Global L2-Boundedness Theorems for Semiclassical Fourier Integral Operators with Complex Phase

0.35 MB·English

by  Vidian Rousse

L2
Global -Boundedness Theorems for
Semiclassical Fourier Integral Operators
with Complex Phase
Vidian ROUSSE and Torben SWART
8
0
Freie Universit¨at Berlin
0
2

n
a Abstract: Inthiswork,aclassofsemiclassical FourierIntegralOperators(FIOs)
J with complex phase associated to some canonical transformation of the phase space
1 T∗Rd isconstructed. Uponsomegeneralboundednessassumptionsonthesymboland
1 the canonical transformation, their continuity (as operators) from the Schwartz class
into itself and from L2 into itself are proven.
]
h
p
1 Introduction
-
h
t
a We consider semiclassical (ε (0,1] will be the small parameter) oscillatory
m integral operators on Rd forma∈lly given by
[ 1
3 ϕ7→[Iε(Φ;u)ϕ](x):= (2πε)(d+D)/2 ZRdZRDeεiΦ(x,y,η)u(x,y,η)ϕ(y)dηdy (1)
v
where
0
0 Φ is a smooth real or complex-valued phase function,
2 •
4 u is a smooth complex-valued amplitude belonging to some symbol class.
•
.
0 An abundant literature is now available about numerous properties of such
1
operators with different assumptions on the phase Φ and the amplitude u, we
7
will mainly quote articles related to our work sometimes disregarding other
0
: interestingproperties. Thegeneralformulation(1)includesastypicalexamples
v
two families of operators with D = d that have been intensively studied since
i
X the ’70s.
r ThefirstfamilycorrespondstothechoiceΦ(x,y,η)=(x y) ηandconsists
a of semiclassical Pseudo-Differential Operators (PDOs) (see−for·instance [24]).
ThoseturnedtobeveryrelevanttoproduceparametricesofPartialDifferential
Equations (PDEs), for instance of elliptic type, with the help of symbolic and
functional calculus. As far as global results are concerned, Caldero´n and Vail-
lancourt [6] showed a fundamental result: PDOs are bounded operators from
L2(Rd;C) into itself if the amplitude and its derivatives up to some order are
globally bounded.
The second family corresponds to the choice Φ(x,y,η) = S(x,η) y η
− ·
where the real-valued function S is a generating function of some canonical
transformation κ of the phase space T Rd i.e.
∗
κ( S(x,η),η)=(x, S(x,η)).
η x
∇ ∇
1
Theyaretheso-calledsemiclassicalFourierIntegralOperators(FIOs)withreal-
valuedphaseassociatedtoa canonicaltranformation(seefor instance[28]) and
are widely used to produce parametrices and semiclassical approximations for
evolution equations like wave or Schr¨odinger equations. In the case where κ is
the identity, we can choose S(x,η) = x η and recover the PDOs. The local
·
existence of a generating function S for κ(y,η)=(Xκ(y,η),Ξκ(y,η)) around a
point (y ,η ) = κ 1(x ,ξ ) completely relies on the invertibility of the matrix
0 0 − 0 0
(∂ Xκ) near (y ,η ). When it is not the case, we usually call (y ,η ) a
yj k jk 0 0 0 0
turning point of κ in the position representation (this is an expression of the
problem of “caustics” originally exhibited in geometrical optics). To give an
ideaofhowseverethisproblemcanbe,wesimplymentionthateveninthecase
of linear canonical transformationsthe entire phase spacemight be constituted
of turning points (think about κ(y,η) = (η, y)). As implicitly suggested and
−
first performedby Maslov [25], a localchange of representation(for example to
the momentum representation) allows to circumvent this difficulty. However,
this procedure leads to FIOs whose representation by a single integral is often
onlylocal. TheextensionoftheCaldero´n-VaillancourtresulttothoseFIOswith
real-valued phase can be found in [28].
ThestudyofevenmoregeneralFIOsoftheform(1)goesbacktothepioneer
works [14] and [10]. As for L2-boundedness results, local results (compactly
supported amplitudes) were first provenin [14] whereas different type of global
results depending on the assumptions on Φ can be found in [15] or [12] for
D =0, [21], [8] or [29] for D =d and finally [1] for general D.
