LECTURE 21:EIGENVALUES AND EIGENFUNCTIONS OFh-PsDOsIn the next several lectures, we apply the theory of semiclassical pseudodiffer-ential operators to study spectral theory of Schrodinger operators on Rn and oncompact Riemannian manifolds (and in particular study the spectral theory of theLaplace-Beltrami operator on compact Riemannian manifolds).Here by“spectraltheory"we means(1) the asymptotic distribution of eigenvalues,(2) the spacial “distribution"of eigenfunctions (in phase space').In particular we would like to prove Weyl law and the quantum ergodicity theoremthatwementioned inLecturel.1. GENERAL SPECTRAL RESULTS OF -PSEUDODIFFERENTIAL OPERATORSDiscrete spectrum.As we have mentioned at the beginning of the course, we would like to studysemiclassical pseudodifferential operators with discrete spectrum, in which case thethe eigenvalues can be viewed as quantum energies (and the eigenfunctions canbe viewed as quantum states whose energies are the corresponding eigenvalues)whose semiclassical behavior should be closelyrelated to the behavior of the classicalsystem. Moreover, we have mentioned at the beginning of Lecture 12 that a veryuseful way to prove the discreteness of spectrum is through compact operators.because for a compact operator A on a separable Hilbert space H,. the spectrum o(A) consists of eigenvalues of finite multiplicities (with theonly possible exception being the origin which could be an accumulationpoint of the spectrum),· moreover if A is also self-adjoint, then the eigenvalues A,'s are all real, andthe corresponding eigenfunctions 's can be taken to be an orthonormalbasisof H, sothatA can bewritten asAu(r) =x(u, Pk)pk(r).Asasimpleapplicationofthisidea,weprovelalthough eigenfunctions are functions defined on the configuration space, it it still possible tostudy its “phase space distribution", as can be seen in today's lecture.1
LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS In the next several lectures, we apply the theory of semiclassical pseudodifferential operators to study spectral theory of Schrodinger operators on R n and on compact Riemannian manifolds (and in particular study the spectral theory of the Laplace-Beltrami operator on compact Riemannian manifolds). Here by “spectral theory” we means (1) the asymptotic distribution of eigenvalues, (2) the spacial “distribution” of eigenfunctions (in phase space1 ). In particular we would like to prove Weyl law and the quantum ergodicity theorem that we mentioned in Lecture 1. 1. General spectral results of ~-pseudodifferential operators ¶Discrete spectrum. As we have mentioned at the beginning of the course, we would like to study semiclassical pseudodifferential operators with discrete spectrum, in which case the the eigenvalues can be viewed as quantum energies (and the eigenfunctions can be viewed as quantum states whose energies are the corresponding eigenvalues) whose semiclassical behavior should be closely related to the behavior of the classical system. Moreover, we have mentioned at the beginning of Lecture 12 that a very useful way to prove the discreteness of spectrum is through compact operators, because for a compact operator A on a separable Hilbert space H, • the spectrum σ(A) consists of eigenvalues of finite multiplicities (with the only possible exception being the origin which could be an accumulation point of the spectrum), • moreover if A is also self-adjoint, then the eigenvalues λk’s are all real, and the corresponding eigenfunctions ϕk’s can be taken to be an orthonormal basis of H, so that A can be written as Au(x) = X k λkhu, ϕkiϕk(x). As a simple application of this idea, we prove 1 although eigenfunctions are functions defined on the configuration space, it it still possible to study its “phase space distribution”, as can be seen in today’s lecture. 1
