LECTURE29-30FIO-SEMICLASSICALFIOs1.GENERATINGFUNCTIONSWITHRESPECTTOAFIBRATION Recall: Generating functions of a horizontal Lagrangian.Let M = T*X be the cotangent bundle of a smooth manifold X. Recall. A horizontal submanifold (=the graph of a 1-form μ)Aμ=((,)IX)is a Lagrangian submanifold of M if and only if dμ = 0..If μ=dp is exact, then we call E C(X)agenerating function of AuFor example, if we take X =IRn× Rn,thenp(r,y) =-r-yis a generating function of the Lagrangian submanifoldA=(a,y,s,n) Is =-y,n= -a),Notethat A=2 oG is the“twisting"of thegraph of thesymplectomorphismF: T*Rn →T*Rn,(c,$)-→ (-s,r). Generating function with respect to a fibration.Unfortunately not all Lagrangian submanifolds are generated (even locally) bythose kind of generating functions: there are many interesting non-horizontal Lagrangian submanifolds.For example,anysmooth map f:X-→Ylifts"toacanonical relation(whichgeneralize the naturality of the cotangent bundle:anydiffeomorphismliftstoasymplectomorphismbetweencotangentbundlesIf :=02(N*Gf) = [(r,y,S,n) I y = f(),S = (df)Tn).1Inwhat followswewillextend theconceptionofgeneratingfunctionsbyintroduc-ing“auxiliaryvariables"(to"separate the non-horizontal directions)so that everyLagrangian submanifold of T*X is locally represented by such a generating functionLetZ.Xaresmoothmanifolds and :Z→X asmoothfibration.ThenT =[(z,S,,) [ =(z),C = (d2)Ts)is a canonical relation in T*Z × (T*X)-. Let A。be a horizontal Lagrangian sub-manifold of T*Z generated by a function E Co(Z), i.e.A = [(z, dp(z)) / zE Z).ICheck this expression!1
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 1. Generating functions with respect to a fibration ¶ Recall: Generating functions of a horizontal Lagrangian. Let M = T ∗X be the cotangent bundle of a smooth manifold X. Recall • A horizontal submanifold (=the graph of a 1-form µ) Λµ = {(x, µx) | x ∈ X} is a Lagrangian submanifold of M if and only if dµ = 0. • If µ = dϕ is exact, then we call ϕ ∈ C ∞(X) a generating function of Λµ. For example, if we take X = R n x × R n y , then ϕ(x, y) = −x · y is a generating function of the Lagrangian submanifold Λ = {(x, y, ξ, η) | ξ = −y, η = −x}. Note that Λ = σ2 ◦ GF is the “twisting” of the graph of the symplectomorphism F : T ∗R n x → T ∗R n y , (x, ξ) 7→ (−ξ, x). ¶ Generating function with respect to a fibration. Unfortunately not all Lagrangian submanifolds are generated (even locally) by those kind of generating functions: there are many interesting non-horizontal Lagrangian submanifolds. For example, any smooth map f : X → Y “lifts” to a canonical relation (which generalize the naturality of the cotangent bundle: any diffeomorphism lifts to a symplectomorphism between cotangent bundles) Γf := σ2(N ∗Gf ) = {(x, y, ξ, η) | y = f(x), ξ = (dfx) T η}. 1 In what follows we will extend the conception of generating functions by introducing “auxiliary variables” (to “separate the non-horizontal directions) so that every Lagrangian submanifold of T ∗X is locally represented by such a generating function. Let Z, X are smooth manifolds and π : Z → X a smooth fibration. Then Γπ = {(z, ζ, x, ξ) | x = π(z), ζ = (dπz) T ξ} is a canonical relation in T ∗Z × (T ∗X) −. Let Λϕ be a horizontal Lagrangian submanifold of T ∗Z generated by a function ϕ ∈ C ∞(Z), i.e. Λϕ = {(z, dϕ(z)) | z ∈ Z}. 1Check this expression! 1