The aim of this article is essentially to show that for some semiclassical
FIOs with complex-valued phase associated to a canonical tranformation the
L2-boundedness property still holds with ε-independent norm bound. More
precisely, we consider D = 2d, set η = (q,p) and make typically the following
assumptions on phases and amplitudes:
Φ is a complex-valued phase function of the form
•
Φκ(x,y,q,p) = Sκ(q,p)+Ξκ(q,p) (x Xκ(q,p)) p (y q)
· − − · −
i i
+ x Xκ(q,p)2+ y q 2
2| − | 2| − |
where(Xκ(q,p),Ξκ(q,p))isthedecompositioninpositionandmomentum
of the canonical transformation κ(q,p) and Sκ(q,p) is some real-valued
function reminiscent of the action of classical dynamics (see Definition 4
for a precise definition and some properties),
u is a smooth complex-valued function in the symbol class S[0;4d] i.e.
•
α N4d, ∂α u = sup ∂α u(x,y,q,p) < ,
∀ ∈ k (x,y,q,p) kL∞ (x,y,q,p) R4d| (x,y,q,p) | ∞
∈
κ is a canonical transformation of class i.e. the Jacobian matrix Fκ is
• B
in S[0;2d] (see Definitions 2 and 3 for more precision).
The main result now reads.
TheoremUponprecedingassumptions,theoperator ε(Φκ;u)definedby (1)
iscontinuousfrom (Rd;C)intoitselfandcanbeuniquelIyextendedtoabounded
S
2
operator on L2(Rd;C) such that
ε(Φκ;u) 6C(κ) ∂α u .
kI kL2→L2 k (x,y) kL∞
|α|6X4d+1
Recent semiclassical contributions closely related to our work are [22], [2]
and [5] where the authors provide uniform approximations to the solution of
the time-dependent Schr¨odinger equation but all of them considered symbols
compactly supported in some of the variables. One of the major improvement
ofourconstructionandproofis the removalofthis restrictivehypothesiswhich
enforced them to consider unitary propagators only in a truncated way.
More general considerations about non-semiclassical FIOs with complex
phase are given in [26], [27] and [16] for local properties, [7], [32]-[33], [3]-[4]
and [31] for global properties.
To summarize, the approach presented here combines two different global
advantages: the amplitude needs not be compactly supported in momentum
and the problem of turning points does not show up in this setting. Moreover,
we get explicit control in ε for the operator norm and the connection with the
underlying geometry of phase space is rather transparent.
An application of those FIOs to approximated propagators of Schr¨odinger
equationswithsubquadraticpotentialwillbeprovidedintheforthcomingarticle
[30] in the spirit of [22], [2] or [5]. To the authors’ knowledge, this would be
the first mathematical proof of a widely spread and used formal result from
Theoretical Chemistry originally named after Herman and Kluk.
The plan of this article is as follows. In Section 2, we recall the definitions
and main properties of symbol classes, we introduce the notion of action asso-
ciated to a canonical transformationwhich will almost play the role devoted to
generatingfunctioninthetheoryofFIOswithrealphase,weremindthenotion
of diffeomorphism of class as introduced in [13] and restrict it to the case of
B
canonical transformation. In Section 3, we present a first construction of FIOs
associatedtoacanonicaltransformationbasedonthe FBItransformandrelate
it to Anti-Wick quantization. In Section 4, we introduce a more general notion
ofFIOsassociatedtoacanonicaltransformation,showthattheyarecontinuous
fromtheclassofSchwartzfunctionsintoitself(seeTheorem1)andfinallyprove
the main result of the article, Theorem 2 which states the L2-boundedness for
bounded symbols. In terms of technicalities, the proofof the Schwartzcontinu-
ity is comparablewith the correspondingresultfor pseudodifferentialoperators
as presented in [9] or [24] whereas the L2-boundedness result uses the strategy
of [12], [1] and [15].
We close this introduction by a short discussion of the notation we use.
Throughout this paper, we will use column vectors. We will denote the inner
product of two vectors a,b RD as a b := D a b which we extend to
∈ · j=1 j j
vectors of CD by the same formula. The Hilbert norm of CD will be denoted
P
by a := (a a)1/2. The transpose of a (real or complex) square matrix A will
be A| |, wher·eas A := A¯ and I stands for the identity matrix. When dealing
† ∗ †
with diffeomorphisms of T Rd or operators on L2(Rd;C), Id stands for the
∗
identity morphism or operator. We will use the standard multi-index notation.
