Pseudospectra of semi-classical (pseudo)differential operators

22 Pages·0.38 MB·English

by  Nils Dencker

PSEUDOSPECTRA OF SEMI-CLASSICAL (PSEUDO)DIFFERENTIAL
OPERATORS
3
0
0
NILS DENCKER,JOHANNESSJÖSTRAND,ANDMACIEJZWORSKI
2

n
a
J
1. Introdu
tion
1
2 The purpose of this note is to show how some results from the theory of partial di(cid:27)erential

equations apply to the study of pseudo-spe
tra of non-self-adjoint operators whi
h is a topi
of
]
P
urrent interest in applied mathemati
s (cid:21) see [4℄ and [26℄.
A We will
oTn∗sRidner operators whi
h arise from the quantization of bounded fun
tions on the
h. phase spa
e . For stronger results in the analyti

ase we will aTs∗sRumne tCha2nt our fun
tions
are holomorphi
and bounded in tubular
omplex neighbourhoods of ⊂ .
t
a Let us present the results in a typi
al example to whi
h they apply. We
onsider
m P(h)= h2∆+V(x),
−
[
a semi-
lassi
al S
hrödinger operator. P(h)
1
We de(cid:28)ne the semi-
lassi
al pseudospe
trum of the S
hrödinger operator as
v Λ(p)= ξ2+V(x) : (x,ξ) R2n, Im ξ,V′(x) =0 ,
2 { ∈ h i6 }
4 Λ(p)
2 npo=tinξg2+thVat(xin) the analyti

ase is either empty or the
losure of the set of all values of
1 .
0 Thefollowingresult(see Ÿ3)showsthattheresolventislargeinside thepseudo-spe
trum. We
3
(cid:28)rst state it in the
ase of S
hrödinger operatorssatisfying the assumptions above:
0 P(h)= h2∆+V(x) V ∞(Rn)
/ Theorem 1. Suppose that − , with ∈C .
h Λ(p) z
t u(hT)henL,2t(hRenre)exists an open dense subset of su
h that for any in that subset there exists
a
∈ with the property
m (P(h) z)u(h) = (h∞) u(h) .
(1.1) k − k O k k
:
v u(h) (x,ξ) p(x,ξ) = z
In addition is lo
alized to a point in phase spa
e, , with . More pre
isely,
i WF (u) = (x,ξ) WF (u)
X h { }, where the wave front set, h , is de(cid:28)ned in (2.5). Finally, for every
K ⋐Λ(p) z K
ar
ompa
t theaboveresulthohl∞dsuniefxoprm( ly1/foCrh)∈ inthenaturalsense. Ifthepotential
is real analyti
then we
an repla
e by − .
This result was proved by Davies [3℄ for S
hrödinger operators in one dimension, but as was
pointed out in [29℄, it follows in great generality from a simple adaptation of the now
lassi
al
results of Hörmander [11℄ and Duistermaat-Sjöstrand [6℄. The main point is that, unlike in the
h 0

ase of normal operators,the resolvent
an be large on open sets as → . That is parti
ularly
P(h)
striking when has only dis
rete spe
trum.
To guaranteethat we
an for instan
e assume that
∂αV(x) C (1+ x)m−|α|, (1+ xm+ ξ 2)/C ξ2+V(x) , (x,ξ) C.
(1.2) | x |≤ α | | | | | | ≤| | | |≥
m > 0
where . This is the simplest example of the behaviour of the potential: we
an make
V
weaker assumptions on (cid:21) see the end of Se
t.3. In the analyti

ase, we assume that (1.2)
x Imx <c α =0
0
holds as | |→∞, | | (and we only need it with | | ).
1
2 N.DENCKER,J.SJÖSTRAND,ANDM.ZWORSKI
p = ξ2 +V(x)
The
lassi
al symbol avoids all su(cid:30)
iently negative values and the Fredholm
P(h) h
theory guarantees that has dis
rete spe
trum for small enough (see Ÿ2).
P(h)
We
an, in pla
e of the S
hrödinger operator, ,
onsider the operator with a bounded
(P(h) z )−1(P(h) z ) z = z
1 2 2 1
symbol, − − , 6 , and this shows that it is su(cid:30)
ient to
onsider
quantization of bounded fun
tions, with all derivatives bounded,
p ∞(T∗Rn)= u ∞(T∗Rn) : α Nn ∂αu L∞(T∗Rn) .
∈Cb { ∈C ∀ ∈ 0 ∈ }
In that
ase we give a more general de(cid:28)nition of the semi-
lassi
al pseudospe
trum:
Λ(p)=p( m T∗Rn : p,p¯ (m)=0 ),
(1.3) { ∈ { } 6 }
where we used the Poisson bra
ket:
n
def
f,g =H g, H = ∂ f∂ ∂ f∂ .
{ } f f ξj xj − xj ξj
Xj=1
p,p¯
The non-vanishing of { } is a
lassi
al equivalent of the operator not being normal (cid:21) see (2.3)
and (2.4) below. We note that in the analyti