2LECTURE29-30FIO-SEMICLASSICALFIOSThen one can think of A as a morphism from “pt" to T*Z. So if F and A aretransversally composable,2 thenA:=o A= ((,S) I(z, S,c, ) I,(z,C) EAg)(1)=[(r,) / =(z) and dp2= (d)T)is a canonical relation from “pt" to T*X, i.e. a Lagrangian submanifold of T*X.Definition1.1.WecallECo(Z)ageneratingfunctionofACT*Xwithrespectto thefibration π:Z→X. Consequence of transversality.Next let's look for conditions so that Fand Aare transversally composable.Let H*Z be the horizontal subbundle of T*Z which is the image of T under theprojection p:I→T*Z × T*X →T*Z. In other words, the fiber of H*Z at z is(H*Z) = [(d2)TE/E T()X),Since H*Z is a subbundle of T*Z, one has a vector bundle short exact sequence(2)0→H*Z-T*Z→V*Z→0,where (V*Z) = T+Z/(H*Z) T*(π-1(π(z)) is the cotangent space to the fiberthrough z. From the exact sequence, any section dp of T*Z gives a section dvertypof V*Z, and H*Z gets projected to the zero section of V*Z.Notethetransversalitycondition of Tand Anowbecomesπ : A。→ T*Z intersect p: F→ T*Z transversallyAintersectp:I→T*ZtransversallyinT*ZAintersectH*ZtransversallyinT*Zduertp intersect the zero section Z transversally in V*ZIt follows that under the transversal intersection assumption, the intersection(3)Cp:= [zEZ / (duertp)z= 0]is a submanifold of Z whose dimension isdimC= dim Z + dim Z - dim V*Z = dim X.Furthermore, the short exact sequence also implies that at any z e Ced= (d)Tfor a unique e T()X, and by (1), A= Io A, is the image of the mapCe → T*X, z-→ (π(z),S).2Recall from Lecture 27 that two canonical relations are transversally composable if the mapsπ and p intersect transversally, which implies that the map q =k ot is of constant rank; moreoverweassume rotis proper with connected fiber
2 LECTURE 29-30 FIO – SEMICLASSICAL FIOS Then one can think of Λϕ as a morphism from “pt” to T ∗Z. So if Γπ and Λϕ are transversally composable,2 then (1) Λ := Γπ ◦ Λϕ = {(x, ξ) | ∃(z, ζ, x, ξ) ∈ Γπ, ∃(z, ζ) ∈ Λϕ} = {(x, ξ) | x = π(z) and dϕz = (dπz) T ξ}. is a canonical relation from “pt” to T ∗X, i.e. a Lagrangian submanifold of T ∗X. Definition 1.1. We call ϕ ∈ C ∞(Z) a generating function of Λ ⊂ T ∗X with respect to the fibration π : Z → X. ¶ Consequence of transversality. Next let’s look for conditions so that Γπ and Λϕ are transversally composable. Let H∗Z be the horizontal subbundle of T ∗Z which is the image of Γπ under the projection ρ : Γπ ,→ T ∗Z × T ∗X → T ∗Z. In other words, the fiber of H∗Z at z is (H ∗Z)z = {(dπz) T ξ | ξ ∈ T ∗ π(z)X}. Since H∗Z is a subbundle of T ∗Z, one has a vector bundle short exact sequence (2) 0 → H ∗Z → T ∗Z → V ∗Z → 0, where (V ∗Z)z = T ∗ z Z/(H∗Z)z ' T ∗ z (π −1 (π(z))) is the cotangent space to the fiber through z. From the exact sequence, any section dϕ of T ∗Z gives a section dvertϕ of V ∗Z, and H∗Z gets projected to the zero section of V ∗Z. Note the transversality condition of Γπ and Λϕ now becomes π : Λϕ → T ∗Z intersect ρ : Γπ → T ∗Z transversally ⇐⇒Λϕ intersect ρ : Γπ → T ∗Z transversally in T ∗Z ⇐⇒Λϕ intersect H ∗Z transversally in T ∗Z ⇐⇒dvertϕ intersect the zero section Z transversally in V ∗Z It follows that under the transversal intersection assumption, the intersection (3) Cϕ := {z ∈ Z | (dvertϕ)z = 0} is a submanifold of Z whose dimension is dim Cϕ = dim Z + dim Z − dim V ∗Z = dim X. Furthermore, the short exact sequence also implies that at any z ∈ Cϕ, dϕz = (dπz) T ξ for a unique ξ ∈ T ∗ π(z)X, and by (1), Λ = Γπ ◦ Λϕ is the image of the map Cϕ → T ∗X, z 7→ (π(z), ξ). 2Recall from Lecture 27 that two canonical relations are transversally composable if the maps π and ρ intersect transversally, which implies that the map α = κ ◦ ι is of constant rank; moreover we assume κ ◦ ι is proper with connected fiber