Following[5]and[22],wewillsometimesusesubscripttodenotedifferentiation.
Thus, for a differentiable mapping F C1(RD,CD), F (x) will denote the
x
∈
transpose of its Jacobian, i.e. (F (x)) = ∂ F (x). As a crucial example,
x jk xj k
3
the factor Xκ in (4) stands for the matrix (∂ Xκ) so that we have the
q qj k 16j,k6d
followingidentity ofcolumnvectors(X.Ξ) =X Ξ+Ξ X. The Hessianmatrix
q q q
of a mapping F C2(RD,C) will be denoted by Hess F(x) and the divergence
x
of a mapping F ∈ C1(RD,CD) by div F.
x
∈
Acknowledgement
Both authors would like to thank Caroline Lasser for fruitful discussions and
comments.
2 Symbol Classes, Canonical Transformations
Following the presentation of [9] and [24], we recall the definition of symbol
classes.
Definition 1 (Symbol class) Let d = (d ) NJ, u(z ,...,z ) a func-
tion of C∞(Rd1 ×···×RdJ;CN) and m =j(m16jj)616Jj6∈J ∈ RJ. 1We sayJthat u is
a symbol of class S[m;d] if the following quantities are finite for any k >0
J
Mkm[u]:=PJjm=1aαxj=kzjs∈uRpdj(cid:12)(cid:12)(cid:12)jY=1hzji−mj∂zαjju(z1,...,zJ)(cid:12)(cid:12)(cid:12)
(cid:12)  (cid:12)
(cid:12) (cid:12)
where z := 1+ z 2. (cid:12) (cid:12)
Weheixtend this d|e|finition to any m R := R + by setting,
p j ∈ {−∞}∪ ∪{ ∞}
for instance with non-finite m ,
1
S[(+ ,m ,...,m );d]= S[(m ,...,m );d]
2 J 1 J
∞
m[1∈R
and
S[( ,m ,...,m );d]= S[(m ,...,m );d]
2 J 1 J
−∞
m\1∈R
and so on.
Remark 1
(i) S[m;d]is naturallyendowed witha Fr´echet space structureand is increas-
ing with m with continuous injection.
(ii) Only few values of m allow collusion of different z : if m= ,0,+
j j
−∞ ∞
S[(m,m);(d,d)]=S[m;d+d].
′ ′
(iii) The class S[( ,..., );d]=S[ ; d] coincides with (Rd;C) the
| |
Schwartz func−ti∞ons on−R∞d. −∞ | | S
| |
(iv) We have S[m;d]S[m;d] S[m+m;d] with continuous injection.
′ ′
⊂
(v) If u(z,z ) S[(m,m);(d,d)], then z mu(z,z ) S[(0,m);(d,d)].
′ ′ ′ − ′ ′ ′
∈ h i ∈
(vi) TocomparewiththesymbolclassesSm introducedbyHo¨rmander, wehave
ρ,δ
Sm (Rd RD)=S[(0,m);(d,D)].
0,0 ×
4
To fix notations, we first recall the definition of a canonical transformation
and the link with symplectic matrices.
Definition 2 (Canonical transformation) Let us consider a smooth diffeo-
morphism κ(q,p) = (Xκ(q,p),Ξκ(q,p)) from T Rd = Rd Rd into itself. We
∗
×
represent its differential by the following Jacobian matrix
Xκ(q,p) Xκ(q,p)
Fκ(q,p)= q † p † . (2)
Ξκ(q,p) Ξκ(q,p)
(cid:18) q † p † (cid:19)
κ is said to be a canonical transformation if Fκ(q,p) is symplectic for any
(q,p) in Rd Rd i.e.
×
0 I
[Fκ(q,p)]†JFκ(q,p)=J where J := I 0 .
(cid:18) − (cid:19)
We specialize here the notion of diffeomorphism of class as presented by
B
Fujiwara in [13].
Definition 3 (Class ) A canonical transformation κ of Rd Rd is said to be
B ×
of class if Fκ S[0;2d]. For any k >0, we set Mκ :=M0[Fκ].
B ∈ k k
Remark 2 The Hamiltonian flow κt associated to a subquadratic Hamiltonian
function h(q,p) (i.e. such that the Hessian matrix Hess h is in the class
(q,p)
S[0,2d]) is of class for any time t (see [13]).