2LECTURE21:EIGENVALUES ANDEIGENFUNCTIONSOF h-PSDOSTheorem 1.1. Suppose m is an order function on R2n withlimm(c,s)=+oo.I(r,5)1-00Suppose p e S(m) is real-valued and almost elliptic in S(m)(i.e. there erists C> 0such that p+C is elliptic). Then for h > O small enough, the operator pw is anunbounded linear operator on L?(Rn) with domain Hr(m), the eigenvalues of pware discrete real numbers with finite multiplicities which diverges to oo, and theeigenfunctions of pw can be chosen to form an orthonormal basis of L?(Rn).Proof. We use many results that we proved earlier:. Proposition 2.1 in Lecture 16: p E S(m) pW is well-defined as a mappW : Hn(m)→ L?(Rn).. Corollary 2.6 in Lecture 16: p+ C is elliptic in S(m) A := p+ChHr(m) → L?(Rn) has an inverse B := 6W : L?(Rn) → Hr(m) with b ES(1/m).. Theorem 1.5 in Lecture 12: lim(r,5)→ m(r,s) = +o0 -→ 6w is a com-pact operator on L?(IRn),whose eigenvalues has tobe discrete (with finitemultiplicity)with 0as the only accumulation point.. Computations at the beginning of Lecture 17: p is almost elliptic → p + iis elliptic..Corollary2.2inLecture16:p+iis ellipticpw,and thusA=p+CHr(m) C L? → L? is self-adjoint.A consequence: B - bw : L? → L? is also self-adjoint.To see this we start with any u,u L?(Rn). Then there existsu', u' e Hr(m) such that p+C" u' = u and p+C"' = w. Thus(Bu,v) =(BAu, Au") =(u,Av') =(Au,v') = (u,Bv)..Garding inequality in Lecture 15:The symbol of B is positive for h smallenough (sincetheleadingtermis1/(p+C)>0)theeigenvaluesofBarenonnegative.Thus by the spectral theory of compact self-adjoint operators, we can write B asBu=x(u, pk)pk(a).kwith Xk → o+. It follows that pW has discrete spectrum (with finite multiplicities)-C which diverges to +oo (which also implies that pW is unbounded on L?(R"),and the eigenfunctions of pw (which are the same as eigenfunctions of the compact口operator B) can be chosen to form an orthonormal basis of L?(Rn).Remark. In the case of compact Riemannian manifold, it is easy to see that thesame conclusion holds if we assume p E sm(T*M) for some m > 0
2 LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS Theorem 1.1. Suppose m is an order function on R 2n with lim |(x,ξ)|→∞ m(x, ξ) = +∞. Suppose p ∈ S(m) is real-valued and almost elliptic in S(m)(i.e. there exists C > 0 such that p + C is elliptic). Then for ~ > 0 small enough, the operator pb W is an unbounded linear operator on L 2 (R n ) with domain H~(m), the eigenvalues of pb W are discrete real numbers with finite multiplicities which diverges to ∞, and the eigenfunctions of pb W can be chosen to form an orthonormal basis of L 2 (R n ). Proof. We use many results that we proved earlier: • Proposition 2.1 in Lecture 16: p ∈ S(m) =⇒ pb W is well-defined as a map pb W : H~(m) → L 2 (R n ). • Corollary 2.6 in Lecture 16: p + C is elliptic in S(m) =⇒ A := p\+ C W : H~(m) → L 2 (R n ) has an inverse B := bb W : L 2 (R n ) → H~(m) with b ∈ S(1/m). • Theorem 1.5 in Lecture 12: lim|(x,ξ)|→∞ m(x, ξ) = +∞ =⇒ bb W is a compact operator on L 2 (R n ), whose eigenvalues has to be discrete (with finite multiplicity) with 0 as the only accumulation point. • Computations at the beginning of Lecture 17: p is almost elliptic =⇒ p + i is elliptic. • Corollary 2.2 in Lecture 16: p + i is elliptic =⇒ pb W , and thus A = p\+ C W : H~(m) ⊂ L 2 → L 2 is self-adjoint. A consequence: B = bb W : L 2 → L 2 is also self-adjoint. To see this we start with any u, v ∈ L 2 (R n ). Then there exists u 0 , v0 ∈ H~(m) such that p\+ C W u 0 = u and p\+ C W v 0 = v. Thus hBu, vi = hBAu0 , Av0 i = hu 0 , Av0 i = hAu0 , v0 i = hu, Bvi. • Garding inequality in Lecture 15: The symbol of B is positive for ~ small enough (since the leading term is 1/(p + C) > 0) =⇒ the eigenvalues of B are nonnegative. Thus by the spectral theory of compact self-adjoint operators, we can write B as Bu = X k λekhu, ϕkiϕk(x). with λek → 0 +. It follows that pb W has discrete spectrum (with finite multiplicities) 1 λek −C which diverges to +∞ (which also implies that pb W is unbounded on L 2 (R n )), and the eigenfunctions of pb W (which are the same as eigenfunctions of the compact operator B) can be chosen to form an orthonormal basis of L 2 (R n ). Remark. In the case of compact Riemannian manifold, it is easy to see that the same conclusion holds if we assume p ∈ S m(T ∗M) for some m > 0.
