( y - y’) + 1 O(e-B1(a’yTy‘)ih) da + 1 O(e-@z(a,y.y’)/h

e(a )S 6dki p ( a ) 2 62’1

with

1 PO (1 - 6) p;i’ + 6 - [(ax - y)’ + (a, - y’)’] , Y , Y’) = ~

2 32Cf 1

32C: @’(a, Y , Y’) = ~ (1 - 6) ,~;[3(1 - 6) Akl + R(a,)]

Therefore I t ( y , y’) - S ( y - y’) is in Cm(lRz”) and satisfies, in this particular case, for any E > O

z t ( y , y’) - s ( y - y’) = f O(e-e&(*-y,y‘)/h) d a , R*”

with (for some B ( E ) > 0)

In particular, using the fact that r is constant and considering separately the parts of the integral where

1 1 lY - %I - R(a,) 9

C - c 1 1

IY’ - a I < - W a x ) 3

* - c - c

lY - a I > - R(g, ) >

IY’ - a I > - R ( 4 1

(with C > 0 large enough)

this gives

(2.13)

for any E > 0 and with 6‘(~) > 0.

point yo , in the sense that

1 I ~ ( ~ , y f ) - s(y - y’) = O ( e ( e - - ” ” ) / h - S ’ ( e ) ( R ( y ) + R ( y ‘ ) ) r / h

Now in the general case ( I non constant) one can assume that y and y’ are near some

where C, is the same as in (1.6).

14*

212 Math. Nachr. 167 (1994)

Then on Supp x,(y) x,(y’) we also have

1 - (2 + N Y O ) ) c,

IRe a, - Yo1

so that by (1.6)

r(Re a,) 4 Y ) - 4 Y ’ ) 3

I f where f’ - g means here that ~ I ~ I C, for some positive constant C, that may depend on 2. cn g

Then for 8 E [0, 11 let

~ o ( C 0 = (1 - 4 ~ b o , + OF(.) 9 Go = QG 9

and define I;(y , y’) as I , (y , y’), starting from Formula (2.1), but replacing p(a) by p0(a) in (2.5), and G by Go in (2.6) (so that J will automatically also be replaced by some Jo) .

As in the proof of [HeSjl] Proposition 4.1, one can then apply the Stokes formula to get

o ( ~ , Y’, a, t, h) d a , i (Y - y’) q i h - qr (a, ax - y ) / h - qr (a, a, - y’)/h k a - I 3 Y , Y’) = s e a8 AtCe

a where k is a symbol in h supported in Supp ~ [Xa(Y) ~ a ( ~ ’ ) l . 8 Re M

Essentially, by the same arguments as before (and also using (1.2)), we then obtain for any c > 0 and for t E IR small enough

with d ” ( ~ ) > 0, and uniformly with respect to h > 0 small enough, y , y’ E IR”, and 0 E [0, 11.

to the case 8 = 0) that for any E > 0 and 1, > 0, if t is small enough, then Integratingfrom 0 to 1 with respect to 8, we deduce from this and (2.13) (which corresponds

1 , (2.14) I , ( ~ , y’) - ~ ( j , - y’) = @(e(‘-“)/k-aofd (r(v)R(yt+rfv‘)R(y’))/k

with So(&) > 0, and uniformly with respect to h > 0 small enough, and y, y’ in IR”. In fact, because of the exponential decay in ( y ) and ( y ’ ) obtained in (2.14), one can

show in exactly the same way that for any y E IN’” (and possibly after having slightly decreased do(&)), one has under the same conditions

(2.15)

This ends the proof of Proposition 2.1.

1 . ~ Y ( I , ( ~ , y’) - d(y - y’)) = @ ( e ( & - s M - S o ( & ) (r(y)R(y)+r(y’)R(y’)) lh

0

3. Estimates at infinity for the F.B.I. transformation

In this section we consider the transformations T, defined in Section 1, but with some additional assumptions on the function a.


For any a E A t G , and il 2 1, let us denote

1 1 I k I n, JRe a, - Re zJ I ~ (A + R(Re a,.))

2Cl We consider the space d,,( of the functions a E C" (AtG x IR") that can be extended to

holomorphic functions with respect to the y-variables, bounded along with all their derivatives, in the set

(3.2) Da = {(a, Y); a E Y E Da,.} . In particular, the function ;defined in (1.12) is in (d,,t)"fl.

and satisfies for some 1 2 0 and for all y, y' E IN" We say that a function m E C"(AtG) is an order function, if it takes its values in (0, + a)

(3.3) m(a) = @((Re a) ' ) ,

Denoting as before @(a) = R(Re a,) + f(Re a), the main goal of this section is to prove:

Proposition 3.1. Assume that a E d,,( and satisfies for some integer v 2 0

uniformly in DI. Then for any order function m, any E > 0, and any s < 0, one has for 6 > 0 and t E IR, both small enough, that

IlmTlu IlLiG ( p ( a r ) t 4 2 k 1 )

uniformly with respect to h > 0 small enough and u E Czw). Here

and S, is defined by

whereK,isauniversalconstant,C, isasin (1.6),andp; = Max ..., pj,"> ( j = 0, 1).

Remark 3.2. Indeed, as it will be seen in the prooA one can take

but this is certainly not optimal.

214 Math. Nachr. 167 (1994)

To prove Proposition 3.1, we first construct an approximate left-inverse for T using the

With the notations of (2.3) and (1.12), let result of Section 2.

(3.4)

where R , is the same as in (1.12) and, for u in (L&)nt1 ,

X,(Y) Y ) . 4.) da h) = l ei (Y-a , )a ( lh-q , (n ,orx- - ) /h

A r c (3.5)

with ''." denoting the usual bilinear product on (En+'. We then have:

Lemma 3.3. For any a E dn, t , m an order function, and E > 0, one has for t E IR small enough

IlmT,(STu - U)llL& - - o ( , ( & - S d / h 11 11 H: (R"))

jor some 6 = a(&) > 0, and uniformly with respect to u E C;(lRn) and h > 0 small enough. Here is defined as in Proposition 2.1.

Proof. By construction, we see that the kernel of ST is exactly Zl(y, y') as defined in (2.6). Therefore Proposition 2.1 shows that STu - u is in CW(IRn) and satisfies for any s, s' E IR

(3.6) I~~JJ(STU - u ) / I H - ; ( R ~ ) = O(e(&-'A)lh) ~lullH;(R.)

for some 6 > 0. It is then easy to extend the action of T, on such a function. Now, on the expression of T,(STu - u), one can make several integrations by parts in the

y-variables, to win as much inverse powers of (Re at). Moreover, because of (1.2)- (1.3), one has

(3.7)

Then one obtains rather easily, setting w = STu - u, and with N E IN arbitrary, that

IG(a)I + IH(a)l = O(r(Re a,) R(Re a,)).

with some positive C and N . Hence, using (3.6) and the fact that r ( y ) - r(Rea,) and R(Re a,) - R ( y ) on Supp x,(y), one obtains for t small enough

with 6, = 6,(s) > 0. Taking N large enough, this gives the result of Lemma 3.3.

of TSu. Let us denote by K(a, p) the integral kernel of T,S, so that we have

(3.8)

0 Now let u = T u . Then Lemma 3.3 reduces the proof of Proposition 3.1 to the study

a(& Y ) X J Y ) X d Y ) s'((P1 Y) dY I K ( ~ , 8) = l ei@D(a,b,y)/h

R"


and let us denote by Tu,s the complex contour

Lemma 3.4. ra,B c DA,, for any a, /-l E AtG and t small enough.

Proof . See Appendix A. 0 Consequently, since Supp xu,@ c {x, = xp = 1) and a E we can make a change of

integration contour in (3.8), and integrate y on ra,p instead of IR". Parametrizing Ta,p by its real part (still denoted by y) , one can then show directly that

(3.13) Im + tH(a) - tH(P) = t[G(Re a) - G(Re P)] - t(Re a, - y ) dxG(Re a)

+ t(Re P, - Y ) dxG(Re P) + (Re Pr - Re a<) XU, p Oil A (a, P ) + 4,("9 Re a, - Y ) + 4JP, Re P, - Y )

- qr(ar td<G(Re a) - X a , p ( Y ) A(a, P)) - 4,(P? td,G(Re PI - X,,p(Y) A(4 PI) .

