LECTURE 20: SEMI-CLASSICAL PsDOs ON MANIFOLDS1.SEMICLASSICALPsDOsUNDER COORDINATECHANGE Pseudolocality.As we have mentioned several times, we want to extend the theory of semiclas-sical pseudodifferential operators from Rn to smooth manifolds. So in some sense,we have to study how to “glue" local pseudodifferential operators to global ones.As we have seen last time, for differential operators it is easy to get global oper-ators from local ones, because of the locality of differential operators. In general,pseudodifferential operators does not satisfy locality.For example, one can look atthe Dirichlet-to-Neumann operator A in PSet 2, which is by definition the map thatsends the Dirichlet boundary value of a Harmonic function on Rn+1 to its Neumannboundary value. It is easy to find f e C(Rn) with f = 0 on an open subset U inIRn, while A(f) + 0 on U.Fortunately, although locality fails, we do have a weaker version of locality forpseudodifferential operators, namely,thepseudolocality.Roughly speaking,localitytells us that the value Au(r) at a point r is determined by the values u(y) for y nearr,while pseudolocality tells us that the value Au(r)is determined, modulo O(h)(which is negligible in semiclassical analysis), by the values u(y) for y near r. Tosee the pseudolocality of A = aw, we just start with the definition of aw, namely1et(r-w)sa(“,)u(y)dyaWu(r) :(2元h)nJR2and noticethat away from the diagonal,namely in theregion r-yl> C (whichcan be produced by multiplying uby a cut-off function that is supported away fromthe point r), one can produce as many h's as we want via integration by parts usinget(c-)s =h(α-) Dset(a-)s.[-y12Pseudolocality can also be explained as follows. Suppose a E S(m) and supposeX1, X2 E Co(IR") are two cut-off functions such thatsupp(x1) n supp(x2) = 0.Then (c.f. PSet 3)XiaWx2 = O(h~).So if we take Xi to be a cut-off function supported near (which equals 1 in a smallneighborhood of ) and take X2 to be a cut-off function supported away,then theeffect of the value of u in the support of x2 is negligible in computing awu(r)
LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS 1. Semiclassical PsDOs under coordinate change ¶ Pseudolocality. As we have mentioned several times, we want to extend the theory of semiclassical pseudodifferential operators from R n to smooth manifolds. So in some sense, we have to study how to “glue” local pseudodifferential operators to global ones. As we have seen last time, for differential operators it is easy to get global operators from local ones, because of the locality of differential operators. In general, pseudodifferential operators does not satisfy locality. For example, one can look at the Dirichlet-to-Neumann operator Λ in PSet 2, which is by definition the map that sends the Dirichlet boundary value of a Harmonic function on R n+1 + to its Neumann boundary value. It is easy to find f ∈ C ∞(R n ) with f ≡ 0 on an open subset U in R n , while Λ(f) 6= 0 on U. Fortunately, although locality fails, we do have a weaker version of locality for pseudodifferential operators, namely, the pseudolocality. Roughly speaking, locality tells us that the value Au(x) at a point x is determined by the values u(y) for y near x, while pseudolocality tells us that the value Au(x) is determined, modulo O(~ ∞) (which is negligible in semiclassical analysis), by the values u(y) for y near x. To see the pseudolocality of A = ba W , we just start with the definition of ba W , namely ba W u(x) = 1 (2π~) n Z Rn e i ~ (x−y)·ξ a( x + y 2 , ξ)u(y)dy and notice that away from the diagonal, namely in the region |x − y| > C (which can be produced by multiplying u by a cut-off function that is supported away from the point x), one can produce as many ~’s as we want via integration by parts using e i ~ (x−y)·ξ = ~ (x − y) · Dξ |x − y| 2 e i ~ (x−y)·ξ . Pseudolocality can also be explained as follows. Suppose a ∈ S(m) and suppose χ1, χ2 ∈ C ∞ 0 (R n ) are two cut-off functions such that supp(χ1) ∩ supp(χ2) = ∅. Then (c.f. PSet 3) χ1ba W χ2 = O(~ ∞). So if we take χ1 to be a cut-off function supported near x (which equals 1 in a small neighborhood of x) and take χ2 to be a cut-off function supported away, then the effect of the value of u in the support of χ2 is negligible in computing ba W u(x). 1