ase we have
Λ(p)= Λ(p)=Σ(p),
∅ or
where we put
Σ(p)=p(T∗Rn).
In that more general setting we
an restate our result as
n 2 p ∞(T∗Rn) p−1(z)
TheorzemC2. SuPpp(ohs)e that ≥ , ∈Cb anpdwt(hxa,thD) is
ompa
t for a dense set of
values ∈ . If has the prin
ipal part given by then the
on
lusions of Theorem
1 hold. p (x,ξ) C2n, Im(x,ξ) 1/C
If in addition has a bounded holomorphi

ontinuation to { ∈ | |≤ }
h∞ exp( 1/Ch)
then the
on
lusions of Theorem 1 hold with repla
ed by − .
n = 1 ′
If then the same
on
lusion holds provided that the assumptions of Lemma 3.2 are
satis(cid:28)ed.
u(h)
Wewill seein Ÿ4that,in general,we
annot
onstru
tan almostsolution atanarbitrary
Λ(p) z
interior point of , .
σ(P(h))
In simple one dimensional examples we
an already see that the spe
trum, , typi-
Λ(p) p

ally lies deep inside the pseudo-spe
trum, (the set of values of in the analyti

ase) (cid:21)
P(h) = (hD )2 +
x
see [2℄,[3℄,[26℄. Consider for instan
e the following non-self-adjoint operator
i(hD )+x2
x
. A formal
onjugation
1
e−x/2hP(h)ex/2h =(hD )2+x2+ ,
x
4
P(h) (2n+1)h+1/4
shows that the spe
trum of is given by , while
Λ(p)= z : Rez (Imz)2 , p=ξ2+iξ+x2.
{ ≥ }
z ∂Λ(p)
0
To see these phenomena for general operators we need to make assumptions on ∈ .
The (cid:28)rst one is the prin
ipal type
ondition,
p(x,ξ)=z = dp(x,ξ)=0, m T∗Rn.
0
(1.4) ⇒ 6 ∈
Then we assume an exterior
one
ondition:
C Λ(p) z
0
There exists a trun
ated
one in \ with vertex at .
PSEUDOSPECTRA OF SEMI-CLASSICAL (PSEUDO)DIFFERENTIAL OPERATORS 3
More pre
isely,
ǫ >0, θ R (z +(0,ǫ )ei(θ0−ǫ0,θ0+ǫ0)) Λ(p)= ,
0 0 0 0
(1.5) ∃ ∈ su
h that ∩ ∅
q =ie−iθ0(p z0)
and a dynami
al
ondition: if − , then
H q−1(0)
Req
(1.6) No traje
tory of
an remain in for an unbounded period of time.
Under these
onditions we have the following
p ∞(T∗Rn) P(h)
Theorem 3. Suppose that ∈ Cb and that the prin
ipal part of is given by
pw(x,hD) z ∂Λ(p) M >0 h<h (M)
0 0
. If ∈ satis(cid:28)es (1.4),(1.5),and (1.6)thenforany ,andfor ,
0<h (M)
0
,
z : z z <Mhlog(1/h) σ(P(h))= .
0
{ | − | }∩ ∅
p Rn
If in addition is a bounded holomorphi
fun
tion in a
omplex tubular of . then there
C >0
0
exists su
h that
z : z z <1/C σ(P(h))= .
0 0
{ | − | }∩ ∅
In Ÿ6 we will show that if (1.6) is violated then, for a large
lass of dissipative operators, the
h 0
spe
trum lies arbitrarily
lose (as → ) to the boundary of the pseudo-spe
trum.
At the boundary of the pseudo-spe
trum we may expe
t an improved bound on the resolvent
when some additional non-degenera
y is assumed. The result below is based on subellipti
p = p +ip C∞
1 2
estimates [14, Chapter 27℄ and we borrow our notation from there. If ∈ with
p
j
real valued then we de(cid:28)ne the repeated Poisson bra
kets
p =H H ...H p
I pi1 pi2 pik−1 ik
I =(i ,i ,...,i ) 1,2 k I =k
1 2 k
where ∈{ } and | | is the order of the bra
ket.
z ∂Λ(p) p z p−1(z )
0 0 0
We say that ∈ is of (cid:28)nite type for if (1.4) holds at , is
ompa
t, and for
(x ,ξ ) p−1(z ) k 1 I 1,2 k
0 0 0
any ∈ there exists ≥ and ∈{ } su
h that
p (x ,ξ )=0.
I 0 0
(1.7) 6
p w =(x ,ξ )
0 0
The order of at is
k(w)=max j Z: p (w)=0 I j .
I
(1.8) { ∈ for | |≤ }
z p (x ,ξ ) (x ,ξ ) p−1(z )
0 0 0 0 0 0
The order of is the maximum of the order of at for ∈ . We say that