Taking formally = 1 in (3.13), and denoting yc(a, P) = (yC,,(a, P), ..., yCJa, P)) the critical point with respect to y of the expression obtained in this way, we find, fork = 1, . . . , n,

216 Math. Nachr. 167 (1994)

y,(a, P ) = Re a, + 0 IRe a, - Re P,I + t ___ I R ~ - Re ~ ~ 1 ) . ax)

F(Re a) (3.15)

Lemma 3.5. For any E > 0 and N 2 1, one has

uniformly,for h > 0 small enough and v E L:G. Here 6 = B ( E ) > 0 and

Proof. See Appendix B. Essentially, the proof exploits the fact that, on {x , ,~ $- l}, either [Re ax - Re P,I stays away from 0, or, when the latter is small, ly - y,(a, P)I stays away from 0. 0

Now it remains to be estimated

Using the fact that

Martinez, Complex Interactibns in Phase Space 217

we get from (3.16) on (~,,~(y) = l} that

(3.17) Im @IT*.# + t H ( 4 - t W P )

= t[G(Re a) - G(Re /?) - (Re a - Rep) V,,<G(Re a)]

As for Lemma 3.5, we first have:

Lemma 3.6. For any E > 0, and i f t is small enough, one has

A uniformly for u E L:G, h > 0 small enough, and with 6 = a(&, 1) > 0 andS’ =

16(PL,+ + P 9 . Proof. See Appendix C. 17

Now when IRe a< - Re /?<I I - + - (F(Re a) + F(Re p)), since also R(Re a,) - R(Re p,) I 1 2 2

on Supp xu,@, we have F(Re a) N ?(Re p), and thus we get from (3.17) on this set:

(3.18) Im @IT..# + tH(a) - tWB)

(Re a, - Re /Ix)’ + t ~ ax) (Re at - Re 83’). ?(Re a)

218 Math. Nachr. 167 (1994)

Taking t small enough, we then obtain in this case

+9. R(Re ax) (Re a5 - Re pF)’ r(Re a)

with some 9 = q(1) > 0. Using the assumption on a ( a , y ) made in Proposition 3.1, and denoting

a 1 (a, /I) E (AtG)2; IRe at - Re PSI I - + - (P(Re a) + i(Re p))

and 3y E IR” such that ~ , . ~ ( y ) = 1

2 2 (3.20)

we deduce from (3.19) that we have on B

e! ( H ( P ) - H (a))/h K ( (3.21) 2 a, P )

Y , e-Q(a.B,Y)/h d la, - y - ia(a, p)I” = ’( R(Rea,)*

R“

where

Using the definition (3.12) of A(a, fl), as well as (3.14), we find

and thus, inserting (3.22) in (3.21),

Y a)”/2

- Q (a, 8, y)/h d i(Re a) R(Re

uniformly for (a, p) in &7 and h > 0 small enough.

the change of variables To treat the first term er(H(8)-’r(a))ih L,(a, p) appearing in the r.h.s. of (3.23), we make


which gives

f ,t(H(B)-W(a))/h L (a p) = 0 - h-" exp [ -9 (Re a, - Re p,)2/h (3.25) 1 ,

1 R - rl 7 (Re a< - Re PJ2/h

and it is then standard to verify that

Moreover, by construction, L , can be assumed to be supported in B, and using (1.6) we find on this set

ah' (3.27) @(a) 2 4Ak' 3 @(P) 2 4.

As a consequence, the estimate (3.26) can in fact be rewritten as

= Cj h"2 - AtG L:, (e(a)> 4 ~ ~ 1 ) ( I/ (;I2 I1 L : G ( Q ( a ) 2 A k l / J .

The two other terms appearing in the r.h.s. of (3.23) also satisfy the same estimate. For instance, let us treat the second one, denoted by et(H(B)-H(a))/h &(a, p). Using the same change of variables (3.24), we obtain

(3.29) ,r(H(B)-H(a))/h L , ( ~ , p)

R = 0 h- ( n I R e " - Re "1') exp (- y

a,)

Writing

and using (3.29), we then obtain, by the new change of variables,

220 Math. Nachr. 167 (1994)

that

Iu(P)I lW)I dP dp’ - i ( i /R)(Re/ l , ~ R e P I P i h - il(R/il(Re/Jc -Re/J i )z /h

with some fj > 0, and where we have also used that m(p) - m(a) N m(p’) on the support of the summand.

Then, using Cauchy-Schwarz Inequality and the same argument as before (in particular the fact that (3.27) is also satisfied on Supp L2), we get

In the same way, one can prove that (3.31) is also valid with L, replaced by L, which corresponds to the last term appearing in the r.h.s. of (3.23). Inserting this and (3.28) in (3.23), we obtain the same estimate (3.31) with now L, replaced by K, .

Summing up, using Lemmas 3.5 and 3.6, we get for K = K , + K ,

with some 6’ = S‘(8) > 0. Now Lemma 3.3 and Formulas (3.32)- (3.33) clearly imply Proposition 3.1. 0

4. Approximate Cauchy-Riemann equations for T

The purpose of this section is to find a generalization of the Cauchy-Riemann equations (0.7) satisfied by the transformation Tdefined in (0.2). Since now p = p(a) is not constant, the equations (0.7) are not satisfied exactly by Tu. Indeed, the error terms come from the region where V p is supported, and by (1.7) this region is included in (@(a) 2 5Ak1}. But thanks to Proposition 3.1, we can now estimate these terms.


More precisely, denoting

'IT = (T1,T') = (Tl, TI, ... 1 T,), and

h a I h a , D = - - D = - - i ay ' a i am

we have:

Proposition 4.1. For j = 1, . . . , n and u in C: (IR'), we have on AtG

BaxjTu = 'IT(D",,u) + Rju h

= ( m r j + 2ipj(m)Da )Tu - ~ (0, ... ) 0, TIM, 0, ...) 0) + R j U , 51 iR,(%)

uniformly with respect to u E C?(lR") and h > 0 small enough, and where E > 0 and s I 0 are arbitrary, 6 = 6(&) > 0, and t E IR is small enough.

Proof. Let us first study Tl. Denoting

%(a, Y) = (a, - Y) q + iq,(a, a, - Y) 3

we have

(4.1) D",xjTlu(4

= I( mtj + 2ipj(m) (axj - yj) + i 2 (a,,,pk(a)) (mxk - yk)z) e-iqo(a,y)/h XAY) 4 Y ) dY k = 1

+ ei+'o(a,y)'h (daxj~,(y)) U ( Y ) d~ 1 hI

def

= A j U + B j U .

Exactly as in the proofs of Proposition 2.1 and Lemma 3.3, one can show that the second term Bju of the r.h.s. of (4.1) satisfies:

(4.2) IlmBjuII L : ~ - - @(e(~-sa.)/h I1 2.l I1 HZ) . Moreover, since

Supp V p = {@(a) 2 5 i k ' } , and

222 Math. Nachr. 167 (1994)

and thus, just as above,

(4.5)

with Ciu satisfying the same estimates (4.4) as Cju.

pj(cr) d a e J ~ l u ( a ) = S pj(c() (axJ - y j ) eiqo(a,y)/h X a ( Y ) U(Y) dY + c ; u >

B a X , ~ 1 u ( a ) = (arj + 2ipj(a) Dq,) ~ 1 u ( a ) + Rj,ou(a) 9

We deduce from (4.3) and (4.5) that

(4.6)

where Rj,,u also satisfies (4.4) as Cju. Moreover, integrating by parts with respect to y, we see that

As before, this gives by (4.5)

(4.7) TI (D”yJu) (a) = (atJ + 2ivj(a) DmJ) ~ l u + ( K j , o - Rj,o) u 3

where Rj,,u also satisfies the estimates (4.4). By (4.6) and (4.7), the statement of Proposition 4.1 is proved for the T,-part of T. Let us now study the case of T I (1 I I I M), where zl = -.