2LECTURE20:SEMI-CLASSICALPSDOSONMANIFOLDSSemiclassical PsDOs under coordinate change.So we still want to study the change of semiclassical pseudodifferential operatorunder coordinate change. Again we start with a diffeomorphismf:UCRn-VCR"wherewealwaysassume18fl, 18f-1]≤Ca,Va,and again we want to use f to“"transplant",nowa pseudodifferential operator Pdefined for r-variables to a pseudodifferential operator for y-variables. However, wecan't define P to be the operator (f-1)*Plu f*, since pseudodifferential operators arenot local and thus Plu makes no sense.The solution to this issue is straightforward:instead of localizetheoperator, we“globalize"thefunctions.Howdowe“globalize"alocally defined function?Multiply it by a cut-offfunction!More precisely,supposex E Co(V), then we have mapsf*Mx : (Rn) -→ Co(U) C Co(Rn) C (Rn)andMx(f-1)* : 9(Rr) CC(Rn) -→ Co(Rn) C S(Rn)which extends to mapsf*Mx : '(R") -→9(Rn)andMx(f-1)*: (R")-→(R).So given any pseudodifferential operator P on Rn, which, for simplicity, weassume its Kohn-Nirenberg symbol is a, namely P = akN (for a in some symbolclass), and given any cut-off function x E Co(V), we can define Px : (Rn)→S(Rr) (and Px:(R)-g(R) by(1)P, = Mx(f-1)*Pf*Mx.IGiven a tempered distribution u e (R,) and a compactly supported function x= x(y), wecan extend Mx to Mx : '(R) - g(R) defined by(Mxu,p) := (u, Mxp),VpE(R).The definition of pull-back of distribution is a bit complicated: Given any diffeomorphism f : U →V, one has the pull-back map on functions, f* : Co(V) -→ Co(U). By duality, we get a linearmap f:(U)→(V),called thepush-forward, definedby(f+u, p) := (u, f*p).Moreover, it can be shown that the restriction of f to Co(U) is a continuous linear map fromCo(U) to Co(V). By duality again, we get a pull-back map, now defined on distributions,f*:'(V)-→'(U).[For pull-back by submersions, c.f.Theorem 6.1.2 in Hormander, Vol 1.]
2 LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS ¶ Semiclassical PsDOs under coordinate change. So we still want to study the change of semiclassical pseudodifferential operator under coordinate change. Again we start with a diffeomorphism f : U ⊂ R n x → V ⊂ R n y where we always assume |∂ α f|, |∂ α f −1 | ≤ Cα, ∀α, and again we want to use f to “transplant”, now a pseudodifferential operator P defined for x-variables to a pseudodifferential operator for y-variables. However, we can’t define Pe to be the operator (f −1 ) ∗P|U f ∗ , since pseudodifferential operators are not local and thus P|U makes no sense. The solution to this issue is straightforward: instead of localize the operator, we “globalize” the functions. How do we “globalize” a locally defined function? Multiply it by a cut-off function! More precisely, suppose χ ∈ C ∞ 0 (V ), then we have maps f ∗Mχ : S (R n y ) → C ∞ 0 (U) ⊂ C ∞ 0 (R n x ) ⊂ S (R n x ) and Mχ(f −1 ) ∗ : S (R n x ) ⊂ C ∞(R n x ) → C ∞ 0 (R n y ) ⊂ S (R n y ). which extends to maps1 f ∗Mχ : S 0 (R n y ) → S 0 (R n x ) and Mχ(f −1 ) ∗ : S 0 (R n x ) → S 0 (R n y ). So given any pseudodifferential operator P on R n x , which, for simplicity, we assume its Kohn-Nirenberg symbol is a, namely P = ba KN (for a in some symbol class), and given any cut-off function χ ∈ C ∞ 0 (V ), we can define Peχ : S (R n y ) → S (R n y ) (and Peχ : S 0 (R n y ) → S 0 (R n y )) by (1) Peχ = Mχ(f −1 ) ∗P f ∗Mχ. 1Given a tempered distribution u ∈ S 0 (R n y ) and a compactly supported function χ = χ(y), we can extend Mχ to Mχ : S 0 (R n y ) → S 0 (R n y ) defined by hMχu, ϕi := hu, Mχϕi, ∀ϕ ∈ S (R n y ). The definition of pull-back of distribution is a bit complicated: Given any diffeomorphism f : U → V , one has the pull-back map on functions, f ∗ : C∞ 0 (V ) → C∞ 0 (U). By duality, we get a linear map f∗ : D(U) → D(V ), called the push-forward, defined by hf∗u, ϕi := hu, f ∗ϕi. Moreover, it can be shown that the restriction of f∗ to C∞ 0 (U) is a continuous linear map from C∞ 0 (U) to C∞ 0 (V ). By duality again, we get a pull-back map, now defined on distributions, f ∗ : D0 (V ) → D0 (U). [For pull-back by submersions, c.f. Theorem 6.1.2 in H¨ormander, Vol 1.]