p (P) qp
satis(cid:28)es
ondition if the imaginarypart of does not
hangesign on the bi
hara
teristi
s
qp 0=q C∞
of the real part of , for any 6 ∈ .
k(w)>k
As shown in [14, Corollary27.2.4℄, if and only if
z C, j k (H )jImzp(w)=0,
Rezp
(1.9) ∀ ∈ ≤
and this provides a reformulation of the assumptions of the following
p ∞(T∗Rn) P(h) pw(x,hD)
Theorem 4. Assume that ∈ Cb , and that the prin
ipal part of is . If
z ∂Λ(p) p k 1 k h<h 0<h
0 0 0
∈ is of (cid:28)nite type for of order ≥ , then is even and for , ,
(P(h) z0)−1 Ch−k+k1 ,
(1.10) k − k≤
c >0
0
In parti
ular, there exists su
h that
k
z : z z0 c0hk+1 σ(P(h))= 0<h h0.
(1.11) n | − |≤ o∩ ∅ ≤
4 N.DENCKER,J.SJÖSTRAND,ANDM.ZWORSKI
In one dimension this result was proved in [31℄, and in some spe
ial
ases by Boulton [1℄ who
also showed that the bounds are optimal. As was demonstrated by Trefethen [27℄ that is also
easy to see numeri
ally.
A simple hWigher d∞im(eRn2s)ional example to whi
h the theorem applies
an be
oxn2s+trux
2te=d a1s
follows. Let ∈ Cb be a non-negative fun
tion, vanishing on the
ir
le 1 2 .
Consider
P(h)= h2∆+iW(x)+i(x2+x2 1)m, m
− 1 2− with even.
z >0 (0, ) k =2m
0
Then the estimate (1.10) holds for uniformly on
ompa
t subsets of ∞ , with .
k
The in
reasein isdue to the(simple) tangen
yofsomebi
hara
teristi
softherealpart to the
set where the imaginary part vanishes.
We
on
lude by pointing outthat we
ould havede(cid:28)ned the semi-
lassi
alpseudospe
trum of
P(h) Λ(P) z
, , as the
losure of the set of points at whi
h (1.1) holds. We have shown that
Σ(p) Λ(P) Λ(p).
⊃ ⊃
An equality is not true in general but we
ould perhaps hope for
Λ(P)◦ =Λ(p),
under suitable assumptions.
Another important topi
not explored in this paper is the behaviour of evolution operators
exp(itP/h) P
for non-normal 's, and its relation to semi-
lassi
al pseudospe
tra.
A
knowledgements. The third author is grateful to the National S
ien
e Foundation for
partialsupportunderthegrantDMS-0200732. HewouldalsoliketothankMikeChristandNi
k
Trefethen for helpful dis
ussion.
2. Review of semi-
lassi
al quantization
Forpsimp∞li(
Tity∗Ronf)presentation wpe will
onsider the
ase of semi-
lassi
al quantization of fun
-
tions ∈Cb , that is that is bounded with bounded derivatives of all orders.
p(x,ξ)
In the analyti
T
∗aRsenweRw2inll asCsu2nme that is bounded and holomorphi
in a tubular
neighbourhood of C≃ ⊂ . As pointed out in(xt,hξe)introdu
tion, the
ase of fun
tions
whi
h omit a value in and whi
h tend to in(cid:28)nity as → ∞
an be redu
ed to this
ase
(see also the remark at the end of Ÿ3).
We use the Weyl quantization,
1 x+y
(2.1) pw(x,hDx)u= (2πh)n Z Z p(cid:18) 2 ,ξ(cid:19)ehihx−y,ξiu(y)dydξ.
p ∞(T∗Rn) L2(Rn)
whi
h for ∈ Cb gives operators bounded on (cid:21) see [5, Chapter 7℄. We
an

onsider more general operators,
∞
P(h) hjpw(x,hD),
∼ j
Xj=0
where in the
ase of analyti
symbols we assume that
∞
P(x,ξ;h) hjp (x,ξ),
j
∼
Xj=0
in the spa
e of bounded holomorphi
fun
tions in a tubular neighbourhood of the real phase
spa
e. Although it is not stri
tly speaking ne
essary for our (cid:28)nal
on
lusions, in the analyti
PSEUDOSPECTRA OF SEMI-CLASSICAL (PSEUDO)DIFFERENTIAL OPERATORS 5