Noticing that

Yl - %I

R,(%)

Supp VR,(a,) c {IRe a,[ 2 %Ik1} c {@(a) 2 5Ak1} ,

we get this time, using the same arguments,

h D a x J ~ , u ( a ) = (a<, + 2ipj(cO DacJ) T,,U - ~ 6j,IT,u + R j J U

iR,(%) (4.8)

= T[ (D~~ I , J ) + Rj,,u 3

where Rj,,u and Rj,,u satisfy the estimate (4.4) as Cju.


(Note that, by (4.5), we also have T,,u = -Buar,Tlu + R;u with R;u satisfying (4.4)). This finishes the proof of Proposition 4.1. 0

The following generalization will also be useful1 in Section 6.

Proposition 4.2. For any f i E IN", denote p . Daz = (plDuz, , .. ., pnDucJ. For any u E C?@In) one then has

(4 D!xTlu = Tl(6!u) + RB,p = (ar + 2ip(a) . DUJB Tlu + l?B,ou

(ii) for any 1 = 1, ..., n,

Proof. We prove (i) by induction on IpI. Since the result is immediate for P = 0, we can assume for instance that f i = (1,0, . .., 0) + y with y E IN". Applying Proposition 4.1 with u replaced by Diu, we get

(4.9) T,@u = BazlTID$ - R,,,D$

= (arl + 2ipl(a) Dab,,) T,D$ + (I?,,, - .

Moreover, looking at the proof of Proposition 4.1, we see that the operators Rj,l and are all of the type

u ~ T , u + K u , with

(4.10) )ImKullL:, = fl(e

and a E

II I1 ";I ( E - S d / h

supp a c {@(a) 2 4 P } ,

Making several integrations by parts, we then obtain that Rj,J6; and Rj,,@ are of the type

u H T,u + K u

224 Math. Nachr. 167 (1994)

with K satisfying (4.11), and o = o1 + ho, E &A,r,

SUPP o = {e(a) 2 41"') ,

1% - Y12 ;(Re a)1y1+'>, R(Re a,)'

0 1 = 0

Applying Proposition 3.1, the identity (4.9) permits to conclude (i). (ii) Proceeding in the same way, we get from Proposition 4.1

i'$!~ = D"a,,T,llJ$ - Rl,lD$

h = (a<, + 2 i / ~ 1 ( a ) B ~ ~ , ) T,,BJu - 7 61, lTlB;~ + (81.1 - R ~ , J D ; u ,

1

where the remainder terms are of the same type as above. Then, the induction assumption and the result for Ti permit to conclude the statement. 0

5. Local exponential weighted estimates

Let now y E C"(A,,; IR) be such that

v(4 = s o

is constant on

As explained in the introduction, we want to study e'+'IhTP where P is some differential operator. For this, in the next section we shall need to know how to estimate ewIh Tbu when b is a function in dA,, compactly supported with respect to the a-variables. This is precisely the goal of this section.

First of all let us fix 6, > 0 arbitrarily small, and denote

Ts, = { a E C'"; IIm a,l < G,R(Re ax), IIm arl < 6,r(Re ax)]

and also (extending y in a natural way to a function in C"(Tao; lR), e.g., by considering it as a function of Re a only)


Moreover, we assume that A is taken sufficiently large so that

(5.2) s, 2 6, + SUPW r60

and for 1 E N we define gA,, as the space of functions b = b(a, y) in C"(T6,, x IR") which can be extended to holomorphic functions with respect to the y-variables, bounded along with all their derivatives, in Tao x D,(y).

The main result of this section is:

Proposition 5.1. Let E > 0, b = b(a, y ) E gA,, be supported in (@(a) < (4 + E ) ,Ik1}, and

Then if. and t are small enough, one has for some 6 > 0 and for s < O arbitrary m E C" (Ts,; IR?) satisfying (3.3).

~lm ew/h %ullL;, = ~m ewih + lIUIIH:( iv i<~w))

uniformly for h > 0 small enough, and u E C$(IR").

Proof. Assume first t = 0. Then the idea is to compare Gu with

Op(b(x , 5, x - 5*)) TIU

where Op denotes the usual (1, 0)-quantification of h-pseudodifferential op6rators on IR'" (in the terminology of [Ho]), and <* is the dual variable of t.

More precisely, let x4 E C$ (I'6o), satisfying

SUPPX4 = {@(a) 5Ak11 and

x4(a) = 1 on {@(a) I (5 - E ) P } .

For (x, 5) E IR'" we then have

(5.3) O P ( W , 5, x - t*)) (x4T14 (x, <)

Since p(a) = po on Supp x4, we also have for (x, y) E IR'"

= x4(x, y) eixqih fi,(y) (2nh)",

where

and fix denotes the usual Fourier transform of ox, with small parameter h. Inserting (5.4) in (5.3), we obtain

e i ( t - d t * / h f i r n / h b(x 5 - y) ( OP(b(X, t, x - <*If (X4TIU) (.% tf = s 9 , 4 x> y) U r f dg d5* !

15 Math. Nachr., Ed. 167

226 Math. Nachr. 167 (1994)

and thus, by the change of variables [* + + y = x - t*,

2 e ( x , '1) - e(x, t) 2 (5 - E ) Ak' - (4 + E ) ilk' 2 + Ak' . Moreover, since 151 I (3 + E ) ,Ik1, we deduce from this

and thus in any case

on SUPP b(x, 4 , Y ) n SUPP (x4(x, rl - 1). We next consider the complex contour:


and we can see that for (x, 5, q) E Supp b(x, 4, y) n Supp (x4(x, q) - l), we haver;,, c DI, (X ,5 ) . Indeed, we have on this set

5 2 . 4 + c < l + - - 5 - &

Therefore, by the holomorphy assumption on b(x, 4, .), we can make a change of contour of integration in (5.8) (interpretated as an oscillatory integral, which can be made convergent as Iy'l -+ 00 by a suitable number of integrations by parts with respect to q), and integrate instead on ri,q. This gives

(5.10)

and thus, integrating first in y', and using (5.9) we obtain

r(x, 4, y ) = O(e-6Ak1/16fi+h 1 9

which implies

(5.11)

locally uniformly on lR3n, and with some 6 > 0. We also get the same kind of estimate for any derivative of r.

r(x, 5, y ) = O(e-(sA+a)/h)

Moreover since, ~(x, 4, y) u,(y) is supported in the set

we get from (53 , (5.7) and (5.11)

(5.12) Op(b(x, 4, x - 4*)) x ~ T ~ u = Tbu(x, 4 ) + Ru(x, 4 ) with (for some 6 > 0 and any s 5 0)

(5.13) I W U I I L 2 ( R Z n ) = O(e-SA'h IIuIIH;(lyl<~kl/aJ .

Ilm eWlh TbUIIL2 = Ilm ewlh OP(b(X7 4, x - <*)) x4~1ullI2 + 07(IIUIIHS,(lyl<Ikl,g)).

In particular, using (5.2), we get

(5.14)

Then for the case t = 0 it now remains to show that the operator

B, = ewIh Op(b(x, 5, x - t*)) e-Vlh

is uniformly bounded on L2(lR2"; m(x, 4)2 dx dr) = L2(m2 dx dt). We can write

(5.15) Y ( X 9 4 ) - v(x, rl) = ( 4 - rl ) @(x, 4, q)

228 Math. Nachr. 167 (1994)

with

(5.16) @ = (@I, ..., @"), pjl- 5 s u p ld5jy)l . This gives for any u in C;(IR2")

(x, 5, x - 5*) v ( x , y) drl d5* (5.17) B , ~ ( ~ , 5) = (2,&-" J e i (5 - , ) ( e ;* - i~ tx , e , s ) ) / f I b

Thanks to the assumptions on b and (5.15)- (5.16), we can make the change of contour of integration in (5.17)

t* H t* + W X , t, y) *

(This can be justified in a usual way, e.g., by considering the integral (5.17) as the limit in 9'(IR") of the integral obtained by multiplying the summand by e-"lc'12, 8 + O+).

