Table Of ContentON GENERALIZED BOHR-SOMMERFELD
QUANTIZATION RULES FOR OPERATORS WITH PT
6 SYMMETRY
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b A.IFA1, N.M’HADHBI1,2 & M.ROULEUX3
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1 February 2, 2016
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h 1 Universite´ deTunisEl-Manar,De´partement deMathe´matiques,1091Tunis,Tunisia
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m email: [email protected]
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2
v
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2 2 DepartmentofMathematics,CollegeofSciences and Arts
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4 KingAbdulazizUniversity,Rabigh Campus,P.O. Box 344,Rabigh 21911,Saudi Arabia
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. email: [email protected]
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1 3 AixMarseilleUniversite´,Centre dePhysiqueThe´orique, UMR7332, 13288Marseille,France
:
v
i & Universite´ deToulon,CNRS, CPT, UMR7332,83957LaGarde, France
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r email: [email protected]
a
Abstract
We give Bohr-Sommerfeld rules corresponding to quasi-eigenvalues in the pseudo-spectrum for a
1-Dh-Pseudodifferential operatorverifying PTsymmetry.
1 Introduction and statement of the result
Let p(x,x ;h)beasmooth (possibly complexvalued) Hamiltonian onT∗R,withtheformalexpansion
p(x,x ;h)∼ p (x,x )+hp (x,x )+h2p (x,x )+.... Assumethatforsomeorderfunctionm, pbelongs
0 1 2
1
tothespaceofsymbolsS0(m),with
SN(m)={p∈C¥ (T∗R):∀a ∈N2,∃Ca >0,∀(x,x )∈T∗R; |¶ (ax,x )p(x,x ;h)|≤Ca hNm(x,x )} (1.1)
[for instance m(x,x )=(1+|x |2)M], and that p+i is elliptic. This allows to take Weyl quantization
P= pw(x,hD ;h)of p
x
x+y
Pu(x;h)=(2p h)−1 ehi(x−y)h p( ,h ;h)u(y)dydh (1.2)
Z Z 2
whichwedenotealsobyP=Opw(p),or p=s w(P). Wecallasusual p theprincipalsymbol, p the
0 1
sub-principal symbol, andassumethroughout that p isreal.
0
Fix some compact interval I = [E ,E ],E < E and assume, following [6] that there exists a
− + − +
topological ring A ⊂T∗R such that ¶ A =A ∪A with A a connected component of p−1(E ).
+ − ± 0 ±
Assumealsothat p hasnocriticalpontinA,andA isincludedinthediscbounded byA (ifthisis
0 − +
notthecase,wecanalwayschange pto−p). WedefinethemicrolocalwellW asthediscboundedby
par A . This includes the case of the standard Hamiltonian p (x,x )=x 2+V(x), but allows also for
+ 0
moregeneral geometries.
ForE ∈I, let g ⊂W be acompact embedded Lagrangian manifold (periodic orbit) in theenergy
E
surface {p (x,x )=E}, and for N =1,2,···, let KN(E) denote the microlocal kernel of P−E up to
0 h
orderN,i.e. thesetoflocalsolutionsof(P−E)u=O(hN+1)inthedistributionalsense,microlocalized
on g . This is a smooth complex vector bundle over p (g ), where p :T∗R→R. Finding the set of
E x E x
E =E(h;N)suchthatKN(E)contains aglobalsection, amountstoconstruct normalized quasi-modes
h
(QM) (u(h;N),E(h;N)) up to order N. In other words, the condition that determines the sequence of
quasi-eigenvalues E(h;N)=E (h;N)isthatthecorresponding quasi-eigenfunction u(h;N)=u (h;N)
n n
be single-valued. It is known as Bohr-Sommerfeld condition (BS for short). In the sequel, we drop
indexN whenunnecessary.