3LECTURE29-30FIO-SEMICLASSICALFIOSWe will denote this map by Pe:(4)Po: C→ A The generating function in local coordinates.Locally assume X is an open subset of Rn and Z = X × Rk. Let (r, s) be thecoordinates on Z so that = p(r, s). Then C C Z is defined by the k equations04=0,(5)i=1,2,...k,siand the transversality condition becomesTransversality Assumption: the differentials of these functions,i= 1,2,.., kare linearly independent.In this case, A C T*X is the image of the embeddingCe→T*X, (r,s) - (c, drp(r,s))Erample. Let Y C X be a submanifold defined by k equationsfi(c) =...= ft(r) = 0and assume that these equations are functionally independent, i.e. dfi, "., dfk arelinearly independent. Let p : X × Rk→ R be the function(6)((r, s) =fi(r)si.=fiWe claim that A = F, o A is the conormal bundle N*Y of Y. In fact, since we seeCe=Y × Rk,and the map C → T*X is given by(r,s) → (r, sidfi(r))The conclusion follows since daf's span the conormal fiber to Y at each r.Erample.In particular, if we let X =Rn × Rn and let Y be the diagonalY =diag(X) = ((r,a) / r eX),then Y C X is defined by the equationsTi-yi=0,i=1,2,.,n.So the function(7)p(r,y,s) = (r-y) -s =(ri-yi)siis the generating function of N*(diag(X))
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 3 We will denote this map by pϕ: (4) pϕ : Cϕ → Λ. ¶ The generating function in local coordinates. Locally assume X is an open subset of R n and Z = X × R k . Let (x, s) be the coordinates on Z so that ϕ = ϕ(x, s). Then Cϕ ⊂ Z is defined by the k equations (5) ∂ϕ ∂si = 0, i = 1, 2, · · · k, and the transversality condition becomes Transversality Assumption: the differentials of these functions, d ∂ϕ ∂si , i = 1, 2, · · · , k are linearly independent. In this case, Λ ⊂ T ∗X is the image of the embedding Cϕ → T ∗X, (x, s) 7→ (x, dxϕ(x, s)). Example. Let Y ⊂ X be a submanifold defined by k equations f1(x) = · · · = fk(x) = 0 and assume that these equations are functionally independent, i.e. df1, · · · , dfk are linearly independent. Let ϕ : X × R k → R be the function (6) ϕ(x, s) = Xfi(x)si . We claim that Λ = Γπ ◦Λφ is the conormal bundle N∗Y of Y . In fact, since ∂ϕ ∂si = fi we see Cϕ = Y × R k , and the map Cϕ → T ∗X is given by (x, s) 7→ (x,Xsidxfi(x)). The conclusion follows since dxfi ’s span the conormal fiber to Y at each x. Example. In particular, if we let X = R n × R n and let Y be the diagonal Y = diag(X) = {(x, x) | x ∈ X}, then Y ⊂ X is defined by the equations xi − yi = 0, i = 1, 2, · · · , n. So the function (7) ϕ(x, y, s) = (x − y) · s = X i (xi − yi)si , is the generating function of N∗ (diag(X))
4LECTURE29-30FIO-SEMICLASSICALFIOS General facts about the generating function.Of course one may ask: Given any Lagrangian submanifold A C T*X, does thereexist any fibration : Z → X and E Co(Z) so that is a generating functionof A? If yes, is it unique? We state without proof the following general results. Fordetails, c.f.Guillemin-Sternberg $5.9 and g5.11:Theorem 1.2 (Existence). For any Lagrangian submanifold A C T*X and anypEA, there erist a fibration T :Z-→X and a smooth function ECo(Z) so that is a generating function of A near p.Theorem 1.3. (Uniqueness up to “Hormander moves") Suppose Pi, i = 1,2, aregenerating functions for the same Lagrangian submanifold A T*X with respect tofibrations ,:Z,X.Then locally one can obtain one description from the otherby applying a sequence of “moves" of the following three types:(1) Adding a constant: replace y by y+c.