This change of contour gives

B,,,u(x, 5 ) = (27ch)-" ei(s-q)T*ih b(x, 5, x - t* - i@(x , t, y)) u(x, y) dy d(* . (5.18)

Then the semiclassical version of the Calderon-Vaillancurt theorem (see e.g. [Ro]) gives

For the case t + 0, we can write for any LX E AtG and y E lR" (denoting (x, 5) = Re a) the required uniform continuity of B,. This finishes the proof of the case t = 0.

"< - 11 + i td,G(x, y) = (1 + i t G , ( x , 5, rl)) ( 5 - r ) I and then, denoting T,(a) the contour of Cz" given by

r,(4 = {8 = (B,, 8,4; Im 8, = - td,G(x, Re P,,), (1 + itG; (x, t, Re P,)) 8,' ER") (where G; denotes the transposed matrix of GI), we have

Im("g - P,)P** = 0

OP(NLX, a x - 8,*)) (x4T14 (co on r,(a) and we can define again the oscillatory integral

(5.19)

x - B,*) x4(ax, B,) TlU(%, B,) dB, d8,*. - - L j ei(a<-Pn)Pv*ih b ( U , a (27ch)"

rt (4

Moreover, we have the following result:

Lemma 5.2. For any y E IR" and a E AtG one has


Proof. One can interpret the oscillatory integral as the limit in 9’ when l3 + 0, of the convergent integral [,(a, y) obtained by multiplying the summand by e-Biy’iz. Then one can apply the Stokes formula to I,(a,y) between its actual contour of integration and the contour { (y‘, P,) E JR”} (this must be made in two steps, first letting y’ become real and then P,). This gives

(5.20) (a, Y) + M a , Y) + G ( a , Y) z ~ ( ~ , y) = , i (aX-y)adh-@y2 b

where IL(a,y) involves 1 - x4(a,,P,) as a factor in its summand, and, because of the holomorphic dependance on y’ of b(a, y’), [:(a, y) involves aB,x4(ax, P,) as a factor in its summand.

Then, using that

while

@(a) 5 (4 + E ) ,Ik1 on Supp b(a, y’) ,

we obtain again on the intersection

la5 - p,I 2 $ / I k l .

Therefore one can prove as for (5.11) (by a new change of contour of integration in y’ such that on the new contour one has Im (a, - y’) (a5 - P,) 2 S , + 6) that 1; and I: are 8(e-‘S”fS)/h) uniformly with respect to 6 > 0, and locally uniformly with respect to (a, y). The result follows by letting 9 tend to 0 in (5.20). 0

Replacing T,u in (5.19) by its expression in terms of u and using Lemma 5.2 and the fact that p(a,, j,) = pLo on Supp x4(ax, P,), one then easily obtains the analogous of Formula (5.12) in that case, with an estimate as (5.13).

Therefore it now remains to estimate in L& the expression given in (5.19) multiplied by the exponential weight ew(a)/h.

Denoting again (x, 5) = Re a, one can write for any y, q in IR” and s E [- Itl, Itl],

where G, and G, are n x n matrices of C“ functions of their arguments. Then we consider the contour of C4” given by

230 Math. Nachr. 167 (1994)

on which (ay - p,) p,* + (a, - p,) by* is real (as previously C) denotes the transposed matrix of Gj) , and we define the oscillatory integral

(5.21) J&)

- - 2 1 ei(at-fln)fin‘jh + i ( n x - f l y ) f i y * / h b(a, M, - p,.) x4(py, p,) TIu(py, &) dp, d& dp, dp,, . (27th)”

r.,t(a)

Using the same kind of arguments as in the proof of Lemma 5.2 (the factor of convergence being e-e(p:* +8:*) in this situation), it is not difficult to see that for s = 0 one has

uniformly (note that Js. t is always compactly supported). Indeed, the term involving ax4 has now its summand supported in lae: - &I + la, - p,I 2 for some positive constant

C C, and the estimate follows by making a change of contour in the variables (by*, &),

In the same way, one can prove by the Stokes formula that for any s

But by construction, one has (by, p,) E AtG on rt , t(a), Then, parametrizing the contour r J a ) by its real part, we see that J t , f ( a ) is again a h-pseudodifferential operator acting on Tlulnc,, the symbol of which is uniformly bounded along with all its derivatives, and holomorphic with respect to the dual variables. Since the domain of this holomorphy is large enough, one can prove as before that the operator is continuous on the space L2(Atti; ezcwPtH)lh da), uniformly for h > 0 small enough. Thanks to (5.22) and (5.2), this ends the proof of Proposition 5.1. 0

6. The main theorem

We now apply the results of the previous sections to obtain global exponential-weighted

More precisely, let estimates in phase space for differential operators with analytic coefficients.


be a differential operator on IR" whose coefficients aY can be extended to holomorphic functions in D,, defined at the begining of Section 5 (see (5.1)) and satisfy for some q E IR (6.2) a,(z) = O ( ( R ~ Z ) ~ )

uniformly for z in D,,w. In fact, because of the particular conic shape of DA,w, and thanks to Cauchy Inequalities,

we can also assume without loss of generality (possibly after having slightly increased the p l , i s ) that for any y' E IN" one has

(6.3) aY'a,(z) = @((Re z ) ~ - - " ' ~ )

uniformly in D,,w, Then one has for any z , z' E D,,w

(6.4) a,(z) - a,(z') = ( z - 2') A,(z, z')

where A, is holomorphic in D,,w x DA,w. Indeed, one has 1

A,(z, z') = 1: (tz' + (1 - t ) Z ) dt 0

and thus A , satisfies for any y' E INz"

(6.5) aY'A,(z, z') = @(((Re z ) + (Re Z ' ) ) ~ - I ~ ' I - ' )

uniformly on {z , z' E DATV; Re z . Re z' 2 0) . We also strengthen Assumption (5.2) by

(6.6)

Our main result is:

s, 2 6, + 2 SUP y .

Theorem 6.1. Under assumption (6.1)- (6.3) and (6.6), there exists 6 > 0 such that for any u, u E C:@"), any functions m, m' E C"O(Tsu; R*,) satisfying (3.3), any s E IR, and any t E R small enough, one has

( m ewih TPu , ewih TV),;~

= (rn(a) p(a, - a,,,w(a), a< + 2i diag (po ) a,,,y(a)) ewlh Tu, ewlh T u ) ~ ~ tG

uniformly with respect to u, v E Czw) and h > 0 small enough. Here

P ( a x , = c ay@A I Y I 5 d

is the symbol of P, diag (po ) denotes the diagonal n x n matrix whose coeflicients are the po,;s (1 5 j < n), and a,,, is a n x 1 matrix of derivations, of the form:

232 Math. Nachr. 167 (1994)

where F , and P, are n x n matrices depending smoothly on a, can he expressed in terms of C and p only, are supported in Supp Hess G, and satisfy,for any y E IN2"

ldYF,I + l a y F 1 - ; l = s(lt l)

locally unijormly in C2".

Remark 6.2. As it will be seen in the proof, the weight P(Re a)d (Re appearing in the right-hand side of the above estimate, can in fact be replaced by

P(Re (Re (r"(Re I%) + (Re a,)).

Remark 6.3. Actually, from the explicit form of F, and F, we find in the proof, one can show that the set

Remark 6.4. In view of the applications, notice that if y is given, one can always increase ,t in such a way that (6.6) becomes satisfied. Then the only problem is that D+ increases also. However, one can always increase p1 so that the conic sector containing DA,v at infinity becomes arbitrarily sharp.

Proof of Theorem 6.1. At first, let us study the operator

_ _ a - ( a _ - , ') on A, , . aa act, acts

Denoting

(6.7) M ( a ) =

it is easy to verify, u ng (1.2), that det M ( a ) = 8(1) and that 1 + i tM(a) is invertible on C2" for t small enough.