AssumethatPisself-adjoint, andE <E =liminf p (x,x ). ThenBSdetermines asymptotically
+ 0 0
|x,x |→¥
alleigenvaluesofPinI,bytheequationS (E (h))=2p nh,toanyorderN. Thesemi-classicalaction
h n
S (E)hasasymptoticsS (E)∼S (E)+hS (E)+h2S (E)+.... S istheclassicalaction x dx=
h h 0 1 2 0 Ig
E
dx ∧dx,S (E)=p − p ((x(t),x (t))dt includesMaslovcorrectionandthesubprincipal
Z{p0≤E}∩W 1 ZgE 1
1-form p dt, where t is the parameter in Hamilton equations. Terms S and S are computed using
1 0 1
Maslovcanonical operator, ormorespecifically inthepresent 1-Dcase, themonodromy operator (see
[11] and references therein). A more systematic method (still in the 1-D case) is based on functional
calculus for h-PDO’s, in particular Moyal’s product, and uses a general formula due to [16] (see also
2
[2]forearlierwork). Thus,with
¶ 2p ¶ 2p ¶ 2p
D = 0 0 −( 0 )2 (1.3)
¶ x2 ¶x 2 ¶ x¶x
wehave
1 d 1 d
S (E)= D dt− p dt− p2dt (1.4)
2 24 dE Zg Zg 2 2 dE Zg 1
E E E
Thismethod wasimplemented indifferent waysby[14],[6]andgivenlater adiagrammatic approach
by [5] and [10], providing an algorithm to compute all higher order terms, in particular the 4th order
term can be computed in a closed form without too much work. Note that all S (E) with j ≥3 odd
j
vanish.
Itisshownfurtherin[6],usingtraceformulas,thatBSgivesactuallyalleigenvaluesinI. Notethat
this approach, in contrast withthe method ofthe monodromy operator, assumes already the existence
ofBS,andtheproblem isaboutthemostefficientwayofcomputing theS ’s.
j
Intherealanalyticcase,whenP=−h2D +V(x)isSchro¨dingeroperator,BScanbeobtainedusing
the exact complex WKB method (see [9], [7] and references therein); it consists first in transforming
theeigenvalue equation −h2u′′(x)+V(x)u(x)=Eu(x)intoaRicattiequation, andthencompute Jost
function whosezeroesareprecisely theeigenvalues ofP.
Consider now a h-PDO P (not necessarily self-adjoint) that satisfies PT symmetry i.e. PPT =
PT P, where PT =X I, X is the parity operator Xu(x)=u(−x) and I the complex conjugation.
At the level of Weyl symbol, this symmetry takes the form p(−x,x ;h)= p(x,x ;h). Such a property
issometimesconsidered inPhysicsasanaturalsubstitute forself-adjointness. Itisknownthatfinding
quasi-modes is in no ways sufficient to get information about the spectrum of P, but only about its
pseudo-spectrum (see [9], [8] and [15] for more recent results). The pseudo-spectrum is symmetric
withrespecttotherealaxis,andoneexpectsgenerallytorecoversomerealeigenvalues. Wespecialize
furtherinthecasewherePhasarealprincipal symbol. Ourmainresultisthefollowing:
Theorem1.1. LetPasaboveenjoyPTsymmetry,and p bereal. Then,foratleastN=4,thereexists
0
b∈ S0(m) defined microlocally in W, such that Q= BPB−1,b =s w(B), is formally self-adjoint (at
least modulo an operator with symbol in SN+1(m)). In particular, there is asequence of quasi-modes
(u (h),E (h)) such that (P−E (h))u (h)=O(hN+1), with E (h)∈I, satisfying S (E (h))=2p nh,
n n n n n h n
N
for an asymptotic series S (E)= (cid:229) S (E)hj+O(hN+1) where S ∈R are real. In particular, the
h j j
j=1
pseudo-spectrum of Plies within adistance O(hN+1)ofI. Thecoefficients S (E)canbecomputed as
j
in[10]fromthesymbolofQ;thus
S (E)= x (x)dx= dx ∧dx
0 IgE Z Z{p0≤E}∩W
3
istheactionintegral,
S (E)=p − Re(p (x(t),x (t))dt,
1 Zg 1
E
and
1 d 1 1 d
S (E)= D dt− Re(p )− {{b ,p },b } dt− (Re(p ))2dt
2 24 dE Zg Zg 2 2 0 0 0 2 dE Zg 1
E E(cid:0) (cid:1) E
withD asin(1.3),and
s
b (x,x )= (1− )Im(p )◦expsH (x,x )ds
0 Ig T(E) 1 p0
E
Denoting by T(E) the period of the flow on g . Again, S = 0, and S (E) can be computed using
E 3 4
([5],Formula(7.3))andtheformulagivings w(BPB−1)modO(h5).