ase we make an additional assumption that
p (z,ζ) Cjjj, (z,ζ) Cn, Im(z,ζ) 1/C.
j
(2.2) | |≤ ∈ | |≤
That allows us exponentially small errors in the expansions.
The produ
t formula of the Weyl
al
ulus says that
pw(x,hD) pw(x,hD)=(p ♯ p )w(x,hD;h),
(2.3) 1 ◦ 2 1 h 2
where (p1#p1)(x,ξ;h)=ei2hω((Dx,Dξ),(Dy,Dη))p1(x,ξ)p2(y,η)|y=x,η=ξ
has the following asymptoti
expansion
∞ 1 ih k
p ♯ p (x,ξ;h) ω((D ,D ),(D ,D )) p (x,ξ)p (y,η) ,
1 h 2 x ξ y η 1 2 y=x,η=ξ
(2.4) ∼ k!(cid:18) 2 (cid:19) |
kX=0
ω = n dξ dx T∗Rn D = (1/i)∂
with j=1 j ∧ j, the symple
ti
form on , and • •. The expansion
Pp ♯ p (h∞) ∞ k 1/(Ch)
determines 1 h 2 uptoaterminO Cb . Intheanalyti

asebysummingupto ∼
(e−1/(Ch))
we
an obtain O errors (cid:21) see [20℄.
A basi
tool of mi
rolo
al analysis is the FBI transform:
T :L2(Rn) L2(T∗Rn),
→
de(cid:28)ned by
Tu(x,ξ)=cnh−34n ehi(hx−y,ξi+i|x−y|2/2)u(y)dy.
ZRn
u L2(Rn)
Roughly speakingits rle
anbe dTeus
ribLe2d(Tas∗Rfonll)ows.hThe0phasespa
epropertiesof ∈
are re(cid:29)e
ted by the behaviour of ∈ as → . In this note we will only deal with
h Dαu = (h−Nα)
-dependentsmoothfun
tionswithatempered behaviour,| | O . The notionofthe
u WF (u)
h
wave front setof , , explainsthelo
alizationstatementinTheorem2(seealsoTheorem
′ ∞ h
2 below). In the C
ase the -wavefrontset is de(cid:28)ned by
(2.5)
(x0,ξ0) / WF (u) N Tu(x,ξ) C h−N (x,ξ) (x0,ξ0)
h N
∈ ⇐⇒ ∀ | |≤ for in a neighbourhood of ,
and in the analyti

ase by
(2.6)
(x0,ξ0) / WF (u) c>0 Tu(x,ξ) e−c/h (x,ξ) (x0,ξ0)
h
∈ ⇐⇒ ∃ | |≤ for in a neighbourhood of .
∞ WF (u)
h
In the C
ase we
an also
hara
terize using pseudodi(cid:27)erential operators:
(x0,ξ0) / WF (u) p ∞(T∗Rn), p(x0,ξ0)=0, p(x,hD)u= (h∞).
∈ h ⇐⇒ ∃ ∈Cc 6 O
P(h)
In the analyti