3LECTURE21:EIGENVALUESANDEIGENFUNCTIONSOFh-PSDOSErample.ConsiderSchrodingeroperatorP=-h△+Von L2(IR"), where V e C(IR") is a real-valued smooth function such that?(1) V has “polynomial growth" in the sense that there exists k > 0 such that[0V(r)/ ≤Ca(c),(2)V is“"almost elliptic"in S((c))in the sense thatV(r) ≥ c(r)*- C.[Notethat this implies V(r)→+oo as|→o.Of courseyou may replace(r)byanyorderfunctionm(r)whichdivergesto+ooasr-→oo.]Then we can apply the Proposition 1.1 to P + C.Id with order functionm(r,)=(s)? +(r)kto concludethat P has discrete spectrum that diverges to oo and has an L?eigenbasis.Erample. Similarly, on a compact Riemannian manifold (M,g), for any potentialfunction V eCo(M),the Schrodinger operatorP=-△+V(and in particular the Laplace operatoritself)has discrete spectrum that divergesto oo and has an L?-eigenbasis.Regularityofeigenfunctions.Although at first glance, the eigenfunctions of pw are only L?-functions, theyhave much better regularity:Theorem 1.2. Suppose m ≥ 1 is an order function on R2n, p E S(m) is a real-valued almost elliptic symbol.Suppose un EL?(Rn)is an eigenfunction of pw, i.e.(1)pWun(r) = Anun(a).Then for any k = O,1,2,.., we have un Hr(mk). Moreover, if h E [a, B] forsome constants a,β, then there erists constants Ck so thatIunllH(mk)≤Ckl|unl/L2.2According to a theorem of Friedrichs, if V() → +oo as [l -→ co, then P has discrete spectrum.So thepolynomial growth condition can beremoved.Fora proof, c.f.Theorem XIIl.16 in M.Reedand B.Simon, Methods of Modern Mathematical Physics, Volume 4
LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS 3 Example. Consider Schr¨odinger operator P = −~ 2∆ + V on L 2 (R n ), where V ∈ C ∞(R n ) is a real-valued smooth function such that2 (1) V has “polynomial growth” in the sense that there exists k > 0 such that |∂ α x V (x)| ≤ Cαhxi k , (2) V is “almost elliptic” in S(hxi k ) in the sense that V (x) ≥ chxi k − C. [Note that this implies V (x) → +∞ as |x| → ∞. Of course you may replace hxi k by any order function me (x) which diverges to +∞ as x → ∞.] Then we can apply the Proposition 1.1 to P + C · Id with order function m(x, ξ) = hξi 2 + hxi k to conclude that P has discrete spectrum that diverges to ∞ and has an L 2 - eigenbasis. Example. Similarly, on a compact Riemannian manifold (M, g), for any potential function V ∈ C ∞(M), the Schr¨odinger operator P = −~ 2∆ + V (and in particular the Laplace operator ∆ itself) has discrete spectrum that diverges to ∞ and has an L 2 -eigenbasis. ¶Regularity of eigenfunctions. Although at first glance, the eigenfunctions of pb W are only L 2 -functions, they have much better regularity: Theorem 1.2. Suppose m ≥ 1 is an order function on R 2n , p ∈ S(m) is a realvalued almost elliptic symbol. Suppose u~ ∈ L 2 (R n ) is an eigenfunction of pb W , i.e. (1) pb W u~(x) = λ~u~(x). Then for any k = 0, 1, 2, · · · , we have u~ ∈ H~(mk ). Moreover, if λ~ ∈ [α, β] for some constants α, β, then there exists constants Ck so that ku~kH~(mk) ≤ Ckku~kL2 . 2According to a theorem of Friedrichs, if V (x) → +∞ as |x| → ∞, then P has discrete spectrum. So the polynomial growth condition can be removed. For a proof, c.f. Theorem XIII.16 in M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume 4