3LECTURE20:SEMI-CLASSICALPSDOSONMANIFOLDSSemiclassical PsDOs under coordinatechange: Schwartz symbols.We have to check that the operator Px defined above is a pseudodifferentialoperator, and calculate its symbol:Theorem 1.1. Suppose a E and P = aw. Then the operator P defined by (1)is a pseudodifferential operator whose Kohn-Nirenberg symbol b E and has theasymptoticerpansion(2)b(y,n) ~ a (f-1(y), (af)Tn) x(y)? +nib;(y,n),j≥1for some b e y.Proof.By definition we havex(y)e(-1()-2):a (f-1(g),E) x(f(2)u(f(2)dzdPxu(y) =(2元h)nx(y)et(-1(v)-)-a (f-1(y),) x(f(2)u(f(z)dzde(2元h)nJUxRrx(y)et(--()-1(u)-a (f-(),5) x(w)u(w)] det af-1]dwds(2元h)n1X(g)et(-(o)-(m);a (f-1(g), ) x(u)u(w)] det af-(w)]dude.(2元h)nSo P, is an operator whose Schwartz kernel Kp is compactly supported (since thesupport is contained in supp(x)× supp(x). It follows (c.f. the computation on page7 of Lecture 14) that P is the Weyl quantization of the symbolwwe-wKp(+)dwT22which can be shown to be Schwartz. To calculate the Kohn-Nirenberg symbol b(y, n)of Px, we use the oscillatory test (PSet 2) to getb(y,n) =e-typ(etun)1e[(-(s)-f-1(w)$+(u-)-mla(f-1(g), s)x(g)x(w)] det 0f-1(w)]dwdE(2元h)n1etpu.nay(w,E)dwde.(2元h)nJR2Soaccordingto the lemmaof stationaryphase (Lecture5),ay(p)b(y,n) ~ Z etpw.n(p)esgn(dpy,r(p)"[det dom()ENL(a)(),dey,n(p)=0where L, is a differential operator in w, s of order 2j and Lo = 1
LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS 3 ¶ Semiclassical PsDOs under coordinate change: Schwartz symbols. We have to check that the operator Peχ defined above is a pseudodifferential operator, and calculate its symbol: Theorem 1.1. Suppose a ∈ S and P = ba W . Then the operator Peχ defined by (1) is a pseudodifferential operator whose Kohn-Nirenberg symbol b ∈ S and has the asymptotic expansion (2) b(y, η) ∼ a f −1 (y),(∂f) T η χ(y) 2 + X j≥1 ~ j bj (y, η), for some bj ∈ S . Proof. By definition we have Peχu(y) = χ(y) (2π~) n Z R2n e i ~ (f−1 (y)−z)·ξ a f −1 (y), ξ χ(f(z))u(f(z))dzdξ = χ(y) (2π~) n Z U×Rn e i ~ (f−1 (y)−z)·ξ a f −1 (y), ξ χ(f(z))u(f(z))dzdξ = χ(y) (2π~) n Z V ×Rn e i ~ (f−1 (y)−f−1 (w))·ξ a f −1 (y), ξ χ(w)u(w)| det ∂f −1 |dwdξ = 1 (2π~) n Z R2n χ(y)e i ~ (f−1 (y)−f−1 (w))·ξ a f −1 (y), ξ χ(w)u(w)| det ∂f −1 (w)|dwdξ. So Pbχ is an operator whose Schwartz kernel KPeχ is compactly supported (since the support is contained in supp(χ)×supp(χ)). It follows (c.f. the computation on page 7 of Lecture 14) that Pbχ is the Weyl quantization of the symbol Z Rn e − i ~ w·ξKPeχ (x + w 2 , x − w 2 )dw which can be shown to be Schwartz. To calculate the Kohn-Nirenberg symbol b(y, η) of Pbχ, we use the oscillatory test (PSet 2) to get b(y, η) = e − i ~ y·ηPbχ(e i ~ y·η ) = 1 (2π~) n Z R2n e i ~ [(f−1 (y)−f−1 (w))·ξ+(w−y)·η] a(f −1 (y), ξ)χ(y)χ(w)| det ∂f −1 (w)|dwdξ =: 1 (2π~) n Z R2n e i ~ ϕy,η ay(ω, ξ)dωdξ. So according to the lemma of stationary phase (Lecture 5), b(y, η) ∼ X dϕy,η(p)=0 e i ~ ϕy,η(p) e iπ 4 sgn(d 2ϕy,η(p)) ay(p) | det d 2ϕy,η(p)| 1/2 X j ~ jLj (ay)(p), where Lj is a differential operator in w, ξ of order 2j and L0 = 1