ase we will need to understand the a
tion of on mi
rolo
ally weighted
H(Λ )
tG
spa
es whose de(cid:28)nition, in the simple setting needed here, we will now re
all (cid:21) see [9℄
for the origins of the method, and [17℄ for a re
ent pTre∗sRenntaTti∗oCn.n
The
omplexi(cid:28)ω
Cation of the symple
ti
manifold , ImωC is eqRuiepωpCed with the
omTp∗Rlenx
symple
ti
form, and two natural realsymple
ti
forms and . We see that
is Lagrangian with respe
t to the (cid:28)rst form and symple
ti
with respe
t to the se
ond one. In
general we
all aGsubm∞an(iTfo∗lRdns)atisfying these two
onditions an IR-manifold.
Suppose that ∈Cc . We asso
iate to it a natural family of IR-manifolds:
Λ = ρ+itH (ρ) : ρ T∗Rn T∗Cn, t R t
tG G
(2.7) { ∈ }⊂ with ∈ and | | small.
Im(ζdz) Λ H Λ
tG t tG
Sin
e is
losed on , there exists a fun
tion on su
h that
dH = Im(ζdz) ,
t − |ΛtG
6 N.DENCKER,J.SJÖSTRAND,ANDM.ZWORSKI
Λ T∗Rn
tG
and in fa
t we
an write it down expli
itely, parametrizing by :
H (z,ζ)= ξ,t G(x,ξ) +tG(x,ξ), (z,ζ)=(x,ξ)+itH (x,ξ).
t ξ G
−h ∇ i
H(Λ ) Tu(x,ξ)
tG
Theasso
iatedspa
es arede(cid:28)ned asfollows. TheFBItransform, , isanalyti
(x,ξ) Λ T u ∞(Λ ) Λ
Tin∗Rn and we
an
ontinueTit tou tG. That de(cid:28)nes ΛtG Λ∈ C tG . Sin
e tG di(cid:27)ers from
on a
omHp(aΛ
t s)et only, ΛtG is square ihntegrable on tG. L2(Rn)
tG
The spa
es are de(cid:28)ned by putting -dependent norms on :
u 2 = T u(z,ζ)2e−2Ht(z,ζ)/h(ω )n/n!.
k kH(ΛtG) Z | ΛtG | |ΛtG
ΛtG
The mainresult relaptesthe ap
tionof apseudodi(cid:27)erentialoperatorto themultipli
atioTn∗Rbynits
1 2
sCy2mnbol. Suppose that tand are bounded and holomorphi
in a neighbourhood of in
(see (2.2)). Then for small enough
(2.8)
hpw1(x,hD)u,pw2(x,hD)viH(ΛtG) =h(p1|ΛtG)TΛtGu,(p2|ΛtG)TΛtGviL2(ΛtG,e−2Ht/h(ω|ΛtG)n/n!)
+ (h) u v ,
O k kH(ΛtG)k kH(ΛtG)
p =p p =p¯ u=v
1 2
see [9℄,[17℄. In parti
ular, by taking and , and we obtain
pw(x,hD)u 2 = p T u 2 + (h) u 2 .
(2.9) k kH(ΛtG) k |ΛtG ΛtG kL2(ΛtG,e−2Ht/h(ω|ΛtG)n/n!) O k kH(ΛtG)
For the use in the next se
tion we also reT
∗aCllnsome basi
fa
ts aλbout positive Lagrangian
submanifoldsofa
omplexsymple
ti
manifold . A
omplexplane ,of(
omplex)dimension
n
is Lagrangianand positive if
X,Y λ ωC(X,Y)=0, iωC(X¯,X) 0.
(2.10) ∀ ∈ ≥
The
ru
ial
hara
terizationis given as follows (see [14, Proposition 21.5.9℄):
λ T∗Cn λ= (z,Az) : z Cn
⊂ is a positive Lagrangianplane ⇐⇒ { ∈ }
(2.11) A=A1+iA2 A1 A2
where is a symmetri
matrix with real, and positive de(cid:28)nite.
3. Semi-
lassi
al pseudo-spe
trum
Λ(p)
InŸ1 wede(cid:28)ned the semi-
lassi
alpseudospe
trum, , asthe
losureofthe setofvaluesof
p
. We de(cid:28)ne some additional sets
Λ (p)= p(x,ξ) : Rep,Imp (x,ξ)>0 p(T∗Rn)
±
{ ±{ } }⊂
(3.1) Σ∞(p)= z : (xj,ξj) lim p(xj,ξj)=z ,
{ ∃ →∞ j→∞ }
Σ (p) p
∞
that is, is the set of limit points of at in(cid:28)nity.
∞
In the C
ase Theorem (2) follows immediately from a semi-
lassi
al reformulation of the
non-propagation of singularities [6℄,[11℄,[12℄ (cid:21) see [14, Se
tion 26.3℄ and [29℄.
Theanalyti

aseisalsowellknown(see[15℄)butsin
eaready-to-usereferen
eisnotavailable
∞
we in
lude a proof. It
an also be adapted to give a self-
ontained proof in the C
ase.
′ n 2 p(x,ξ)
Theorem2. Suppose that ≥ and satis(cid:28)es the assumptions inŸ2 inthe analyti

ase,
Λ (p)
−
and that is given by (3.1). Then
Λ (p) Λ(p) Σ (p),
− ∞
⊃ \
z Λ (p) (x0,ξ0) T∗Rn
−
and for every ∈ , and every ∈ with
p(x0,ξ0)=z, Rep,Imp (x0,ξ0)<0,
{ }
PSEUDOSPECTRA OF SEMI-CLASSICAL (PSEUDO)DIFFERENTIAL OPERATORS 7
0=u(h) L2(Rn)
there exists 6 ∈ su
h that su
h that
(P(h) z)u(h) = (e−1/Ch) u(h) , WF (u(h))= (x0,ξ0) .
h
(3.2) k − k O k k { }
n=1 ′
If thenthesame
on
lusionholdsprovidedthattheassumptionsofLemma3.2 aresatis(cid:28)ed.
p
In dimension one the theorem holds as well but further assumptions need to be made on (cid:21) see
the remark after Lemma 3.2.
Λ (p)
−
Before the proof we wantto stressthe need for an open dense subset . One
ould ask if
Λ(p) Σ (p)
∞
any interiorpoint of \ is an (cid:16)almost eigenvalue(cid:17) or(cid:16)quasimode(cid:17) in the senseof (3.2).
That is not so as shown by
T∗R
Example. Consider the following bounded analyti
fun
tion on :
ξ2 1+iξx2(1+x2)−1
p(x,ξ)= − .
1+ξ2+iξx2(1+x2)−1
p−1(0) = (0,1),(0, 1) 0
We see that { − }, and that is an interior point of the pseudospe
trum,
0 Λ(p)◦ 0 (0, 1) p
∈ . Also, is a boundary point of images of neighbourhoods of ± under .
(0, 1) p ξ + ix2
Near ± , is mi
rolo
ally equivalent to a non-vanishing multiphle−32of . An ex-
pli
it
om†putation shows that the inverseof the models are bounded by , and a lo
alization
argument then shows that
pw(x,hD)−1 L2→L2 h−23 .
k k ≤
0
Hen
e, is not a quasi-mode. It should be stressed that the vanishing of the Poisson bra
ket
Rep,Imp p−1(0)
{ } (whi
h o