Now, for any function f holomorphic on CC'. and c( E A,,, one can write

f ( a ) = f(Re a, + itd,G(Re a), Re a< - ita,G(Re a)) = f(Re a) , (6.8) so that

and thus


af am If now f~ Cm(A,,), Formula (6.9) will be the definition of -. In particular, i f f is

holomorphic on C'", one has on AtG

(6.10) a aa - f ( i ) = (1 + itM(a))-' (1 - i tM(a))f '(c?).

Now to prove the theorem one can assume without loss of generality that P = a ( x ) bt

Let us denote with a satifying all the assumptions required on the a,'s.

(6.11) I = (rn ewIh TPu, e"Ih To),;G

- - S m(a) eio(a ,y3 z)lh (4 Y ) 3% 4 XAY) X a ( 4 4 Y ) &(Y) vo da dY dz > A t e x R2"

where

(6.12) d a , Y , 4 = (ax - Y ) a< - (a, - 4 q + iq&, ax - Y )

+ iq,(a, Ex - z ) - 2iyl(a) + 2itH(a).

Using (6.9) and (6.10), we find on {@(a) 5 5Ak1} (where in particular p(a) = po)

~ = (1 + i tM(a)) ( aH av

a(P at + 2i diag (po) (a, - (6.13) a R e a ax - Y

ti, - 2i diag (po) (Ex - z ) - (1 - itM(a)) ) + 2it - 2it ~

a R e a '

and one can verify easily that

so that, insering this in (6.13), we get for @(a) s 5Ak'

~-

'1) - 2it =. ayl

2i diag (po) (a , - (6.14) - (1 + i tM(a)) (

a R e a a, - Y

- (1 - itM(a)) -2i diag (po) (a, -

Denoting

we deduce from (6.14) that

acp acp ayl av % E X at% a t d X ate<

(6.16) ~ + 2i diag (po) ~ = -2i ~ + 4 diag (po) __ + &(a) (a, - y) ,

234 Math. Nachr. 167 (1994)

where B,(a) is the n x n matrix defined by

for all z E (c" and denoting

Using the fact that (1 - itM(ct))-' is the transposed of the co-matrix of 1 - itM(ct), we see that (1 - itM(a))-' is also of the type

and thus by (6.17)

(6.19) B t ( 4 = 4i diag (PM) + QWcl(a)l).

In particular, B,(a) is uniformly invertible if t is small enough, and denoting

a a at% a,%

(6.20) a,,, = 2iB,(a)-' ~ - 4B,(01)-' diag ( ~ ( a ) ) -

we see that Formula (6.16) can be rewritten as

h 2

(6.21)

for any c1 E & satisfying @(a) I Saki.

(a, - y - i?p,,tp(a)) eip'h = - - dp,t(eip'h)

Moreover, using (6.9), (6.15) and (6.19), we also have

Applying (6.4), (6.5) to a, we get in particular

(6.23) 4 Y ) = 4% - a,,,w(M)) + (a, - Y - a,,,w(a)) Y) 5

where A is holomorphic with respect to y in D,,,, and satifies for any y E IN2"

(6.24) (iy A(a, y ) = O(R(Re aJ-IyI-' 1

uniformly for (01, y) satisfying y E D,,,(w), and

uniformly for (01, y) satisfying @(a) I 5Ak1.


Formulas (6.21) and (6.23) will be essential for controlling the part of I involving the

Now let us consider a cut-off function x5 E Cz(rao) such that region where @(a) I 5,1k1.

SUPPXS = { e ( 4 (4 + Wkl)

xs(a) = 1 on {@(a) I 4dk1},

and

where E > 0 will be fixed small enough later. Using (6.23), we can write

I = I , + I , + I 3

with

(i) Estimation of Z,

On Supp (1 - xg), y = So is constant and thus a,,,y = 0 there. Moreover, one can see as for (6.14), but now with p(a) not constant, that we have on this set

As for (6.21), we deduce from this and (6.22)

236 Math. Nachr. 167 (1994)

Using the operator a,,,, we can make an integration by parts in I , with respect to M.

Making also several integrations by parts with respect to y and using Proposition 3.1, we deduce easily from (6.24) and (6.26)

x [Ilm' e S o ' h ~ ~ I l , . : G ( P ( 0 ) > i k 1 / 4 ) + ll4 H3R") l 9 ) where b > 0 is small enough.

Making again integrations by parts with respect to y and using the fact that yi(a) = So i k i

for @(a) 2 -, we get finally from Proposition 3.1 4

(6.27)

(ii) Estimation of 12.

On the support of x 5 the relation (6.21) holds. Therefore, denoting A = (Al, ..., A,) and d W , , = (a;, 1, . . . , a;,,), we get by an integration by parts

where the last term R(u, u) involves 8,,,(x,(y) x,(z)) as a factor in the summand, and where we have denoted

Denotef'(y) = (C + y')" -4 ) i 2 where C = C(A) is so large thatfis an holomorphic function in DA,v Then by (6.24'), and i f we choose E small enough in the definition of xs, we can apply Proposition 5.1 to Tfbl, Tfbz,J and Tfba,j. In fact, we have first to integrate by parts with


respect to y , so that the derivations on u disappear (this multiplies the summand by some powers of Re a, and Re at, but this has no importance here since the summand is compactly supported).

Using also estimates on R(u, v) as in the proof of Proposition 2.1, and denoting

g,-,(y) = l / f ( y ) = (C + yz)(4- '"2,

we then obtain

(iii) Estimation of ZE

We have

I, = (m(a) a(a, - dp,ty(a)) ewih T&, epih TV),;~

and thus applying Proposition 4.2

(6.29) I, = (rn(a) a(a, - dp,ty) evih (ar + 2ip(a) . BJ Tu, eWih TU)~:,

where R , (u, v) satisfies

\

[ I l r n ' epih 'ITVll L:, + l l u l l H;(R")l ' 1 Using again Proposition 4.2 (i), we get from (6.29)

I, = (m(a) u(a, - d,L,ty) ewih (ar + 2i diag (p(a)) Tu, ewih Tu),.;~

+ RZ(4 0) 9

where Rz(u, v) satisfies the same estimate (6.30) as R,(u, v). Making several integrations by parts with respect to y , this gives as before

(6.31)

where R,(u, u) also satisfies (6.30).

I , = (m(a) U(M, - d,,,y) evih (a5 + 2i diag (p(a)) Tu, ewih T v ) ~ ; ~ + R,(u, v) ,

238 Math. Nachr. 167 (1994)

In the particular case p = 0, we get (summing up the estimates of (i), (ii), (iii))

(6.32)

(m evIh T(au), evih Tu),;,

= (ma(a, - dr,(y) eW/" ?Tu, evih Tu),:,

for any a holomorphic as before, and growing at most as (x), at infinity. Taking u = au, m' = 1, replacing m by m2 in (6.32), and applying twice the estimate, this gives by elementary algebraic manipulations

/ / m evIh Tau// 2 2 = //ma(a, - ar,ry) ewIh Tu// 2 2 LtC tC

+ @(h[llm(Re a x 1 ev'hT~ll;:G + l l ~ l l $ ( R " ) l ) + ~(h[Ilmeu"hTg,-,ullL:, 2 + IIuIIH;(R")l). 2

Applying this estimate with a replaced by the function g,(y) = (C + y2)4i2 with C as in the definition of g,- 1, we then get after q iterations

Ilm ev'hTg,ull~;G = 0(h[llm(Re a,), evihTu1122 L,G + Ilull$(an)l).