B
Once again, for later use, we establish biLipschitzian estimates and proper-
ties with respect to composition of canonical transformations.
Lemma 1 The subset of canonical transformations of class is a subgroup
(for composition) of diffeomorphisms of Rd Rd. Moreover, ifBκ is a canonical
×
transformation of class , then there exist two strictly positive constants c and
κ
C such that for any (qB,p ) and (q ,p ) in Rd Rd
κ 1 1 2 2
×
c (q ,p ) (q ,p ) 6 κ(q ,p ) κ(q ,p ) 6C (q ,p ) (q ,p ) . (3)
κ 2 2 1 1 2 2 1 1 κ 2 2 1 1
k − k k − k k − k
Remark 3 (3) is a particular case of the notion of tempered diffeomorphism
introduced in [3].
Proof: For κ and κ two canonical transformations of class , we have
′
Fκ′ κ = (Fκ′ κ)Fκ and Fκ−1 = (Fκ κ 1) 1 = J(Fκ κ 1B) J. Thus
◦ − − − †
◦ ◦ − ◦
Mκ−1 = Mκ and Mκ−1 (respectively Mκ′ κ) are bounded by polynomials in
0 0 k k ◦
Mκ (respectively (Mκ,Mκ′)). Finally, with C = Mκ and c = [Mκ−1] 1, (3)
l l l′ κ 0 κ 0 −
directly follows from the Mean Value Inequality applied to κ and κ 1. (cid:3)
−
In analogy with the situation of FIOs with real phase, we introduce now
the notion of an action associated to a canonical transformation as already
suggested in [31]. It will play the role usually devoted to a generating function
Sκ associated to κ (see Section 5.5 (b) of [24]) i.e. such that
(x, Sκ(x,η))=κ( Sκ(x,η),η).
x η
∇ ∇
5
Definition 4 (Action) Let κ = (Xκ,Ξκ) be a canonical transformation of
Rd Rd. A real-valued function Sκ is called an action associated to κ if it
×
fulfills
Sκ(q,p)= p+Xκ(q,p)Ξκ(q,p), Sκ(q,p)=Xκ(q,p)Ξκ(q,p). (4)
q − q p p
Remark 4
(i) The function Sκ always exists and is uniquely defined up to an additive
constant. Wheneverpossibleandunambiguous,wewillchoosetheconstant
of κ=Id so that SId =0. If κ and κ are two canonical transformations,
′
then Sκ′ κ =Sκ′ κ+Sκ and Sκ−1 = Sκ κ 1 where both equalities hold
◦ −
◦ − ◦
up to an additive constant.
(ii) If κ is of class then Sκ is S[2;2d], more precisely Sκ is S[1;2d].
(q,p)
B ∇
Fromnow on,allcanonicaltransformationsconsideredareassumedto be of
class and ε will denote a small parameter such that 0<ε61.
B
3 Anti-Wick Calculus and FIOs
We combine here the presentations of Lerner [23] and Tataru [31] with the
additionalsemiclassicalparameterεinthespiritofSection3.4of[24]. First,we
introduceGaussianwavepacketscenteredinphasespaceandtheFBItransform.
Definition 5 (FBI transform) Let(q,p) Rd Rd andΘ beacomplex sym-
∈ ×
metric (i.e. Θ =Θ) d d matrix with positive definite real part Θ. Wedefine
†
× ℜ
the normalized coherent state gε,Θ on Rd
• q,p
(det Θ)1/4
gqε,,pΘ(y):= (πℜε)d/4 eεip·(y−q)e−2Θε(y−q)·(y−q),
the FBI transform of a function ϕ (Rd;C)
• ∈S
[Wε(Θ)ϕ](q,p) := (2πε)−d/2 gqε,,pΘ|ϕ L2y
(det Θ)1(cid:10)/4 (cid:11)
= ℜ e−εip·(y−q)e−2Θε(y−q)·(y−q)ϕ(y)dy,
2d/2(πε)3d/4 ZRd
the inverse FBI transform of a function Φ (R2d;C)
• ∈S
[Wε (Θ)Φ](y) := (2πε) d/2 gε,Θ(y) Φ
inv −
· L2
D (cid:12) E (q,p)
(det Θ)1/4 (cid:12)
= ℜ e(cid:12)εip·(y−q)e−2Θε(y−q)·(y−q)Φ(q,p)dqdp.
