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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 expli 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 omplexi(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