Ofcourse, weconjecture thatTheorem1.1holdsforallN.
Example 1.1. Consider the operator Q(x,hD )=(hD )2+p(x)hD +q(x) with smooth, real coeffi-
x x x
cients. Then Q can be mapped into P(x,hD )=(hD )2+q(x)− 1(p(x))2+ih p′(x) by the unitary
x x 4 2
transformation Q=BPB∗, Bv(x;h) =exp(− i xp(t)dt)v(x). Assume Q verifies PT symmetry, i.e.
2h
R
p and q are even on R, then the same holds for P. The microlocal wellW ={(x,x )∈T∗(R);x 2+
E
q(x)− 1(p(x))2 ≤E} for P projects onto the potential wellU ={x∈R;q(x)− 1(p(x))2 ≤E}, so
4 E 4
Theorem1.1holdsprovidedV(x)=q(x)−1(p(x))2 hasnocriticalpointinI.
4
If pandqanalytic, thenthespectrumofPisinfactrealinI andgivenby(exact)BS.Infact,using
alsosomeofthetechnicselaboratedin[12],[4]showedthatifP=−h2D +V(x)+ie W(x)isasmall
perturbation oftheself-adjoint Schro¨dingeroperatorP =−h2D +V(x),thenthesemiclassicalaction
0
is a real analytic function and the roots of BS are real eigenvalues of P. This implies Theorem 1.1 by
choosing e =h,buttheargumentof[4]heavilyreliesuponthatparticular formofP.
2 Proof of the Theorem
Since we know S (E) for 1 self-adjoint operator ([14],[16]), it suffices to conjugate P by an elliptic
h
(but non-unitary) h-PDO so that the resulting operator becomes formally self-adjoint up to order N.
Weproceedinseveralsteps.
4
2.1 Birkhoff normal form (BNF)
LetP˜beself-adjointasin(1.3)with(real)Weylsymbol p˜∈S0(m),andassumethatitsprincipalsymbol
p˜ = p has a periodic orbit g at non critical energy E =0. Then there exists a smooth canonical
0 0 0
transformation (s,t ) 7→ k (s,t ) = (x,x ), s ∈ [0,2p ], defined in a neighborhood of g and a smooth
0
function t 7→ f (t ), f (0)=0, f′(0)6=0 such that p ◦k (s,t )= f (t ). Energy E and momentum t
0 0 0 0 0
arerelatedbythe1-to-1transformationE= f (t ). ThistransformationcanbequantizedtotakeP˜inits
0
semi-classicalBNF.Namely,thereisamicrolocallyunitaryh-FIOoperatorU associatedwithk ,anda
classical symbol f(t ;h)= f (t )+hf (t )+h2 f (t )+···, such thatU∗P˜U = f(hD ;h). Seee.g. [3]
0 1 2 s
attheleveloftheprincipalsymbol,and[13]forthefullsymbol;BNFturnsouttobeconvergent inthe
1-Dcase. Inthecanonical(action-angle) variables(s,t ),s∈[0,2p ],theparityoperatorX:x7→−xon
thereallinetakestheformX:s7→p −sonthecircle. Moreover,wecanchooseU sothatitcommutes
withPTsymmetry:UPT =PTU.