2d/2(πε)3d/4 ZR2d
Remark 5 Defining Tεϕ(y) := εd/4ϕ(√εy) (which is unitary on L2(Rd;C)),
d
we have the scaling formulas
gε,Θ =(Tε) g1,Θ ,
q,p d ∗ q/√ε,p/√ε
Wε(Θ)=(Tε ) W1(Θ)Tε, Wε (Θ)=(Tε) W1 (Θ)Tε .
2d ∗ d inv d ∗ inv 2d
6
We recall, with proof, elementary properties of the FBI transform as pre-
sented in [23].
Proposition 1 If ϕ (Rd;C), Φ (R2d;C), then Wε(Θ)ϕ (R2d;C)
and Wε (Θ)Φ (R∈d;CS). Moreover∈WSε(Θ) : (Rd;C) (R2d∈;CS) is con-
tinuousi,nvextends∈bSy duality to a continuous opeSrator (R→d;SC) (R2d;C)
′ ′
and to a norm preserving map L2(Rd;C) L2(R2d;CS). Finally→weShave the
→
reconstruction formula
1
ϕ(y)= [Wε(Θ)ϕ](q,p)gε,Θ(y)dqdp (5)
(2πε)d/2 ZR2d q,p
which holds pointwise whenever ϕ (Rd;C).
∈S
Remark 6 (5) corresponds to the fact that (gqε,,pΘ)(q,p) R2d is an “overcomplete
set of vectors” of L2(Rd;C) which reads in the “bra-ke∈t” notation
1
(2πε)d ZR2d gqε,,pΘ gqε,,pΘ dqdp=IdL2(Rd;C) =Wiεnv(Θ)Wε(Θ).
(cid:12) (cid:11)(cid:10) (cid:12)
However, thecomposi(cid:12)tion Wε(Θ)(cid:12)Wε (Θ)is onlytheorthogonalprojection onto
inv
the image of L2(Rd;C) under Wε(Θ) (see [24]).
Proof: Remark 5 shows that it is enough to treat the case ε=1.
The Schwartz property follows from the fact that W1(Θ)ϕ is the partial
Fouriertransform(q,y) (q,p)ofaSchwartzfunction(SchwartzinyandGaus-
→
sian in q) and an analogoustreatment for W1 (Θ)Φ. The isometry property is
inv
provenbyintroducinganextraGaussianfactortomaketheintegralsabsolutely
convergent and allow several applications of Fubini’s Theorem. W1(Θ)ϕ 2 is
k k
the limit δ 0 of the integral
→
e−δ22|p|2[W1(Θ)ϕ](q,p)[W1(Θ)ϕ](q,p)dqdp
ZR2d
e−hδ22|p|2−ip·(y1−y2)+Θ2(y1−q)2+Θ2(y2−q)2i
= ϕ(y )ϕ(y )dy dy dqdp
ZR4d 2dπ3d/2(detℜΘ)−1/2 1 2 1 2
e−hδ−22|y1−y2|2+Θ2(y1−q)2+Θ2(y2−q)2i
= δ d ϕ(y )ϕ(y )dy dy dq
− ZR3d 2d/2πd(detℜΘ)−1/2 1 2 1 2
e−|w2|2e−hℜΘq2+δ42ℜΘw2+iδℑΘq·wi δ δ
= ϕ y+ w ϕ y w dydwdq
ZR3d 2d/2πd(detℜΘ)−1/2 (cid:18) 2 (cid:19) (cid:18) − 2 (cid:19)
|w|2
= e− 2 e−δ42[ℑΘ(ℜΘ)−1ℑΘ+ℜΘ]w2ϕ y+ δw ϕ y δw dydw
ZR2d (2π)d/2 (cid:18) 2 (cid:19) (cid:18) − 2 (cid:19)
with the abuse of notation Θz2 for z Θz. Hence
·
W1(Θ)ϕ 2 = (2π)−d/2e−|w2|2dw ϕ(y)ϕ(y)dy = ϕ 2.
k k ZRd ZRd k k
7
Finally,thereconstructionformulafollowsfromthepolarizationofthepreceding
identity:
ψ ϕ = W1(Θ)ψ W1(Θ)ϕ
h | i h | i
= (2π)−d/2 g1,Θ(y)ψ(y)dy W1(Θ)ϕ
(cid:28)ZRd · (cid:12) (cid:29)
(cid:12)
= (2π)−d/2 gq1,,pΘ(y)ψ(y)(cid:12)(cid:12)[W1(Θ)ϕ](q,p)dydqdp
ZR2dZRd
= ψ(y) (2π) d/2 g1,Θ(y)[W1(Θ)ϕ](q,p)dqdp dy.