(2) Equivalence: For a diffeomorphism g :Z-→Z, replace (π, p) by (g*, g*).(3) Increasing the number of fiber variables: replace Z by Z = Z × Rd and p byp(z) +(Az, z), where A is a non-degenerate d x d matrir.In Guillemin-Sternberg Chapter 5,many nice facts wereproven for the generat-ing functions (with respect to fibrations).We list several of them without proof:. If e Mor(T*X, T*Y) is a canonical relation, π : Z → X xY a fibration, and a generating function of T with respect to this fibration. Suppose locallyp = p(r,y,s). Then the function (y,r,s) = -p(r,y,s) is a generatingfunctionforthetransposecanonicalrelationIT = (y, n, a,S)l(r,S, y, n) e F) e Mor(T*Y,T*X),. If I; Mor(T*X, T*X+1), i = 1, 2 are canonical relations which are transver-sally composable, πi : Zi → X, × Xi+1 are fibrations and pi E Co(Zi) aregenerating functions for I, with respect to i, then one can construct a fi-bration Z→Xi×X3with(8)Z = (π1 × 2)-1(Xi × △x × X3),Let be the restriction to Z of the function(9)(z1, 22) -→ (P1(z1) + (P2(22),then is a generating function for 2oTi with respect to the fibrationZ →Xi × X3.. Suppose that the fibration π : Z → X can be factored as a succession offibrations = Ti o o, where πo : Z→ Zi and Ti : Zi → X are fibrations.Moreover, suppose that the restriction of the generating function to eachfiber -'(z)has a unique non-degenerate critical point(zi), so thatwegeta section :Z→Z.Then the function i =i@ is a generating functionof A with respect to T1
4 LECTURE 29-30 FIO – SEMICLASSICAL FIOS ¶ General facts about the generating function. Of course one may ask: Given any Lagrangian submanifold Λ ⊂ T ∗X, does there exist any fibration π : Z → X and ϕ ∈ C ∞(Z) so that ϕ is a generating function of Λ? If yes, is it unique? We state without proof the following general results. For details, c.f. Guillemin-Sternberg §5.9 and §5.11: Theorem 1.2 (Existence). For any Lagrangian submanifold Λ ⊂ T ∗X and any p ∈ Λ, there exist a fibration π : Z → X and a smooth function ϕ ∈ C ∞(Z) so that ϕ is a generating function of Λ near p. Theorem 1.3. (Uniqueness up to “H¨ormander moves”) Suppose ϕi, i = 1, 2, are generating functions for the same Lagrangian submanifold Λ ⊂ T ∗X with respect to fibrations πi : Zi → X. Then locally one can obtain one description from the other by applying a sequence of “moves” of the following three types: (1) Adding a constant: replace ϕ by ϕ + c. (2) Equivalence: For a diffeomorphism g :Z →Ze, replace (π, ϕ) by (g ∗π, g∗ϕ). (3) Increasing the number of fiber variables: replace Z by Z = Z × R d and ϕ by ϕ(z) + 1 2 hAz, zi, where A is a non-degenerate d × d matrix. In Guillemin-Sternberg Chapter 5, many nice facts were proven for the generating functions (with respect to fibrations). We list several of them without proof: • If Γ ∈ Mor(T ∗X, T∗Y ) is a canonical relation, π : Z → X×Y a fibration, and ϕ a generating function of Γ with respect to this fibration. Suppose locally ϕ = ϕ(x, y, s). Then the function ψ(y, x, s) = −ϕ(x, y, s) is a generating function for the transpose canonical relation Γ T = {(y, η, x, ξ)|(x, ξ, y, η) ∈ Γ} ∈ Mor(T ∗Y, T∗X). • If Γi ∈ Mor(T ∗Xi , T∗Xi+1), i = 1, 2 are canonical relations which are transversally composable, πi : Zi → Xi × Xi+1 are fibrations and ϕi ∈ C ∞(Zi) are generating functions for Γi with respect to πi , then one can construct a fi- bration Z → X1 × X3 with (8) Z = (π1 × π2) −1 (X1 × ∆X2 × X3), Let ϕ be the restriction to Z of the function (9) (z1, z2) 7→ ϕ1(z1) + ϕ2(z2), then ϕ is a generating function for Γ2 ◦ Γ1 with respect to the fibration Z → X1 × X3. • Suppose that the fibration π : Z → X can be factored as a succession of fibrations π = π1 ◦ π0, where π0 : Z → Z1 and π1 : Z1 → X are fibrations. Moreover, suppose that the restriction of the generating function ϕ to each fiber π −1 0 (z1) has a unique non-degenerate critical point γ(z1), so that we get a section γ : Z1 → Z. Then the function φ1 = γ ∗ 1φ is a generating function of Λ with respect to π1
5LECTURE29-30FIO-SEMICLASSICALFIOS2. OSCILLATORY HALF DENSITIESBohr-Sommerfeld conditions.Now assume X is a smooth manifold, A C T*X a Lagrangian submanifold. Let E C(Z)be a (global!)generating functionforAwith respect to a fibration : Z → X. In developing the global theory, we need to assume that A satisfies thefollowing Bohr-Sommerfeld condition:In whatfollows,wewill assumethatA iseract inthe sense that(10)LAaT+X=dAfor some PA E Co(A), where aT*x is the canonical 1 form on T*X.One major application of the Bohr-Sommerfeld assumption on A is to fix thearbitrary constant in the generating function, which need to be kept tract of inapplications. Let : C, Z be the inclusion and p : C→ A be the map (4).Lemma 2.1. d(t* - pp^) = 0.Proof. In fact, by definition of phase function pA,d(t*p -p.pA) =t*dp- ( ope)*aT*x,As we have seen, in local coordinates Z = X × S c IRn × Rk, then1%(r, s) =0,1≤i≤k),C= [(α,s) IOsand the map Pe: Ce→A is the mapaPpe(r, s) = (r, r((r, s).It follows0 da++dst) =0dri*dp =t*(Lari+asioriOn the other hand, since t o p(r, s) = (r, ),(rA 0 Pp)arx = (rA 0 pe)Eedr =drt.or口Inwhatfollows,we will fixachoice ofsuch an eract phase function PA,and we will fix the constant in the generating function by requiring(11)'=opn
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 5 2. Oscillatory half densities ¶ Bohr-Sommerfeld conditions. Now assume X is a smooth manifold, Λ ⊂ T ∗X a Lagrangian submanifold. Let ϕ ∈ C ∞(Z) be a (global!) generating function for Λ with respect to a fibration π : Z → X. In developing the global theory, we need to assume that Λ satisfies the following Bohr-Sommerfeld condition: In what follows, we will assume that Λ is exact in the sense that (10) ι ∗ ΛαT ∗X = dϕΛ for some ϕΛ ∈ C ∞(Λ), where αT ∗X is the canonical 1 form on T ∗X. One major application of the Bohr-Sommerfeld assumption on Λ is to fix the arbitrary constant in the generating function, which need to be kept tract of in applications. Let ι : Cϕ ,→ Z be the inclusion and pϕ : Cϕ → Λ be the map (4). Lemma 2.1. d(ι ∗ϕ − p ∗ ϕϕΛ) = 0. Proof. In fact, by definition of phase function ϕΛ, d(ι ∗ϕ − p ∗ ϕϕΛ) = ι ∗ dϕ − (ιΛ ◦ pϕ) ∗αT ∗X, As we have seen, in local coordinates Z = X × S ⊂ R n × R k , then Cϕ = {(x, s) | ∂ϕ ∂si (x, s) = 0, 1 ≤ i ≤ k}, and the map pϕ : Cϕ → Λ is the map pϕ(x, s) = (x, ∂ϕ ∂x (x, s)). It follows ι ∗ dϕ = ι ∗ ( X ∂ϕ ∂xi dxi + ∂ϕ ∂si dsi) = X ∂ϕ ∂xi dxi . On the other hand, since ιΛ ◦ pϕ(x, s) = (x, ∂ϕ ∂x ), (ιΛ ◦ pϕ) ∗αT ∗X = (ιΛ ◦ pϕ) ∗Xξidxi = X ∂ϕ ∂xi dxi . In what follows, we will fix a choice of such an exact phase function ϕΛ, and we will fix the constant in the generating function ϕ by requiring (11) ι ∗ϕ = p ∗ ϕϕΛ.