4LECTURE20:SEMI-CLASSICALPSDOSONMANIFOLDSSo we calculate the critical points of the phase functionPy,n(w,s)=(f-1(g)-f-1(w))-E+(w-y) ·n,which is given byOgpyn=0—y=wandQupyn=0n=(f-1)In other words, the phase function Py,n admits a unique critical point=y,=(af)TnWe can also calculate the Hessian of the phase function, which, at the critical point.hastheform( dyn-(of-1)Tdpy.n0-(of-1)and thussgn(dpy,n) = 02 and Idet py.nl/2 =Idet af-(y)Thus we concludeb(y, n) ~ a (f-1(u), (f)n) x(g)2 1+EL;(ag)(y, (f)n)≥1Itremainstoproveb-a(f-(y),(of)Tn) x(y)? - hb, etg.1<j≤k-1This can be proved inductively by a similar argument to the function (y,n)aag.,bWe omit the details.[Another way to see P, is a semiclassical pseudodifferential operator: AssumingV is star-shaped.Let B be the n x n matrix whose (k,l)-entry is the integralo (w + t(y-w)dt.Then we have the formula (c.f. Lemma 2.6 in Lecture 5)f-(y)-f-(w) =B(y-w)and as a result, we can rewrite Px asx(g)e(w-m):BTa (f-1(g),) x(w)u(w)af-1()]dudePru(y)=(2元h）nJRx(u)e(w-m)5a (f-(g),(BT)-E) x(w)u(w)[af-(w)(BT)-dwds(2元h)n口from whichwe can calculatethesymbol.l2Reason: easy to show that the matrix is congruent to
4 LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS So we calculate the critical points of the phase function ϕy,η(w, ξ) = (f −1 (y) − f −1 (w)) · ξ + (w − y) · η, which is given by ∂ξϕy,η = 0 =⇒ y = w and ∂wϕy,η = 0 =⇒ η = (∂f −1 ) T ξ. In other words, the phase function ϕy,η admits a unique critical point w = y, ξ = (∂f) T η. We can also calculate the Hessian of the phase function, which, at the critical point, has the form d 2ϕy,η = d 2 wϕy,η −(∂f −1 ) T −(∂f −1 ) 0 and thus sgn(d 2ϕy,η) = 02 and | det d 2ϕy,η| 1/2 = | det ∂f −1 (y)|. Thus we conclude b(y, η) ∼ a f −1 (y),(∂f) T η χ(y) 2 " 1 +X j≥1 ~ jLj (ay)(y,(∂f) T η) # . It remains to prove b − a f −1 (y),(∂f) T η χ(y) 2 − X 1≤j≤k−1 ~ j bj ∈ ~ kS . This can be proved inductively by a similar argument to the function (y, η) α∂ β y,ηb. We omit the details. [Another way to see Pbχ is a semiclassical pseudodifferential operator: Assuming V is star-shaped. Let B be the n × n matrix whose (k, l)-entry is the integral R 1 0 ∂f−1 k ∂xl (w + t(y − w))dt. Then we have the formula (c.f. Lemma 2.6 in Lecture 5) f −1 (y) − f −1 (w) = B(y − w) and as a result, we can rewrite Pbχ as Peχu(y) = 1 (2π~) n Z R2n χ(y)e i ~ (y−w)·BT ξ a f −1 (y), ξ χ(w)u(w)|∂f −1 (w)|dwdξ = 1 (2π~) n Z R2n χ(y)e i ~ (y−w)·ξ a f −1 (y),(B T ) −1 ξ χ(w)u(w)|∂f −1 (w)|(B T ) −1 dwdξ from which we can calculate the symbol.] 2Reason: easy to show that the matrix is congruent to 0 I I 0 .