2.2 The homological equation
Westartwiththefollowingelementary result(seee.g. ([15],p.93)):
Lemma2.1. Letqand pberealHamiltonians. Thentheequation
q+{b ,p}=0 (2.1)
hasa(global) realsolutionb alongg iff
E
q◦exptH (r )dt =0 (2.2)
Ig p
E
foranyr ∈g . Itisgivenby
E
t
b (r )=− (1− )q◦exptH (r )dt (2.3)
Ig T(E) p
E
Lemma2.2. Assume p= p asaboveandqisoddwithrespecttoPT ;then(2.2)holds.
0
¶
Proof. Using action-angle coordinates (s,t ) we have p (s,t ) = f (t ), hence H (t,t ) = f′(t ) ,
0 0 p0 0 ¶ t
where f′(t )=w (E)istheenergydependent frequency.
0
Forr =(s,t ),exptH (r )=f (r )=(s+w (E)t,t ). Then,usingtheperiodicity ofq
p0 t
T(E) 1 s+2p
q◦exptH (r )dt = q(s+w (E)t,t )dt = q(s′,t )ds′
IgE p Z0 w (E)Zs
whichis0sinceq(.,t )isoddasafunctiononthecircle.
5
2.3 Reducing to a formally self-adjoint operator
Proposition 2.1. Let p(x,x ;h)∼ p (x,x )+hp (x,x )+h2p (x,x )+···∈S0(m)satisfy PTsymmetry
0 1 2
withreal p . Then at least for N =4, there exists b∈S0(m)elliptic such that Weyl symbol of BPB−1,
0
s W(B),isrealmodO(hN+1). Moreover,BPB−1isagainPT-symmetric(uptothatorder).
Proof. To shorten the exposition, we content to the lower order accuracy O(h4). First we carry BNF
P+P∗
for the self-adjoint part P˜ = of P, which has real Weyl symbol and verifies PT symmetry.
2
P−P∗
SinceU commutes with PT , the anti-self adjoint part also satisfies PTsymmetry (but isnot
2
necessarily inBNF).
Check firstthe Proposition for N =1. LetB have Weyl symbol b , which wewrite as s w(B )=
0 0 0
b0. Let b0 = eb0, with real b 0. By h-Pseudodifferential Calculus (i.e. Moyal product), B0PB0−1 =
[B ,P]B−1+PhasWeylsymbol
0 0
h2 h3 h4
s W B PB−1 = p−ih{b ,p}+ {b ,p},b +i R (b ,a (p))+ R (b ,a (p))+O(h5)
0 0 0 2 0 0 8 5 0 48 8 0
(cid:0) (cid:1) (cid:8) (cid:9)
(2.4)
with
a (p)={b ,p} (2.5)
0
and
¶ 2b ¶ 2b ¶ 2a (p)
R5(b 0,a (p))= (¶ x b 0)2− ¶x 20 × 2¶ xa (p)¶ xb 0+a (p) ¶ x20 + ¶ x2 +a (p)(¶ xb 0)2
(cid:0) (cid:1) (cid:0) (cid:1)
¶ 2b ¶ 2b ¶ 2a (p)
+ (¶ xb 0)2− ¶ x20 × 2¶ x a (p)¶ x b 0+a (p) ¶x 20 + ¶x 2 +a (p)(¶ x b 0)2
(cid:0) (cid:1) (cid:0) (cid:1)
¶ 2b ¶ 2a (p) ¶ 2b
−2 ¶ xb 0¶ x b 0−¶ x¶x 0 × ¶ x¶x +¶ x a (p)¶ xb 0+¶ xa (p)¶ x b 0+a (p)¶ xb 0¶ x b 0+a (p)¶ x¶x 0