6LECTURE29-30FIO-SEMICLASSICALFIOSTOscillatory half densities.Let d = dimZ-dimX be the fiber dimension.For anyk Z, we defineI(X, A), the space of compactly supported oscillatory half densities on X associatedwithA,tobe(X,A)= [μ=k-2*(a(z,h)eiT) / a E C(Z × R),(12)where T is a nowhere vanishing half-density on Z. (Obviously the space is indepen-dent of the choice of .)SimilarlywedefineIk(X,A),the space of oscillatoryhalfdensities on X associated with A, to be the set consists of those half densities μ sothat pμ E I(X,A) for all p E Co(X).Locally we may assume Z = X x S, where S is an open set in Rd. We maychoose ourfiber half-densitytobethe Euclidean one dsand choosetobeTodswith To a nowhere vanishing half-density on X. Then μ I(X, A) is of the form[a(r, s,h)etp(a,)dshk-2 Independence of generating function.We must show that the above definition is also independent of the choices ofgeneratingfunctions.Letπ:Z,→X,i=l,2betwo fibrations,and ibeagenerating function of A with respect to Ti.It is enough to do this locally.Recall that the two generating functions Pi andP2 are related by(a)Replaceby+c.(b) For a diffeomorphism g : Z -→ Z, replace by g* and by g*p.(c) Replace Z by Z = Z × Rd and by (z) +(Az,z), where A is a non-degenerate d x d matrix.We have already get rid of type (a)by requiring Ato satisfy the Bohr-Sommerfeldcondition (10) and fixing the constant in the generating function via the normal-ization condition (11). If two densities are related by a type (b) change, then by achange of variableargument it is not hard to prove(2)+(ae2g1)=(1)+(gae表P1T1)so the spaces defined via i and via P2 are the same.Now suppose 1 and p2 are related by a type (c) change. Without loss ofgenerality,we may assume Z2=Zi × S, where S is an open subset of Rm, andLLSAS,(P2(z, s) =(P1(z) +2where A is a symmetric non-degenerate m × m matrix. Let d be the fiber dimensionof Zi→X,then the fiber dimension of Z2 X is d +m.Let Ti be a nowhere
6 LECTURE 29-30 FIO – SEMICLASSICAL FIOS ¶ Oscillatory half densities. Let d = dim Z − dim X be the fiber dimension. For any k ∈ Z, we define I k 0 (X,Λ), the space of compactly supported oscillatory half densities on X associated with Λ, to be (12) I k 0 (X,Λ) = {µ = ~ k− d 2 π∗(a(z, ~)e i ϕ(z) ~ τ ) | a ∈ C ∞ 0 (Z × R)}, where τ is a nowhere vanishing half-density on Z. (Obviously the space is independent of the choice of τ .) Similarly we define I k (X,Λ), the space of oscillatory half densities on X associated with Λ, to be the set consists of those half densities µ so that ρµ ∈ I k 0 (X,Λ) for all ρ ∈ C ∞ 0 (X). Locally we may assume Z = X × S, where S is an open set in R d . We may choose our fiber half-density to be the Euclidean one ds 1 2 and choose τ to be τ0⊗ds 1 2 with τ0 a nowhere vanishing half-density on X. Then µ ∈ I k 0 (X,Λ) is of the form ~ k− d 2 Z S a(x, s, ~)e i ~ ϕ(x,s) ds τ0. ¶ Independence of generating function. We must show that the above definition is also independent of the choices of generating functions. Let π : Zi → X, i = 1, 2 be two fibrations, and ϕi be a generating function of Λ with respect to πi . It is enough to do this locally. Recall that the two generating functions ϕ1 and ϕ2 are related by (a) Replace ϕ by ϕ + c. (b) For a diffeomorphism g : Z → Z˜, replace π by g ∗π and ϕ by g ∗ϕ. (c) Replace Z by Z = Z × R d and ϕ by ϕ(z) + 1 2 hAz, zi, where A is a nondegenerate d × d matrix. We have already get rid of type (a) by requiring Λ to satisfy the Bohr-Sommerfeld condition (10) and fixing the constant in the generating function via the normalization condition (11). If two densities are related by a type (b) change, then by a change of variable argument it is not hard to prove (π2)∗(ae i ~ ϕ2 g∗τ1) = (π1)∗(g ∗ ae i ~ ϕ1 τ1) so the spaces defined via ϕ1 and via ϕ2 are the same. Now suppose ϕ1 and ϕ2 are related by a type (c) change. Without loss of generality, we may assume Z2 = Z1 × S, where S is an open subset of R m, and ϕ2(z, s) = ϕ1(z) + 1 2 s TAs, where A is a symmetric non-degenerate m×m matrix. Let d be the fiber dimension of Z1 → X, then the fiber dimension of Z2 → X is d + m. Let τ1 be a nowhere