urs in this example at ) is not enough to guarantee the absen
e of
Ψ
a quasi-mode. A violationof the
ondition (see [14, Se
t.26.4℄)
an produ
e quasi-modeswith
p(x,ξ) = ξ ixk
the simplest example
oming from adapting [14, Theorem 26.3.6℄ as in [29℄: − ,
k >1
with odd.
′ Λ (p) Λ (p)
± −
We start the proof of Theorem 2 with the dis
ussion of . To establish that is
dense we need the following result of Melin-Sjöstrand [19, Lemma 8.1℄:
n 2 dRep dImp p−1(z)
Lωemma 3.1. Suppose that T∗≥Rn and that , are linearly independent on . If
is the symple
ti
form on then
ωn−1
Rep,Imp λ = ,
{ } p,z (n 1)! p−1(z)
− (cid:12)
(cid:12)
λ p−1(z) λ dRep dImp=ωn/n!
p,z p,z
where is the Liouville measure on : ∧ ∧ .
p−1(z) Γ
In parti
ular, for any
ompa
t
onne
ted
omponent of , , we have
Rep,Imp λ (dρ)=0.
p,z
Z { }
Γ
Λ (p)=Λ(p)
−
As an immediate
onsequen
e we see that if the assumptions of Theorem 2 are
satis(cid:28)ed and that in general we have the following
p n 2 Λ (p) Λ (p)
+ −
Lemma 3.2. If the assumptions on are satis(cid:28)ed, ≥ , and either or are
Λ (p) Λ (p) Λ(p)
+ −
non-empty, then ∪ is dense in , and
Λ (p) Λ(p) Σ (p).
± ∞
⊃ \
†
ItisaneasyonedimensionalversionoftheargumentinŸ5.
8 N.DENCKER,J.SJÖSTRAND,ANDM.ZWORSKI
Rep,Imp 0 H d=ef ρ T∗Rn; Rep,Imp (ρ)=0
Proof. Assumethat{ }6≡ . Then { ∈ { } }isananalyti
z =p(ρ) ρ H
hypersurfa
e without anyz in=tepr(iρor)pointsR. C2nonHsequeρntly, ρevery vΛalu(ep) Λ (p)with ∈
an be
j j j + −
apppr(oTx∗iRmna)tedby values with \ ∋ → , and ∪ isopen anddense
in . (p(T∗Rn))◦ =
Sin
eunderourassumption 6 ∅,anelementaryversionoftheMorse-Sardtheorem
dRep dImp p−1(z) z Ω Λ(p)
implies that and are independent on for in a dense open set ⊂ \
Σ (p) Λ (p) Ω = Λ (p) Ω
∞ + −
. Lemma 3.1 then shows that ∩ ∩ ,
ompleting the proof of the
(cid:3)
lemma.
Remark. In the
ase of dimension one a di(cid:27)erent argument, based on elementary topologi
al
p

onsiderations, is needed and some assumptions have to be made on . To see that
onsider for
instan
e (ξ+ix)2
p(x,ξ)= , Rep,Imp (x,ξ)>0, (x,ξ)=(0,0).
1+x2+ξ2 { } 6
p
For 's arising from S
hrödingeroperators
onsidered in Ÿ1 we alwayshave
sgn Rep,Imp (m)=0,
(3.3) { }
m∈Xp−1(z)
z p(x,ξ)=p(x, ξ)=z z
for a dense set of values . In fa
t, − , and the set of values
orresponding
ξ = 0
to 6 is dense in the set of values for whi
h the bra
ket is non-zero. Now we simply noti
e
that
Rep,Imp ((x,ξ))= Rep,Imp ((x, ξ)).
{ } −{ } −
′ n = 1
LeCmmΣa 3(.p2). Suppose that and in additio∁nΛ(tpo)the general assumptions, ea
h
omponent
∞
of \ has a non-empty interse
tion with . Then the
on
lusions of Lemma 3.2 and
(3.3) (for a dense set of values) hold.
Ω C Σ (p)
∞
Proof. Let be a
omponent of \ . Then
def
ι = var arg (p z)
γ(z) −
z Ω γ(z) (x,ξ) = R(z) R(z)
is independent of ∈ if is the positively oriented
ir
le | | with large
z Ω Λ(p)ι z Ω
enough. For ∈ \ iszero(formappingdegreereasons)andhen
eit iszeroforall ∈ .
z Λ(p) Ω
If ∈ ∩ is a regular value, we get
0=ι=2π sgn Rep,Imp (m),
{ }
m∈Xp−1(z)
z Λ (p) Λ (p) (cid:3)
+ −
so belongs to both and .
In the remainder of the proof there is no restri
tion on the dimension.
′ z = 0
Proof of Theorem 2: We
an assume that and we follow the now standard pro
edure
of the
omplex WTK∗RBn
onstru
tion asso
iated to a positive Lagrangian submanifold of the
om-
plexi(cid:28)
ation of . We start with the geometri