(cid:0) (cid:1) (cid:0) (cid:1)
¶ 2b ¶ 3b
R (b ,a (p))=F (b ,a (p)) 3¶ b 0 − 0 −(¶ b )3
8 0 5 0 x 0 ¶ x2 ¶ x3 x 0
(cid:0) (cid:1)
¶ 2b ¶ 3b
−F˜5(b 0,a (p)) 3¶ x b 0 ¶x 20 − ¶x 30 −(¶ x b 0)3
(cid:0) (cid:1)
¶ 2b ¶ 3b ¶ 2b
+3G5(b 0,a (p)) 2¶ x b 0¶ x¶x 0 −¶ x¶x 02 −¶ xb 0(¶ x b 0)2+¶ xb 0 ¶x 20
(cid:0) (cid:1)
¶ 2b ¶ 3b ¶ 2b
−3G˜5(b 0,a (p)) 2¶ xb 0¶ x¶x 0 −¶ x2¶x0 −¶ x b 0(¶ xb 0)2+¶ x b 0 ¶ x20
(cid:0) (cid:1)
where
¶ 2b ¶ 3b ¶ 2a ¶ 3a ¶ 2b
F5(b 0,a (p))=3¶ x a ¶x 20+a ¶x 30+3¶ x b 0 ¶x 2 +¶x 3 +3¶ x a (¶ x b 0)2+3a¶ x b 0 ¶x 20+a (¶ x b 0)2
6
¶ 2b ¶ 3b ¶ 2a ¶ 3a ¶ 2b
F˜ (b ,a (p))=3¶ a 0+a 0+3¶ b + +3¶ a (¶ b )2+3a¶ b 0+a (¶ b )2
5 0 x ¶ x2 ¶ x3 x 0 ¶ x2 ¶ x3 x x 0 x 0 ¶ x2 x 0
¶ 2a ¶ 2b ¶ 2b ¶ 3b ¶ 3a
G5(b 0,a (p))=2¶ xb 0¶ x¶x +2¶ xa ¶ x¶x 0 +¶ x a ¶ x20 +a ¶ x2¶x0 +¶ x2¶x
¶ 2b ¶ 2b ¶ 2a
+¶ x a (¶ xb 0)2+2a¶ xb 0¶ x¶x 0 + 2¶ xa¶ xb 0+a ¶ x20 + ¶ x2 +a (¶ xb 0)2 ¶ x b 0
(cid:0) (cid:1)
¶ 2a ¶ 2b ¶ 2b ¶ 3b ¶ 3a
G˜5(b 0,a (p))=2¶ x b 0¶ x¶x +2¶ x a ¶ x¶x 0 +¶ xa ¶x 20 +a ¶ x¶x 02 +¶ x¶x 2 +¶ xa (¶ x b 0)2
¶ 2b ¶ 2b ¶ 2a
+2a¶ x b 0¶ x¶x 0 + 2¶ x a¶ x b 0+a ¶x 20 + ¶x 2 +a (¶ x b 0)2 ¶ xb 0
(cid:0) (cid:1)
HereR (b ,a (p ))isaHamilton-Jacobipolynomialwithintegercoefficients,polynomialinthederiva-
5 0 0
tivesofb 0 upto order 2, homogeneous ofdegree 5(total degree in(¶ x,¶ x ))whencounting altogether
products andderivatives; andsimilarly for R (b ,a (p )). Notethatthese Hamilton-Jacobi polynomi-
8 0 0
alsdependlinearlyona (p). Thefirstordertermofthesymbolisrealiff
{b ,p }=Im(p ) (2.6)
0 0 1
andbyLemmas2.1and2.2thisequation canbesolvedong ,and
E
T(E) t
b (s,t )= (1− )Im(p )(s+w (E)t,t )dt (2.7)
0 Z T(E) 1
0
with
w (E)T(E)=2p (2.8)
Wenoticethatb (.,t )isanevenfunctiononthecircle. Soin(2.4)weareleftwith
0
1
s W(B PB−1)= p +hRe(p )+h2 Re(p )− {b ,p },b +O(h3)
0 0 0 1 2 2 0 0 0
(cid:0) (cid:8) (cid:9)(cid:1)
Chek now the proposition for N = 2. Let B1 have Weyl symbol s W(B1) = ehb1 with b 1 real, and
computeWeylsymbolofB B PB−1B−1. AgainbyMoyalproduct, wegetmodO(h3)
1 0 0 1
1
s W B B PB−1B−1 ≡ p +h p −i{b ,p })+h2 p −i{b ,p }−i{b ,p }+ {{b ,p },b }
1 0 0 1 0 1 0 0 2 0 1 1 0 2 0 0 0