− q,p
ZRd (cid:20) ZR2d (cid:21)
(cid:3)
WedefineanotionofaFIOwithcomplexphasethatgeneralizestheso-called
Anti-Wick quantization of pseudodifferential operators.
Proposition-definition 6 (Anti-Wick FIO) Let u L (Rd Rd;C), κ a
∞
∈ ×
canonical transformation and Θx, Θy two symmetric d d complex matrices
×
with positive definite real part. We define the semiclassical FIO associated
to κ with symbol u as the linear operator ε (κ;u;Θx,Θy):L2(Rd;C)
L2(Rd;C) such that IAWick →
hψ|IAεWick(κ;u;Θx,Θy)ϕiL2y := [Wε(Θx)ψ]◦κ eεiSκu[Wε(Θy)ϕ] L2 . (6)
D (cid:12) E (q,p)
(cid:12)
ε (κ;u;Θx,Θy) is bounded and (cid:12)
IAWick
ε (κ;u;Θx,Θy) 6 u . (7)
kIAWick kL2→L2 k kL∞
Proof: For any fixed ϕ L2(Rd;C), the right-hand side of (6) is a contin-
uous antilinear form in ψ ∈L2(Rd;C) with norm bounded by u ϕ , so
L∞ L2
∈ k k k k
the Riesz representation theorem applies. (cid:3)
Remark 7
(i) We have
Θx+Θy −1/2
ε (Id;1;Θx,Θy)=(det Θx)1/4(det Θy)1/4det Id
IAWick ℜ ℜ 2
(cid:18) (cid:19)
where the choice of the square root is explained in the appendix.
(ii) ε (Id;u;I,I)is exactly the Anti-Wick quantization of pseudodifferen-
IAWick
tial operators.
(iii) The situation with u = 1 and κ linear has been investigated in detail in
Section 3.4 of [24].
(iv) If κ is a canonical transformation of class , u S[0;2d] and ϕ is in
(Rd;C), then ε (κ;u;Θx,Θy)ϕ (RdB;C) a∈nd pointwise
S IAWick ∈S
[ ε (κ;u;Θx,Θy)ϕ](x)
IAWick
= 1 eεiSκ(q,p)u(q,p) Wε(Θy)ϕ (q,p)gε,Θx (x)dqdp.
(2πε)d/2 ZR2d κ(q,p)
(cid:2) (cid:3)
8
Moreover, ifu S[(0,mp);(d,d)]withmp < d,thislastexpressionequals
∈ −
the absolutely convergent integral
(det Θx)1/4(det Θy)1/4
ℜ ℜ eεiΦκ(x,y,q,p;Θx,Θy)u(q,p)ϕ(y)dydqdp
2−d/2(2πε)3d/2 ZR3d
where
Φκ(x,y,q,p;Θx,Θy)=Sκ(q,p)+Ξκ(q,p) (x Xκ(q,p)) p (y q)
· − − · −
i i
+ (x Xκ(q,p)) Θx(x Xκ(q,p))+ (y q) Θy(y q). (8)
2 − · − 2 − · −
(v) The result also holds if Sκ is not an action associated to κ. However,
the presence of Sκ in the oscillating phase is motivated by stationary
phase arguments. Indeed, if u is compactly supported the integral kernel of
ε (κ;u;Θx,Θy) is given by
IAWick
(det Θx)1/4(det Θy)1/4
ℜ ℜ eεiΦκ(x,y,q,p;Θx,Θy)u(q,p)dqdp
2−d/2(2πε)3d/2 ZR2d
and provides a contribution bigger than O(ε ) if and only if there exists
∞
(q ,p ) with u(q ,p )=0,
0 0 0 0
6
Φκ(x,y,q ,p ;Θx,Θy)=0 and Φκ(x,y,q ,p ;Θx,Θy)=0.