4LECTURE21:EIGENVALUESANDEIGENFUNCTIONS OFh-PSDOSProof.We have(pW +C)un=(An+C)unBy ellipticity, there exists symbol b e S(1/m) so that (pw + C)-1 = bw. It followsthat(6W)*L? c Hr(mk).Sinceun = (An+C)*(6W)un,口the conclusion follows.Erample. For the Schrodinger operator H = -?+ V(r) on Rn whose potential isalmost elliptic and of polynomial growth as described above, then the eigenfunctionun E (Rn). [We have seen this for the Harmonic oscillator in Lecture 3.]Concentrationofeigenfunction inphasespace.Next weprove thefollowing classical-quantum correspondence phenomena:theeigenfunctions associated to eigenvalues close to a given “energy level" E will “con-centrate"on the corresponding energy surface in the phase space:Theorem 1.3. Suppose m is an order function satisfying lim(r,)l-→o m(r, E) = +00,p E S(m) is a real-valued almost elliptic symbol. Suppose un E L?(Rn) is such thatpWu= Ahun. Suppose ce S(1) is a symbol satisfying[(r,s) I p(r, s) =E) nsupp(c) = 0,where E>0. Then there erists sufficiently small >0 such that if [n-El<s,thenIW unl/L2(R") =O()unllL2(R")Proof. Since the order function m is proper and p is almost elliptic, p is also properand thus the level set p-(E) =[(r,s) Ip(r,)=E) is compact. So we can find acut-off function X E Co(IRn) such that0≤x≤1, x=1 on p-1(E), x=0 on supp(c)Consider the symbolb(r,) =p(,) - +ixThen b is elliptic in S(m). ThusBw=pW-入n+ixWis invertible with inverse (6w)-1 = bW for some bi e S(1/m),. Since supp(c) nsupp(x)=0, we have (c.f. Corollary 1.4 in Lecture9)wb"x"xW = O(h).ItfollowsWun=WhW(w-入n+ixw)un=O(h)口
4 LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS Proof. We have (pb W + C)u~ = (λ~ + C)u~. By ellipticity, there exists symbol b ∈ S(1/m) so that (pb W + C) −1 = bb W . It follows that (bb W ) kL 2 ⊂ H~(mk ). Since u~ = (λ~ + C) k (bb W ) ku~, the conclusion follows. Example. For the Schrodinger operator H = −~ 2∆ + V (x) on R n whose potential is almost elliptic and of polynomial growth as described above, then the eigenfunction u~ ∈ S (R n ). [We have seen this for the Harmonic oscillator in Lecture 3.] ¶Concentration of eigenfunction in phase space. Next we prove the following classical-quantum correspondence phenomena: the eigenfunctions associated to eigenvalues close to a given “energy level” E will “concentrate” on the corresponding energy surface in the phase space: Theorem 1.3. Suppose m is an order function satisfying lim|(x,ξ)|→∞ m(x, ξ) = +∞, p ∈ S(m) is a real-valued almost elliptic symbol. Suppose u~ ∈ L 2 (R n ) is such that pb W u = λ~u~. Suppose c ∈ S(1) is a symbol satisfying {(x, ξ) | p(x, ξ) = E} ∩ supp(c) = ∅, where E > 0. Then there exists sufficiently small δ > 0 such that if |λ~ − E| ≤ δ, then kbc W u~kL2(Rn) = O(~ ∞)ku~kL2(Rn) . Proof. Since the order function m is proper and p is almost elliptic, p is also proper and thus the level set p −1 (E) = {(x, ξ) | p(x, ξ) = E} is compact. So we can find a cut-off function χ ∈ C ∞ 0 (R n ) such that 0 ≤ χ ≤ 1, χ ≡ 1 on p −1 (E), χ ≡ 0 on supp(c). Consider the symbol b(x, ξ) = p(x, ξ) − λ~ + iχ Then b is elliptic in S(m). Thus bb W = pb W − λ~ + iχb W is invertible with inverse (bb W ) −1 = bb1 W for some b1 ∈ S(1/m). Since supp(c) ∩ supp(χ) = ∅, we have (c.f. Corollary 1.4 in Lecture 9) bc W bb1 W χb W = O(~ ∞). It follows bc W u~ = bc W bb1 W (pb W − λ~ + iχb W )u~ = O(~ ∞).
LECTURE21:EIGENVALUESANDEIGENFUNCTIONS OFh-PSDOS52.WAVEFRONTSETPROPERTIESWavefront set of a semiclassical family of states.Now we introduce the conception of wavefront set associated with a family ofwavefunctions un (not necessarily eigenfunctions). Roughly speaking, the wavefrontset of a family of functions un is the region in the phase space where un are microlo-calized:Definition 2.1. Let un be a family of L?-normalized functions on Rn. We definethesemiclassical wavefrontset (whichis alsoknownasthefreguencyset.asitwasfirst studied by V. Guillemin and S. Sternberg in their classical book GeometricAsymptotics) of un to be the set WFr(u) C T*Rn characterized by(ro, So) WFn(u) 3a E S(1) with [a(ro, o)/ ≥ c > 0 for all h, such that(2)aw unll 2 = O(h~).By definition, the complement of WFr(u) is always open, thus WFr(u) must bea closed subset of R2n.Erample (truncated plane wave). Fix So e R" and x E Co(R") such that IxllL2 = 1.ConsiderthefollowingfamilyofL2-normalizedfastoscillatingfunctionseco(r,h) := x(r)ei50-2/hThenWFr(e) =[(r, So) / E supp(x)To see this we calculateet(a-)sa(r,E)x(y)esodydeaWu(r) :(2元h)n/The critical point of the phase function pr(y, E) := (r - y) - + So - y isy=,=so.So for any (ci, si) [(r, So) I E supp(x)), if we can take a such that a(t1,Si) + 0and (r,So) supp(a),then by“non-stationary phase lemma"(Proposition 1.2 inLecture 5) we get awu(r) = O(h) and thus (ri,si) W Fh(ego).Conversely suppose r is a point with x(r) > 0 and suppose a(r, So) > 0. Thenby the lemma of stationary phase,aWu(r) =Ca(r, So)x(r) +O(h)for some nonzero constant C. So (r, So) e WFr(ego). Since WFr(e) is closed, theconclusionfollows.It turns out that in the definition of wavefront set, we may replace a e S(1) bya E Co(IR2n):
LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS 5 2. Wavefront set properties ¶Wavefront set of a semiclassical family of states. Now we introduce the conception of wavefront set associated with a family of wavefunctions u~ (not necessarily eigenfunctions). Roughly speaking, the wavefront set of a family of functions u~ is the region in the phase space where u~ are microlocalized: Definition 2.1. Let u~ be a family of L 2 -normalized functions on R n . We define the semiclassical wavefront set (which is also known as the frequency set, as it was first studied by V. Guillemin and S. Sternberg in their classical book Geometric Asymptotics) of u~ to be the set WF~(u) ⊂ T ∗R n characterized by (2) (x0, ξ0) 6∈ WF~(u) ⇐⇒ ∃a ∈ S(1) with |a(x0, ξ0)| ≥ c > 0 for all ~, such that kba W u~kL2 = O(~ ∞). By definition, the complement of WF~(u) is always open, thus WF~(u) must be a closed subset of R 2n . Example (truncated plane wave). Fix ξ0 ∈ R n and χ ∈ C ∞ 0 (R n ) such that kχkL2 = 1. Consider the following family of L 2 -normalized fast oscillating functions eξ0 (x, ~) := χ(x)e iξ0·x/~ . Then WF~(eξ0 ) = {(x, ξ0) | x ∈ supp(χ)}. To see this we calculate ba W u(x) = 1 (2π~) n Z R2n e i ~ (x−y)·ξ a(x, ξ)χ(y)e i ~ ξ0·y dydξ The critical point of the phase function ϕx(y, ξ) := (x − y) · ξ + ξ0 · y is y = x, ξ = ξ0. So for any (x1, ξ1) 6∈ {(x, ξ0) | x ∈ supp(χ)}, if we can take a such that a(x1, ξ1) 6= 0 and (x, ξ0) 6∈ supp(a), then by “non-stationary phase lemma” (Proposition 1.2 in Lecture 5) we get ba W u(x) = O(~ ∞) and thus (x1, ξ1) 6∈ W F~(eξ0 ). Conversely suppose x is a point with χ(x) > 0 and suppose a(x, ξ0) > 0. Then by the lemma of stationary phase, ba W u(x) = Ca(x, ξ0)χ(x) + O(~) for some nonzero constant C. So (x, ξ0) ∈ WF~(eξ0 ). Since WF~(eξ0 ) is closed, the conclusion follows. It turns out that in the definition of wavefront set, we may replace a ∈ S(1) by a ∈ C ∞ 0 (R 2n ):
6LECTURE21:EIGENVALUESANDEIGENFUNCTIONSOFh-PSDOSProposition2.2.Suppose（ro,So)WFr(u),thenforanyb ECo(IR2n)withsupport sufficiently close to (ro, So), we have 6w unll2 = O(h).Proof. Suppose a E S(1) is a symbol satisfying (2). Take a cut-off x E Co(R2n)with x =1 in a neighborhood of supp(b), such thatIx(c, )(a(r, ) -a(ro, So)) + a(ro, So)/ ≥ c/2 > 0i.e. it is elliptic in S(1) (here we used“"supp(b)is sufficiently close to (ro,So)"). Soby Theorem 1.7 in Lecture 14, for h small enough, there exists c E S(1) such that=[W (aw-a(ro, So)Id) +a(ro, S0)Id)-1.So we havew-bwawxwaw+a(ro,so)6wawi-xwBy condition (2), the first term is O(h). Since supp(b)nsupp(1-x) = 0, Corollary口1.4 in Lecture 9 tells us that the second term is O(h)Wavefront set has the following remarkable property:Theorem 2.3.Supposea=a(r,S;h)eS(m) forall h.ThenWFr(aWu) C WFr(u).Proof. Suppose (ro, So) WFr(u). Choose b E Co(R2n) such that b(ro, So) + 0 and6wu = O(h). ThenVWaw=b*a=aw+o(h)for some symbol c E Co(R2n) satisfying supp(c) C supp(b). It follows6WaWull/2 = Wull2+(h)=(h)口which implies (ro,So)gWFr(aWu),As a consequence, we immediately get the following nice property of ellipticoperators:Corollary 2.4.Ifa is elliptic in S(m), then WFr(aWu) =WFr(u)Remark. Wavefront set is invariant under coordinate change: If f : Rn → Rn is adiffeomorphism that is identity outside a compact set, thenWFr(f*u) = [(r, (df)TE) : (f(r),E) E WFr(u))As a consequence, it is well-defined as a subset in T*M for compact manifolds M