onstru
tion of that submanifold. Sin
e
Rep,Imp = 0 d p = 0 ∂ p(x0,ξ0) = 0 φ
{ } 6 we have ξ 6 , and we
an ays0sumRent−h1atx0ξ1=(x0,y0)6 . Let 0, be a real
analyti
fun
tion de(cid:28)ned in a neighbourhood of ∈ , 1 , with the properties
φ (y0)=0, dφ (y0)=η0, ξ0 =(ξ0,η0), Imd2φ (y0) 0,
0 0 1 0 ≫
d2Imφ y0 dImφ =0
0 0
wheretheHessian iswellde(cid:28)nedat as . Wewillmakefurtherassumptions
φ
0
on later.
PSEUDOSPECTRA OF SEMI-CLASSICAL (PSEUDO)DIFFERENTIAL OPERATORS 9
Λ T∗Cn (x0,ξ0)
0
We then de(cid:28)ne ⊂ , lo
ally near , as follows
Λ = (x0,y;ξ (y),d φ (y)) : p(x0,y;ξ (y),d φ (y))=0, ξ0(y0)=ξ0 ,
0 { 1 1 y 0 1 1 y 0 1 1}
ξ (y) ∂ p=0
whereweknowthatthefun
tion 1 islo
allyde(cid:28)nedandanalyti
fromourT
∗oCnnditΛion Tξ1∗R6n =.
0
Using holomorphi

ontinuation we obtain a lo
ally de(cid:28)ned submanifold of , ∩
(x0,ξ0)
{ }. This submanifold is isotropi
with respe
t to the
omplex symple
ti
form and its
‡
tangent spa
es are positive, in the sense that (2.10) is satis(cid:28)ed without the
ondition on the
dimenstion.C t <ǫ Φ
t
For ∈ , | | , the
omplex (cid:29)ow, , exists by the Cau
hy-KovalevskayaTheorem:
Φ (z,ζ)=(z(t),ζ(t)), z(0)=z, ζ(0)=ζ,
t
z′(t)=∂ p(z(t),ζ(t)), ζ′(t)= ∂ p(z(t),ζ(t)),
ζ z
−
d/dt(z(t),ζ(t))=Hp(z(t),ζ(t)) ωC( ,Hp)=dp
that is , • .
We then de(cid:28)ne
Λ= Φ (Λ ) T∗Cn,
t 0
⊂
t∈C[,|t|<ǫ
ωC
whi
h is Lagrangianwith respe
t to .
Λ
We now want to guarantee that tangent spa
es to are positive in the sense of (2.10). We
(cid:28)rst note that
iωC(tHp,tHp)=it2 p¯,p = 2 Rep,Imp t2 >γ t2 >0,
| | { } − { }| | | |
by the assumptions of the theorem. We have
T Λ=T Λ +span H (x0,ξ0),
(x0,ξ0) (x0,ξ0) 0 C p
iωC(X +tHp,X +tHp)=iωC(X,X)+2Im(tdp∗(X))+ t2iωC(Hp,Hp),
| |
p∗(ρ) = p(ρ¯) φ
0
where . Hen
e, for positivity, we need to show that we
an
hoose so that for
X T Λ
(x0,ξ0) 0
∈ ,
dp∗(X)2 <αiωC(X,X), α= 2 Rep,Imp (x0,ξ0).
| | − { }
(z ,z′;ζ ,ζ′)
1 1
A
al
ulation in lo
al
oordinates shows that this follows from
A+ΦB 2 <αminSpec(ImΦ),
k k
Φ=φ′0′, A=|p′ζ1|−1i pζ1pz′ −pζ1pz′ , B =|p′ζ1|−1i p′ζ1pζ′ −pζ1pζ′ .
(cid:0) (cid:1) (cid:0) (cid:1)
A B Φ A+
The ve
tors and are real, and hen
e we
an
hoose the
omplex matrix so that
(ReΦ)B =0
. This leaves us with
(ImΦ)B 2 <αminSpec(ImΦ),
k k
ImΦ ⋆
whi
h
an be arrangedby making su(cid:30)
iently small .
Λ p = 0
From (2.11) we see that ⊂ { } is lo
ally a graph, and sin
e it is Lagrangian, a graph
φ
of a di(cid:27)erential of a phase fun
tion . Sin
e the tangent plane is positive (2.11) shows that the
Hessian of that phase fun
tion has a positive de(cid:28)nite imaginary part:
Λ= (z,d φ(z)) , p(z,d φ)=0, φ(x0)=0, d φ(x0)=ξ0, Imd2φ(x0) 0.
{ z } z z z ≫
‡ TρΛ0 TρCn
⋆Thesetangent spa
es, ,are
omplexlinearsubspa
esof .
Analternative,andsli
ker,wayofpro
eedingisby(cid:28)rstobservingthatthepositivityisinvariantundera(cid:30)ne
lpin=earξn
a−noinxin
a+ltOra(n(sxf,oξr)m2)ations. Usin(gx0t,hξa0t),a=nd(0a,0m)ultipli
ationbyanon-vanishingfa
toφr,(xw)e
anassumethat
and that . It is then straightforward to (cid:28)nd with the desired
properties.
10 N.DENCKER,J.SJÖSTRAND,ANDM.ZWORSKI
Λ T∗Rn = (x0,ξ0) Imd φ = 0
z
We also note that ∩ { } whi
h
orresponds to the fa
t that 6 for
z =x0
6 .
On
e the phase fun
tion has been
onstru
ted we apply the usual WKB
onstru
tion:
∞
v(z,h) eiφ(z)/h a (z)hj,
j
∼
Xj=0
a z = x0
j
where we will want the
oe(cid:30)
ients, 's to be holomorphi
near , and to satisfy bounds
a (z) Cjjj
j
| |≤ . They are
onstru
ted so that
 pwj (z,hDz)hjeiφ(z)/h aj(z)hj=O(e−1/Ch).
j<X1/Ch j<X1/Ch
  