As a consequence, the estimate (6.28) can now be modified into

(6.33)

1, = 8 (" [ (( (Re a,)q-1 ev/hTu + IlullHi(Rn) [Ilm' ev/hTullL;, + llvllHS,(R*)l . m I1 L:G 1 )

In view of (6.27), (6.33), and (6.31) (and remembering that p(a) = ,uo on Supp vy), it becomes clear that we finish the proof of Theorem 6.1 once we establish the following:

Lemma 6.4. One has for any j E (0,1, ..., n} and any f i EN" (denoting T~ = 1)

(m elvih (ar + 2i diag (,u(tl))

= (m ewih (ar + 2i diag @(a)) d,,,y(a))B T,u, evih Tju),fG T j u , evih T,,u),:,

Proof. We proceed by induction on [PI and we first study the case j = 0. Since the result is immediate for = 0, we can assume for instance

p = ( L O , ..., 0) -I- y


where Rl(u) is a remainder term satisfying estimates as in Proposition 4.2, Moreover, a direct computation on formula (1.10) gives

Therefore, using the same procedure as above for the estimations of I , and I,, and in particular using formula (6.21), we get

In particular, applying this last estimate with u = up this gives easily

and thus, by iteration,

240 Math. Nachr. 167 (1994)

for any p in W". Now, applying (6.37) to y instead of p, and inserting it in (6.36), we get

(6.38) ( m ewih TluB, Tlu)L;G

= ( m ewih (ar + 2ip1(a) (d,,r~)J T1u,, evih Tlu)L;G

Again iterations of this formula give

(6.39) ( m eVih Tlus, evIh T,V),:~

= ( m ewih (ar + 2i diag (@(a)) ( a , , , ~ ) ) ~ Tlu, ewih Tl~)L;G

which implies the required result for TI by a new application of Proposition 4.2. When j =+ 0, the computations are very similar except for the extra term appearing in

Proposition 4.2 (ii). But since this term involves only TI , the previous case applies to it and the required estimates follows along the same lines. We leave the details to the reader.

This ends the proof of Lemma 6.4, and therefore of Theorem 6.1.

We end this section with a useful corollary:

0


7. An application

In this section we apply Theorem 6.1 (or rather its Corollary 6.5) to get an upper bound on the width of resonances for systems on L2(lR") @ L 2 w ) of the form

where R(x , hD,) is a symmetric 2 x 2 matrix of second order differential operators with analytic coefficients, V,V, are real analytic functions, and they satisfy the five following assumptions (denoting p j ( x , 5) = 5' + l$(x)):

(Hl) There exist I , 5 1, E IR, such that the coefficients of R(x , hD,) are C ~ ( ( X ) ~ ' ~ and for some C > 0 and any 1x1 2 C :

1 C - ( x ) 2 ' J I Il$(x)l I C(X)2", 0' = 1,2).

(H2) v, 2 0 , v; '(0) = (0) , V i ( 0 ) > 0 .

(H3) There exists a global escape function for p1 at the energy level 0, i.e., there exists G E Cm(IR2") satisfying (1.2) with r(x) = (x)'l, and such that for some positive constant C ,

ac av, ac 1 ax ax at c, H,,G(x, 5 ) = 25 - - - - 2 - (x)2ll

uniformly on {p l ( x , 5 ) = O}.

(H4)

This last assumption is not really necessary, but when it is not satisfied the usual techniques (involving in particular Agmon-type inequalities in the x-variables) can be applied, and the result follows more easily (see e.g. [Kl]). Moreover, (Hl) can be replaced by a more general assumption involving general weight-functions rj (x) instead of (x)"' (e.g. in the example (0.1) one must take r l ( x ) = (x,)) .

Assumption (H3) implies in particular that P , = - h 2 n + V, admits no resonances near 0 (see [HeSjl]). We also assume

(H5) V,, V, and the coefficients of R(x, hD,) can be extended to holomorphic functions on a domain of the type

( X E IR"; V, (x) = V,(X)) = 8 .

Do = (zEC"; IImzl I C, + 6,(Rez)),

where 6, > 0 is arbitrary, and C, > 0 is large enough. Moreover, V,, V, and the coefficients of R(x, hD,) satisfy the estimates of (Hl) on Do.

The way in which C, must be taken large enough will be specified later. Modifying the escape function G outside { p l ( x , 5 ) = 0}, one can also assume that

SUPP G = {IPl(x, 511 I 4 5 , + (x)2z1))

for some E > 0 arbitrarily small. In particular (1.3) becomes satisfied.

16 Math. Nachr., Ed. 167

242 Math. Nachr. 167 (1994)

Thanks to the ellipticity of P, = -h2A + V, outside {0}, one can define as in [HeSjl] the resonances of P near 0 as the eigenvalues of the realization of P given by

P : f f ( / i t G , 5, + (x)'") x f f ( A , G ; 5' + (x)2'2) H ( / i t G ; 1) x H(AtG; 1) 3

where t > 0 is small enough and H(AtG, rn) can be defined as the completion of C;(lR") for the norm u H IlrnT~ll~;~, ?r being defined as in Section 1 with k , = I , (Actually, these spaces don't depend on the choice of the various parameters p,, pl and A: see [HeSjl].).

It is also possible to show that P admits resonances near 0, and that the resonances of P which are in some box [ - E, Ch] + i [ - E, 01 (with E > 0 small enough and C > 0 arbitrarily large) are close (up to 0(h3'') terms) to the eigenvalues of P , inside [0, Ch]. This can be proved in a very similar way as in [Mall, using also the pseudodifferential calculus on H(A,,) developed in [HeSjl].

Now let d,(x) denote the Agmon distance from x to 0 associated to the potential V, (i.e., the distance associated to the pseudometric V,(x) dx').

For a = (ax, a<) E IR'" small enough and p, = (P,,~, ..., p0,J E (IR*,)n we define n

Y O ( " ; P O ) = c.v.y [(ax - Y ) + i C po,j(axj - yj)' + id,(y)I 7

j = 1 (7.2)

where c.v.), means that one takes the critical value with respect to y of the function between the brackets. Here the critical point y = y(u) is given by

(7.3) 2 diag (p,) y + V d , ( y ) = 2 diag (p,) a, - ia,

Since d , is analytic near 0 and Vd,(O) = 0 (see e.g. [HeSj2]), we see that (7.3) admits a

Denoting unique solution when a is small.

we also have, using the fact that y(a) is critical, -

(7.5)

This means that the function

d r o ~ o = f diag (po)- a t .

i n 1 $,(a) = W O ( 4 - - c - a;,

4 j = 1 p0. j (7.6)

depends holomorphically on the complex variable

z(a) = 2 diag (p,) a, - ia,

for a E R2" small enough. We then denote by 9 the family of the connected bounded open sets Q c R" satisfying

0 E Q = {Pl(X, 5) < 0)

and $, can be extended to an holomorphic function in {2 diag (p,) x - i t ; (x, 5) E a}. Our choice of the function yo is justified by the following.


Proposition 7.1. Let y 1 = Im yo and assume that the constant C , of assumption (H5) is large enough. Then ,for any R E F and any a E R one has

PAM, - a,,y,,(a), + 2i diag (pol d,,,w,(a)) = 0 .

1 . a a 2 aa, aa, Here a,, = - diag (p0)-l - + i -.

Proof. Assume first that a is small enough. Using that y(a) is critical, we get

(7.7) a,,wo(a) = 3diag(p0)-' + W a x - ~ ( 4 ) and thus

a,,G0(a) = diag (p0)- l a{ + 2i(a, - y(a)) .

Moreover, since G0(a) is holomorphic with respect to z(a), we have

depends holomorphically on z(a). As a consequence, the identity (7.10) extends to all of R belonging to 9. 0

Notice that, by (7.9) and (7.11), the function CL H y(a) can also be extended to all of 52 belonging to 9, as an holomorphic function of z(a).

244 Math. Nachr. 167 (1994)

We also set for any Q E .F

e(Q, po) = SUP e > 0; VS E [o, el, 3c > 0 such that va E Q:

PI(^, - S & W ~ (4,

~ ~ ( a , - S ~ ~ , , W ~ ( ~ ,

1

C

i + 2is diag (p0) ~povl(4)l 2 - ,

+ 2is diag (pol ap0w1(a))l 2 - laI2 C l i

so that e(R, po) E (0, 11 by PI Jposition 7.1, and a straightforward calculation shows that

(7.12) H(R, po) + 1 as diam (Q) + 0 (Q E 9).