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)
(2.9)
Theequation forb reads
1
{b ,p }=Im(p )−{b ,Re(p )} (2.10)
1 0 2 0 1
and we need to check the solvability condition (2.2). It is fulfilled when q = Im(p ), since this is
2
an odd function on the circle; consider now q = {b ,Re(p )}, in action-angle coordinates we have
0 1
¶b
Re(p )(t,t )= f (t ),so{b ,Re(p )}=−f′(t ) 0. Sinceb is2p -periodic
1 1 0 1 1 ¶ t 0
T(E)
{b ,Re(p )}(s+w (E)t,t )dt =−f′(t )(b (s+2p ,t )−b (s,t ))=0
Z 0 1 1 0 0
0
7
soagainthecompatibilityconditionholdsforq={b ,Re(p )},(2.10)canbesolved,and(2.9)reduces
0 1
to
1
s w(B B PB−1B−1)= p +hRe(p )+h2 Re(p )− {b ,p },b +O(h3) (2.11)
1 0 0 1 0 1 2 2 0 0 0
(cid:0) (cid:8) (cid:9)(cid:1)
Wenoticethatb (.,t )isanevenfunctiononthecircle.
1
NextwelookforB2=OpW(eh2b2),b 2 realsothatB2B1B0PB−01B−11B−21becomesself-adjointupto
O(h3);theequationforb reads
2
{b ,p }=Im(p )−(cid:229) 1 {b ,Re(p )}+1{{b ,Im(p )},b }+1R (b ,a (p )) (2.12)
2 0 3 k 2−k 0 1 0 5 0 0
2 8
k=0
We need to check the compatibility condition for solving (2.12), by the previous work it suffices to
considerq={{b ,Im(p )},b },andq=R (b ,a (p )). Usingagainaction-angle variables, wehave:
0 1 0 5 0 0
¶b ¶ 2b ¶b ¶ 2b ¶b ¶ 3b ¶b
{{b ,Im(p )},b }=3f′′(t )( 0)2 0 + f′(t ) 0( 0)2+ f′(t )( 0)2 0 + f′′′(t )( 0)3
0 1 0 0 ¶ t ¶t¶ t 0 ¶ t ¶t¶ t 0 ¶ t ¶t 2¶ t 0 ¶ t
¶ 2b ¶b ¶ 2b ¶b ¶b ¶ 3b ¶b ¶b ¶ 2b ¶b ¶ 3b
− f′(t ) 0 0 0 −2f′(t ) 0 0 0 −3f′′(t ) 0 0 0 + f′(t )( 0)2 0
0 ¶t 2 ¶ t ¶ t2 0 ¶t ¶ t ¶t¶ t2 0 ¶t ¶ t ¶ t2 0 ¶t ¶ t3
andthecompatibilityconditionisfulfilledforthattermsinceallfunctionstobeintegratedonthecircle
areodd.
For R (b ,a (p )) we proceed similarly. We check again that b is an even function on the circle
5 0 0 2
(i.e. underthetransformationt 7→p −t).
Onceweknowb ,wecomputes w(B B B PB−1B−1B−1)modO(h4),thisgives:
2 2 1 0 0 1 2
1
s W(B B B PB−1B−1B−1)≡ p +hRe(p )+h2 Re(p )− {b ,p },b
2 1 0 0 1 2 0 1 2 2 0 0 0
(cid:0) (cid:8) (cid:9)(cid:1)
1
+h3 Re(p )− {b ,p },b − {b ,Re(p )},b }
3 1 0 0 0 1 0
2
(cid:0) (cid:8) (cid:9) (cid:8) (cid:1)
LetnowB3haveWeylsymbols W(B3)=eh3b3 withb 3real,andcomputeWeylsymbolofB3B2B1B0PB−01B−11B−21B−31.