LECTURE20:SEMI-CLASSICALPSDOSONMANIFOLDS5TSemiclassicalPsDOsunder coordinatechange:invariant symbolsThe theorem we just proved also holds for invariant symbols:Theorem 1.2. Suppose a E Sm. Then the operator Px defined by (1) is a pseudo-differential operator whose symbol b E sm has the asymptotic erpansion(3)b(y,n) ~ a (f-1(y), (of)Tn) x(y)? + nib;(y, n),j≥1for somebje Sm-j.Proof.Theproof is lengthyand we will omit it.We need to apply a more complicatedversion of lemma of stationary phase, e.g. Theorem 7.7.7 in Hormander, The Anal-ysis of Partial Differential Operators Vol I. See also Theorem 18.1.7 in Hormander,TheAnalysis of Partial Differential Operators Vol III.Infact,according toTheorem18.1.7in thebook,onehasthefollowing somewhatdifferentexpressionof b:ga((),())(()(D)(e())=),b(y,n)~Zwhere pr(z)=f(r)-f(z)-of(z)(r-z).It is crucial to notice that pr(r)=0 and Opr()=o, so thatto produceonen-factor (and oneh-l-factor)from(hD)(epr()s)le=r=f-() you need at least two derivatives. As a result, the righthand side is an asymptotic expansion, but the Q-term is NOT in tlalsm-lal, but inhllal/2] sm-[lal/2]linstead.口Remark. Usually an asymptotic expansion in Sm has the forma(r,E)~a,(r,E),j=0where a e Sm, and aj e Sm-j. In this setting, negligible symbols are symbols inhoS-o2.SEMICLASSICALPSEUDODIFFERENTIALOPERATORONMANIFOLDS↑ Semiclassical pseudodifferential operator on manifoldsRecall that for each m, the symbol class Sm consisting of symbols satisfying[ogagal ≤ Ca,b(s)m-I8l for all a, βis invariant under coordinate changes in phase space, namely(c,) f(c,s) = (y = f(r),n =(of-1)Te)where f : Rn → Rn is a diffeomorphism that has bounded derivatives (of any order),e.g. f is the identity map outside a compact set
LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS 5 ¶ Semiclassical PsDOs under coordinate change: invariant symbols. The theorem we just proved also holds for invariant symbols: Theorem 1.2. Suppose a ∈ S m. Then the operator Peχ defined by (1) is a pseudodifferential operator whose symbol b ∈ S m has the asymptotic expansion (3) b(y, η) ∼ a f −1 (y),(∂f) T η χ(y) 2 + X j≥1 ~ j bj (y, η), for some bj ∈ S m−j . Proof. The proof is lengthy and we will omit it. We need to apply a more complicated version of lemma of stationary phase, e.g. Theorem 7.7.7 in H¨ormander, The Analysis of Partial Differential Operators Vol I. See also Theorem 18.1.7 in H¨ormander, The Analysis of Partial Differential Operators Vol III. In fact, according to Theorem 18.1.7 in the book, one has the following somewhat different expression of b: b(y, η) ∼ X α 1 α! ∂ α ξ a(f −1 (y),(∂f) T η)(χ(y))2 (~Dz) α (e i ~ ρx(z)·η )|z=x=f−1(y) , where ρx(z) = f(x) − f(z) − ∂f(z)(x − z). It is crucial to notice that ρx(x) = 0 and ∂ρx(x) = 0, so that to produce one η-factor (and one ~ −1 -factor) from (~Dz) α (e i ~ ρx(z)·ξ )|z=x=f−1(y) you need at least two derivatives. As a result, the right hand side is an asymptotic expansion, but the α-term is NOT in ~ |α|S m−|α| , but in ~ [|α|/2]S m−[|α|/2] instead. Remark. Usually an asymptotic expansion in S m has the form a(x, ξ) ∼ X∞ j=0 ~ j aj (x, ξ), where a ∈ S m, and aj ∈ S m−j . In this setting, negligible symbols are symbols in ~ ∞S −∞. 2. Semiclassical pseudodifferential operator on manifolds ¶ Semiclassical pseudodifferential operator on manifolds. Recall that for each m, the symbol class S m consisting of symbols satisfying |∂ α x ∂ β ξ a| ≤ Cα,βhξi m−|β| for all α, β is invariant under coordinate changes in phase space, namely (x, ξ) fe(x, ξ) = y = f(x), η = (∂f −1 ) T ξ , where f : R n → R n is a diffeomorphism that has bounded derivatives (of any order), e.g. f is the identity map outside a compact set.