In fact, it suffices to work with the second order Taylor expansions of V, and dz, and to rewrite the computations made in [Ma31 Section 4. The same argument also gives the existence of some positive constant C such that

1

C (7.13) V l ( 4 2 - 14

uniformly for a small enough. We now define (denoting a 0 the boundary of a)

So = Sup Sup Inf B ( 0 , po )w l (a ) . ,L~E(R: )~ Q E F

By (7.13), we see that So is a positive number, and the main result of this section is:

Theorem 7.2. Under assumptions (Hl) to (H5) with C2 large enough, all the resonances e(h) o j P which tend to 0 as h + O , , satisfyfor any c > 0

JIm e(h)l = O(e-z(so-E)/h )

uniformly for h > 0 small enough.

Proof . Let e = e(h) be such a resonance, and let

u = (241, u2) E H(&; r z + (x>z”) x H(&; t2 + ( X p )

such that Pu = eu, u + 0. For E > 0 arbitrarily small, we fix po E (IR*,)” and R E .F in such a way that

~ n f Po) wl(a) 2 so - E a e a o

as well as

Pl = (&, ...&) (lR*,)”,

and we define ‘IT as in Section 1 with this choice of p,,, pl, and with A 2 1 satisfying

SA > 2S0.


We then normalize u in such a way that

(7.14) II~luIIL:G@L:G = 1

(this also insures that IITullL;G8L:G = O(1)). We also take xo E C$(sZ) such that 0 I xo I 1, xo(0) = 1 and

In fact, setting

and introducing another small constant 6 > 0, we can take xo of the form

where d denotes the Euclidian distance, and xa E C$([O, E ) ) satisfies xa = 1 on [ 0 , ~ / 2 ] , xb < 0 and Isxb(s)l 5 6 everywhere (such a function xs can be constructed as a regulariza-

tion of the function f - - 1 where f = 1 on R-, f = 0 on [l - e-'/', + a), and

f(s) = - 6 In ( y + e- '1') on [O, 1 - e- '4). We then have (using the fact that O(Q, po) w,(a) - So + 28 = O(d(cr, 0,)) on Supp Vxo)

(7.15)

(: )

IO(Q, Po) Y1 - so + 24 . IVxol =

uniformly with respect to E and 6. We can also assume

(7.16) Supp xo A Supp G = QI.

Now we set

(7.17) w ( 4 = (1 - 4 [xo(a) w4 Po) V l ( 4 + (1 - x o ( 4 (So - 241

and, assuming that the constant C2 of (H5) is large enough (compared with ,Ik1, k, = Max {l, 11)), we can apply Corollary 6.5 to this choice of y and to u (this is allowed thanks to the density of C;(R") in IT(&)). This gives, denoting also f2(ct) = (Retxe) + (Re ax)12,

246 Math. Nachr. 167 (1994)

for some constant C, > 0 depending on E (but not on 6). Taking 6 small enough compared

with -, we then get from (7.19), (7.20) on Supp Vlc, 1

c,

On the other hand, because of (Hl) and (H3), we have on A,, \ Supp v y ) (and this is also the place where the assumption 1, 5 l 2 is used)

Inserting (7.21) and (7.22) in (7.18), we then obtain

(7.23) 11%42 ew/hTulll;;G + II laI2 e”h~U211;~G(,al),,) + llr;(co2 ew’hTu2112 L:G(lalzl)

2 = @(Ile(h)12 + h) [ I IW’ eoih T u , 1122 tG + IIr“z(.)’ ewIh Tu211t;G + I I ~ I I ~ ~ ~ ~ ~ I

and thus, separating the domains

m4 2 c(imi2 +

m4 I c(tmt2 + h))

and

(C > 0 large enough), and using the fact that y(a) --+ 0 as lal-, 0, we deduce easily from (7.23)

(7.24)

with ~ ( h ) --+ 0 as h + 0 (here, we also used the fact that IlullH; = O ( ~ ~ T u ~ ~ L ~ G ) uniformly for t small enough and u E H(A,,): see [HeSjl]).

Ilf(a)’ e”hTulllt;G + (le(h)12 + h) ev’hTu211;;G = @(le(h)12 + h) es(”)Ih


Since xo = 1 on f@(Q, po) tpl I So - 2~3, we get in particular from (7.24)

(7.25)

with 8 ( ~ , h) + 0 as ( E , h) --, 0.

containing Supp xo. Then, defining 8 as in ( 3 4 , we set

u = S(f(Re a) Tu) E C$(lR").

Ilf(a)' (1 - X ~ ( G ( ) ) ' I T U ~ I I ~ ; ~ + l1f2(a)' (1 - ~ ~ ( a ) ) T u , I l ~ ; ~ = @(e-(So-S(E,h))/h 1

Now let X"E C;(lR2") be another cut-off function equal to 1 in a large enough ball B

Using (7.25), Lemma 3.3 and the fact that 'ITS is uniformly bounded on L& (see [HeSjl] Section 4), we get

(7.26) I l W U - u)I/L;, = IlTS(1 - f)'ITullL:, + o(e-sA/k) IlTulL:, - - Q ( ~ - (So-d(e,h))/h) .

Moreover, if we take the radius of B large enough, but independently of A, one can prove in a very similar way as for Proposition 3.1 that

- ~ ( ~ - 2 S o / h ) . Il'ITS(1 - f ) ~ u l l L 2 ( S U P p X O ) -

Indeed, by Lemmas 3.5 and 3.6, the kernel K(a, /3) of IS has its main contribution on

1 1 \Re a< - Re Pel < ~ + - (f(Re a) + f(Re p))

2 2 and

1 ]Re a, - Rep I < - ( I + R(Re a,) + R(Re p,)) , - c ,

where C, is large. Moreover, if a E Supp xo, then

R(Re/3,) + f(Rep) I 31k' + C'

for some C' > 0 independent of 1. Therefore, possibly by increasing I , we find that

PL(P) = Po(= P ( 4 )

there. Then (3.18) permits to conclude easily. Modulo O(e-z(so-a)/h)-error terms (with 6 arbitrarily small), we can then write

(7.27) e(h) = (TlPU, Tlu)L;G = (TlPU, ~ l ~ ) L 2 ( s u p p X o )

= (TlPU, TIv)L2(suppxo).

n

Now we introduce the Bargman transform of u denoting q,&) = po,jxf ( j = 1

T(u)(x, 5 ) = f e'(~-~)r/k-9ro(~-y)/h O ( Y ) dY '

II TlU - m L2(SUPPXO) = o(e-Sa'h)

Since p(a) = po is constant on Supp xo, we have

(7.28)

248 Math. Nachr. 167 (1994)

We also have by construction

= O(e-SA/h)

uniformly, and since u is also compactly supported, it is then rather easy to prove (possibly by increasing again 2, and thus also the constant C , of (H5)) that

(in a region where x is too large, the estimation follows directly from the definition of 2 while in a region where only t becomes large, it suffices to make a change of contour of integration in the y-variables and to apply the Stokes formula).

On the other hand, if we set

with d ( t , e, h) + 0 as ( t , E, h) -+ 0. We deduce from (7.29)- (7.30)

1 1 To I I ~ ~ ( ~ ~ ~ , ~ ~ ~ ~ = o(e- (So-d(t,Ea h ) ) / h 1 .

(7.31) d h ) = (TPU, ~U)L2(Suppxo)

The same estimate can be proved for Pu, and we deduce from this and (7.27)-(7.28)

= (TPU, Tv)L2(RZn)

= Chm( Pv, u ) L ~ ( w n )

= Chm(u, P u ) L z ( R n )

~

= e(h) 9

where C = C(n) > 0 and m = m(n) are some constants. The equivalence of Q(h) and e(h) given in (7.31) finishes the proof of Theorem 7.2. 0

Remark 7.3. In the special case of the example (0.1) where V,(x) = -x, - 1 and

. Moreover, it is , which corresponds to = ~ V,(x) = (XI', we find So = ~

proved in this case that 8(Q, ,u0) = 1 for any SZ included in Supp xo (see [Ma31 for the details). Unfortunately, we didn't succeed in generalizing this last property to the situation of Theorem 1.2 although we suspect this should be possible.