AgainbyMoyalproduct, wegetmodO(h5)
s w B B B B PB−1B−1B−1B−1 ≡ p−ih(cid:229) 3 hj{b ,p}−h3 (cid:229) 1 hj b ,{b ,p} +h2 (cid:229) 1 h2j {b ,p},b
3 2 1 0 0 1 2 3 j j+1 0 2 j j
(cid:0) (cid:1) j=0 j=0 (cid:8) (cid:9) j=0 (cid:8) (cid:9)
h3 h4 h4
+i R (b ,a (p))+ R (b ,a (p))−i b , {b ,p},b
5 0 8 0 1 0 0
8 48 2
(cid:8) (cid:8) (cid:9)(cid:9)
Theequation forb nowreads
3
{b ,p }=Im(p )−(cid:229) 2 {b ,Re(p )}+1{{b ,Im(p )},b }−1 {b ,p },b ,b +1R b ,a Re(p )
3 0 4 k 3−k 0 2 0 0 0 0 1 5 0 1
2 2 8
k=0 (cid:8)(cid:8) (cid:9) (cid:9) (cid:0) (cid:0) (cid:1)(cid:1)
8
ThecompatibilityconditionisfulfilledforIm(p )and{b ,Re(p )},0≤k≤2asbefore,forthedou-
4 k 3−k
blePoissonbracket{{b ,Im(p )},b },thetriplePoissonbracket {b ,p },b ,b andR b ,a Re(p )
0 2 0 0 0 0 1 5 0 1
(cid:8)(cid:8) (cid:9) (cid:9) (cid:0) (cid:0) (cid:1)(cid:1)
weproceedsimilarly. WededuceS (E)=0.
3
Onceweknowb ,wecomputes w(B B B B PB−1B−1B−1B−1)modO(h5)thisgives:
3 3 2 1 0 0 1 2 3
1
s w(B B B B PB−1B−1B−1B−1)= p +hRe(p )+h2 Re(p )− {{b ,p },b }
3 2 1 0 0 1 2 3 0 1 2 2 0 0 0
(cid:0) (cid:1)
1
+h3 Re(p )− {b ,p },b − {b ,Re(p )},b
3 1 0 0 0 1 0
2
(cid:0) (cid:8) (cid:9) (cid:8) (cid:9)(cid:1)
+h4 Re(p )+(cid:229) 1 {b ,Im(p )}+1{{b ,Re(p )},b }+1{{b ,p },b }
4 k 3−k 0 2 0 1 0 1
2 2
(cid:0) k=0 (cid:1)
1 1
+h4 {b ,Re(p )},b − R b ,a Re(p ) + R b ,a (p )
0 1 1 5 0 1 8 0 0
8 48
(cid:0)(cid:8) (cid:9) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1)(cid:1)
To prove the Theorem, we observe eventually that the knowledge of the symbol of BPB−1 mod
O(h5) ([10],Formula(7.3)) is sufficient to compute S (E) (although this formula was derived in the
4
particular casewherethesymbolofPcontains only p ).
0
3 Extension to operators with periodic coefficients
Asin[6],wereplaceT∗RbyT∗S1,andthehypothesis onPbythefollowing:
• thereisatopologicalringA,homotopictothezerosectionofT∗S1,suchthat¶ A =A ∪A with
− +
A aconnected component of p−1(E ).
± 0 ±
• p hasnocriticalpointsinA.
0
• A is”below”A .
+
Then Theorem 1.1 holds; the only change is that S (E)=0. Again we reduce P to f(hD ;h) as an
1 t
operator onS1.
Acknowledgments: Thesecondauthor(N.Mhadhbi)acknowledges withthankstheDeanshipofScien-
tificResearchDSR,KingAbdulazizUniversity(Jeddah)foritssupport. Thethirdauthor(M.Rouleux)
cheerfully thanks S.Dobrokhotov, for his kind hospitality at Ishlinskiy Institute for Problems of Me-
chanics(Moscow).
9
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