0 0 (q,p) 0 0
ℑ ∇ ℜ
The equation on the imaginary part is equivalent to
x Xκ(q ,p )=0 and y q =0
0 0 0
− −
whereas the one on the gradient of the real part reads
Sκ(q ,p ) = p +Xκ(q ,p )Ξκ(q ,p )
q 0 0 − 0 q 0 0 0 0
[Ξκ+Xκ Θx](q ,p )(x Xκ(q ,p )) Θy(y q )
− q qℑ 0 0 − 0 0 −ℑ − 0
Sκ(q ,p ) = Xκ(q ,p )Ξκ(q ,p )
p 0 0 p 0 0 0 0
[Ξκ+Xκ Θx](q ,p )(x Xκ(q ,p ))+(y q )
− p pℑ 0 0 − 0 0 − 0
whose relation with (4) is obvious.
(vi) With the formal “bra-ket” notation, we have
IAεWick(κ;u;Θx,Θy)= (2π1ε)d ZR2du(q,p)(cid:12)eεiSκ(q,p)gκε,(Θq,xp)EDgqε,,pΘy(cid:12)dqdp.
(cid:12) (cid:12)
(cid:12) (cid:12)
4 FIOs with Complex Phase
Trivialcompositionsofthe FIO ofthe precedingsectionwiththe positionoper-
ator (either on the left or on the right) show that it could be useful to consider
FIOs with symbols depending not only on (q,p) but also on (x,y).
9
4.1 Definitions and Continuity
S
We define the main object of this article: semiclassical FIOs with quadratic
complex phase.
Proposition-definition 7 (FIO) Let Θx and Θy be two complex symmet-
ric matrices with positive definite real part. For u S[(+ ,mp);(3d,d)],
ϕ (Rd;C) and a positive integer k > mp +d, we d∈efine th∞e action of the
∈ S
semiclassical FIO associated to κ with symbol u as the absolutely con-
vergent integral
[ ε(κ;u;Θx,Θy)ϕ](x):=
I
1
(2πε)3d/2 ZR3edεiΦκ(x,y,q,p;Θx,Θy)(L†y)k[u(x,y,q,p)ϕ(y)]dqdpdy
where Φκ is a complex-valued phase function given by (8), L is the first order
y
differential operator
1
L = 1 iε Φκ(x,y,q,p;Θx,Θy)
y 1+ Φκ(x,y,q,p;Θx,Θy)2 − ∇y ·∇y
y
|∇ | h i
and L stands for its symmetric given by
†y
v(y)[L u](y)dy = [L v](y)u(y)dy.
†y y
ZRd ZRd
If mp < d, its integral kernel is given by the absolutely convergent integral
−
1
Kε(κ;u;Θx,Θy)(x,y):= eεiΦκ(x,y,q,p;Θx,Θy)u(x,y,q,p)dqdp.
(2πε)3d/2 ZR2d
Remark 8
(i) As already noticed in [1] or [28], the following property can be alternatively
used as a definition. If σ (Rd Rd;C) is suchthat σ(0,0)=1, we have
∈S ×
[ ε(κ;u;Θx,Θy)ϕ](x)= lim [ ε(κ;uλ;Θx,Θy)ϕ](x) (9)
I λ + I σ
→ ∞
where uλ(x,y,q,p):=σ(q/λ,p/λ)u(x,y,q,p) S[(+ , );(2d,2d)].
σ ∈ ∞ −∞
(ii) For any u S[+ ;4d], the operator ε(κ;u;Θx,Θy) is clearly continuous
from (Rd∈;C) in∞to its dual (Rd;C)I.
′
S S
(iii) For (x,y)-independent symbols u, we have
ε(κ;u;Θx,Θy)=2 d/2(det Θxdet Θy) 1/4 ε (κ;u;Θx,Θy).
I − ℜ ℜ − IAWick
In particular, ε(Id;1;Θx,Θy)= det[Θx+Θy]−1/2 Id.
I
(cid:16) (cid:17)
(iv) To justify the presence of the action Sκ in the oscillating phase, we notice
that in general eεiSκ does not belong to any symbol class:
∂α(eεiSκ) Cε−|α| (q,p)Sκ |α| C′ε−|α| (q,p) |α|.
| |≃ |∇ | ≃ h i
10