1 + p 3 - p 4 4

Martinez, Complex Interactions in Phase Space

Appendix A

249

We give the proof of Lemma 3.4. First of all, notice that:

?(Re a) 1 2 R(Rea,)

, and thus If @(a) 2 6Ak1, then pk(a) 2 - pi,k

If @(a) 5 61k1, then &(a) 2 $ p o , k , and thus

We deduce from this (distinguishing 4 cases):

Case 1: @(a) 2 6Ak1 and e(P) 2 6Ak' In that case, we have by (A.l)

1 /Re a, - yl ~ (2 + R(Re a,))

4C 1

and

so that (using the fact that 3 (1 + 1x1) 2 R(x) 5 (1 + 1x1))

and thus by (A.3)

Taking t small enough, we get

250 Math. Nachr. 167 (1994)

Case 2: @(a) 2 61k1 and Q(P) I 6Lk1 In that case, applying also (A .2 ) to P, we have on the support of ~ . , ~ ( y ) (and for t small

enough)

6Lk1 L 3 I-+- + ~ "1.

P 0 , k 2clPl,k p1.k

Case 3: @(a) I 61k' and e(P) 2 61k1 This is just similar to Case 2, exchanging OL and P.

Case 4: @(a) 5 6Ak1 and e(P) 5 61k1 Using (A.2), we have in that case

In every case, one can see from (3.11) and (3.1) that we have: ToI,B c DA,u . 0

Appendix B

We give the proof of Lemma 3.5. On

{ X U J * 1 1 n SUPP X A Y ) X S W we have either

(B. 1 )

or

(B.2) ( 4 A i (Re P, - yl 2 ~ - 6 (A + R(Re B,)) .

Assume for instance that ( B . l ) is satisfied. By (3 .14) we have

(B.3)


We then study the problem separately in two regions.

First region: IRe a,, - Re p,,I I (2 + R(Re a,)) for all k = 1, ..., n. We then get by (B.4)

- I4 @@(Re ax) r(Re a,) + N R e P,) r(Re P,) + X,,B(Y) M a ) ld,G(Re .)I + A P ) IdrG(Re P)I) In(% 8)l) 9

where we have also used (1.3) which implies in particular that

F(a) = @(r(Re a,))

on Supp G.

enough Using the fact that _< 1 and that a H p ( a ) is elliptic, this gives in that case for t small

Im + tH(a) - tH(P) 2 n ~ - 26 (2 + R(Re a,))’ @(a) + p - ( P ) ) .

Since R(Re a,) - R(ReP,) on Suppx,(y)xp(y), and using (2.10), (2.11), we get in this

(B.5) (8Al Y region for some y = y(6, A) > 0 and E -+ 0 as 6 + 0

(B.6) et(H(P) -H(4 ) lh K , p) = O(e(E-Si-VR(Re ux)?(a)- t lR(ReB,) i (P)) lh) .

Second region: IRe a,, - Re p,,I 2 (A + R(Re a,)) for some k E (1, ..., n]. 2 4c,

Then by (3.16)

252 Math. Nachr. 167 (1994)

Since r(Re a,) - r(Re 0,) and R(Re a,) - R(Re 8,) on Supp x,(y) xB(y) , we can write

and thus, using the fact that

We distinguish four cases:

Case 1: @(a) and e(P) 5 6Ak1.

In that case Min (pk(a), p k ( P ) ) 2 -, and thus, for t small enough we get from (B.7) PLg 2

Case 2 : @(a) I 6Ak1 and e(P) 2 6Ak1. On Supp x,(y) xp8y) we see that

R(Re Px) I 4(1 + R(Re a,))

and thus we get here

Inserting this in (B.7), we get for t small enough

(B.9) lm@lra,a + tH(cc) - tH(P) 2 Min


Case 3: @(a) 2 6Ak1 and e(P) I 6Aki It is analog to Case 2.

Case 4: ,o(ol) and e(P) 2 6Ak' Then

and as in Case 2

Therefore, we get from (B.7), and still for t small enough

Min (F(Re a); i(Re 8)) (A + R(Re a,)),

Now let us see how to estimate K1(rx, p) in each of these cases.

In Case 1, we get immediately from (B.8):

In Case 2, la5[ is bounded, and thus Ipc - a51 - ]Pel as lpcl + + co on &. Then making several integrations by parts in the definition of K , ( a , 8) and using (B.9), we get for any N 2 0:

In Case 3, we just get the same as (B.lO) with at instead of j3< in the right-hand side.

In Case 4, we have to distinguish two subcases: (i) If la5 - 2 ((Re CX<) + (Re Pr)), then, integrating again by parts several times,

and using (B.10), we get for any N 2 0

with some 6, > 0.

(ii) If lag - f15 i I$ ((Reat) + (Rep,)), then F(Rea) - k(Re,G), and thus for some 6; > 0

Min (F(Re a); ?(Re p)} 2 Max (1; 6',(P(Re CI) + k(Re p))} .

Using directly (B.10), this gives in particular that (B.13) is also verified in this case.

254 Math. Nachr. 167 (1994)

Summing up all the cases (and noticing that E and P remain bounded in Case 1; E and /?, remain bounded in Case 2), we get for any v EL:, and any order function m

Appendix C

We give the proof of Lemma 3.6.

In the region IRe at - Re Prl 2 - + - (F(Re E) + F(Re p)), taking t small enough and I 1 2 2 using (1.3) we get from (3.17),

(C. 1) Im + t H ( 4 - W P ) 1

2 ( P - ( E ) + P ~ ( P ) ) (Y - Y&, PN2 + Min { P - ( E ) , P-(P)) (Re a, - Re PA2

(Re eC - Re P J 2 ( I + i(Re a) + F(Re ,8))2 + 6 + (1 - 6)

P+ (4 + /J+ (PI 8 ( P + ( 4 + P+(P))

for any S > 0. The result follows easily from ((2.1). 0

References

B. HELFFER, J. SJOSTRAND: Resonances en Limite Semi-classique, Bull. S.M.F., Memoires no 24/25, 1986, 114 B. HELFFER, J. SJOSTRAND: Multiple Wells in the Semiclassical Limit I, Comm. P.D.E.,

L. HORMANDER: The Analysis of Linear Partial Differential Operators, T. I , Springer- Verlag 1983 M. KLEIN: On the Mathematical Theory of Predissociation, Ann. of Physics, Vol. 178, no 1, 1987, 48-73 M. KLEIN, A. MARTINEZ, R. SEILER, X. P. WANG: On the Born-Oppenheimer Expansion for Polyatomic Molecules, Comm. Math. Phys. 143, 1992, 607-639 A. MARTINEZ: Resonances dans 1’Approximation de Born-Oppenheimer I, J. Diff. Eq., Voi. 91, no 2, 1991, 204-234 A. MARTINEZ: RCsonances dans I’ Approximation de Born-Openheimer I1 - Largeur des rksonances, Comm. Math. Phys. 135, 1991, 517-530 A. MARTINEZ: Estimations sur 1’Effet Tunnel Microlocal, Seminaire E.D.P. de 1’Ecole Polytechnique, 1991 - 92 A. MARTINEZ: Precise Exponential Estimates in Adiabatic Theory, Preprint Univ. Paris- Nord 1993 D. ROBERT: Autour de 1’Approximation Semi-classique, Birkauser, 1987 J. SJOSTRAND: Singularilks Analytiques Microlocales, AstCrisque n “95, 1982 J. SJOSTRAND: Graduate Lecture at the University of Lund, 1985 - 86 (Manuscript)

Vol. 9, (4), 1984, 337-408.

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Estimates on Complex Interactions in Phase Space

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