pw p(z,ζ)
Here denotes the Weyl quantization of a holomorphi
symbol a
ting on holomorphi
fun
tions (
ompare to (2.1)):
1 z+w
pw(z,hDz)u= (2πh)nZZ p(cid:18) 2 ,ζ(cid:19)ehihz−w,ζiu(w)dwdζ,
Γz
Γ
z
where the
ontour is suitably
hosen (cid:21) see [20, Se
tion 4℄ for a dis
ussion of the general
ase.
a
j
The transport equations for 's then are:
n
∂ p (z,d φ(z))∂ a (z)+ip (z,d φ(z))a =A (z),
ζk 0 z zk j 1 z j j
Xk=1
A (z) a l < j a (x0,y) = 1
where j depends on l's with , and we put 0 1 . It is now
lassi
al that the
a Cjjj x0
j
solutions satisfy | | ≤ near (cid:21) see [20, Theorem 9.3℄. The real quasi-mode is obtained
v(z,h)
by restri
ting to the real axis and by trun
ating :
u(x,h)=χ(x)v(x,h), χ(x)=1, x x0 <δ, suppχ B(x0,2δ)
| − | ⊂
δ Imφ x x0 2/C
where is small. Sin
e the
onstru
tion has shownthat ≥| − | , the
ut-o(cid:27) fun
tion
χ
does not destroy the exponential smallness of the error.
For
ompleteness, and lateruse in Ÿ6, wein
lude aresult on the dis
retenessof the spe
trum.
p ∞(T∗Rn) Ω h
Proposition 3.3. Suppose that ∈ Cb . Let be an open
onne
ted ( -independent)
set, satisfying
Ω Σ (p)= , Ω ∁Σ(p)= .
∞
∩ ∅ ∩ 6 ∅
(pw(x,hD) z)−1 0<h<h (Ω) z Ω
0
Then − , , ∈ , is a meromorphi
family of operators with poles
of (cid:28)nite rank.
h pw(x,hD)
In parti
ular, for su(cid:30)
iently small, the spe
trum of is dis
rete in any su
h set.
Ω C > 0
Proof. If satis(cid:28)es the assumptions of the propostion then there exists su
h that for
z Ω p(x,ξ) z >1/C (x,ξ) >C Ω ∁Σ(p)=
every ∈ ,wzehavΩe|(p(x,ξ−) |z )−1 if∞| (T∗R|n) . Thχeass∞um(Tp∗tiRonn;t[h0a,1t]) ∩ 6 im1plies
that for some 0 ∈ , − 0 ∈Cb . Let ∈Cc be equal to in a
su(cid:30)
iently large bounded domain. The remarksabove show that
r(x,ξ;z)=χ(x,ξ)(z p(x,ξ))−1+(1 χ(x,ξ))(z p(x,ξ)−1,
0
− − −
∞(T∗Rn)
is in Cb . The symbol
al
ulus reviewed in Ÿ2 then gives
rw(x,hD,z)(z pw(x,hD))=I + (h)+K (z),
L2→L2 1
− O
(z pw(x,hD))rw(x,hD,z)=I + (h)+K (z),
L2→L2 2
− O