6 LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS Proposition 2.2. Suppose (x0, ξ0) 6∈ WF~(u), then for any b ∈ C ∞ 0 (R 2n ) with support sufficiently close to (x0, ξ0), we have kbb W u~kL2 = O(~ ∞). Proof. Suppose a ∈ S(1) is a symbol satisfying (2). Take a cut-off χ ∈ C ∞ 0 (R 2n ) with χ ≡ 1 in a neighborhood of supp(b), such that |χ(x, ξ)(a(x, ξ) − a(x0, ξ0)) + a(x0, ξ0)| ≥ c/2 > 0, i.e. it is elliptic in S(1) (here we used “supp(b) is sufficiently close to (x0, ξ0)”). So by Theorem 1.7 in Lecture 14, for ~ small enough, there exists c ∈ S(1) such that bc W = [χb W (ba W − a(x0, ξ0)Id) + a(x0, ξ0)Id]−1 . So we have bb W = bb W bc W χb W ba W + a(x0, ξ0)bb W bc W 1[− χ W . By condition (2), the first term is O(~ ∞). Since supp(b)∩supp(1−χ) = ∅, Corollary 1.4 in Lecture 9 tells us that the second term is O(~ ∞). Wavefront set has the following remarkable property: Theorem 2.3. Suppose a = a(x, ξ; ~) ∈ S(m) for all ~. Then WF~(ba W u) ⊂ WF~(u). Proof. Suppose (x0, ξ0) 6∈ WF~(u). Choose b ∈ C ∞ 0 (R 2n ) such that b(x0, ξ0) 6= 0 and bb W u = O(~ ∞). Then bb W ba W = b ? a dW = bc W + O(~ ∞) for some symbol c ∈ C ∞ 0 (R 2n ) satisfying supp(c) ⊂ supp(b). It follows kbb W ba W ukL2 = kbc W ukL2 + O(~ ∞) = O(~ ∞) which implies (x0, ξ0) 6∈ WF~(ba W u). As a consequence, we immediately get the following nice property of elliptic operators: Corollary 2.4. If a is elliptic in S(m), then WF~(ba W u) = WF~(u) Remark. Wavefront set is invariant under coordinate change: If f : R n → R n is a diffeomorphism that is identity outside a compact set, then WF~(f ∗u) = {(x,(df) T ξ) : (f(x), ξ) ∈ WF~(u)}. As a consequence, it is well-defined as a subset in T ∗M for compact manifolds M.
7LECTURE21:EIGENVALUESANDEIGENFUNCTIONSOFh-PSDOSWavefront set of a semiclassical family of eigenfunctions.Foreigenfunctions, we haveTheorem 2.5. Suppose An are eigenvalues of pW such that [An-Eol 1/2 for (r,E) E B(Ck,S),rk). We definea = ark.kThen a > 1/2 in a neighborhood of p-'(Eo).Replace aak.sk byak../a and let a =. Then a = 1 in a neighborhood of p-(Eo). By Theorem 1.3,WaWun=n+1-a"un=uh+(h),口acontradiction.3According to Weyl's law that we will prove later, as h→ 0 there are many eigenvalues satisfyingthis condition
LECTURE 21: EIGENVALUES AND EIGENFUNCTIONS OF ~-PSDOS 7 ¶Wavefront set of a semiclassical family of eigenfunctions. For eigenfunctions, we have Theorem 2.5. Suppose λ~ are eigenvalues of pb W such that |λ~−E0| 1/2 for (x, ξ) ∈ B((xk, ξk), rk). We define ea = Paxk,ξk . Then ea > 1/2 in a neighborhood of p −1 (E0). Replace axk,ξk by axk,ξk /ea and let a =. Then a ≡ 1 in a neighborhood of p −1 (E0). By Theorem 1.3, ba W u~ = u~ + 1[− a W u~ = u~ + O(~ ∞), a contradiction. 3According to Weyl’s law that we will prove later, as ~ → 0 there are many eigenvalues satisfying this condition.