LECTURE29-30FIO-SEMICLASSICALFIOS7vanishing half density on Zi, then Ti ds is a nowhere vanishing half density onZ2. Using the generating function p2 we get the expressionshk-(2)a2(z, s, h)e2(2,s)T1 @ ds2.Let T2.1 : Z2 → Zi be the projection on to the first factor so that (π2)= (πi) o(π2,1)*. Then by definition, (π2,1) acts as/a2(z,s,h)esTAsds)DekpiTi(2,1)(a2(z, s,)ep2(2,8)1 d) =Now the conclusion follows from the lemma of stationary phase (with quadraticphase).In conclusion, we provedTheorem 2.2. The space I(X,A) (and thus I*(X,A) is intrinsically defined (pro-vided A is eract and we fir a choice of px on A).3.SEMICLASSICALFOURIERINTEGRALOPERATORSTThe definition.Now suppose Xi,X, are smanifolds. We will denote M, = T*Xi, i = 1,2.Suppose F C Mi × M2 is an eract canonical relation. ThenA=0201is an exact Lagrangian submanifold of T*X, whereX=X×X2.Associated with Awe have the space of compactly supported oscillatory half densities I(X,A).If wefix a nowhere vanishing one density dci on Xi and a nowhere vanishing one densitydr2 on X2, then a typical element in I(X, A) is of the formμ=hk-号[a(r1, 2, s, h)et(1,m2,)ds) dredrWith someabuse of notion welet L?(X)be theHilbert spaceof L?half densitieson X,. Then associated to each μ = u(r1, 2,h)de dr e Ib(X,A) we can define anintegral operator Fμ : L?(Xi) -→ L?(X2) viaFu(fdat) = (/ f(ai)u(ri,r2, h)dai) das(13)Definition 3.1. Such operators are called compactly supported semi-classical Fouri-er integral operators of order m = k + , where n2 = dim X2. The space of theseoperators is denoted by F"(T).Remark. We could loose the conditions on u by requiring only u(ri, 2,h)dr eL?(Xi), or more generally, with distributional coefficients. In this case we drop thesubscript 0
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 7 vanishing half density on Z1, then τ1 ⊗ ds 1 2 is a nowhere vanishing half density on Z2. Using the generating function ϕ2 we get the expressions ~ k− d+m 2 (π2)∗a2(z, s, ~)e i ~ ϕ2(z,s) τ1 ⊗ ds 1 2 . Let π2,1 : Z2 → Z1 be the projection on to the first factor so that (π2)∗ = (π1)∗ ◦ (π2,1)∗. Then by definition, (π2,1)∗ acts as (π2,1)∗(a2(z, s, ~)e i ~ ϕ2(z,s) τ1 ⊗ ds 1 2 ) = Z a2(z, s, ~)e i 2~ s T Asds e i ~ ϕ1 τ1. Now the conclusion follows from the lemma of stationary phase (with quadratic phase). In conclusion, we proved Theorem 2.2. The space I k 0 (X,Λ) (and thus I k (X,Λ)) is intrinsically defined (provided Λ is exact and we fix a choice of ϕΛ on Λ). 3. Semiclassical Fourier integral operators ¶ The definition. Now suppose X1, X2 are smanifolds. We will denote Mi = T ∗Xi , i = 1, 2. Suppose Γ ⊂ M1 × M− 2 is an exact canonical relation. Then Λ = σ2 ◦ Γ is an exact Lagrangian submanifold of T ∗X, where X = X1×X2. Associated with Λ we have the space of compactly supported oscillatory half densities I k 0 (X,Λ). If we fix a nowhere vanishing one density dx1 on X1 and a nowhere vanishing one density dx2 on X2, then a typical element in I k 0 (X,Λ) is of the form µ = ~ k− d 2 Z S a(x1, x2, s, ~)e i ~ ϕ(x1,x2,s) ds dx 1 2 1 dx 1 2 2 With some abuse of notion we let L 2 (Xi) be the Hilbert space of L 2 half densities on Xi . Then associated to each µ = u(x1, x2, ~)dx 1 2 1 dx 1 2 2 ∈ I k 0 (X,Λ) we can define an integral operator Fµ : L 2 (X1) → L 2 (X2) via (13) Fµ(f dx 1 2 1 ) = Z f(x1)u(x1, x2, ~)dx1 dx 1 2 2 Definition 3.1. Such operators are called compactly supported semi-classical Fourier integral operators of order m = k + n2 2 , where n2 = dim X2. The space of these operators is denoted by F m 0 (Γ). Remark. We could loose the conditions on u by requiring only u(x1, x2, ~)dx 1 2 1 ∈ L 2 (X1), or more generally, with distributional coefficients. In this case we drop the subscript 0
8LECTURE29-30FIO-SEMICLASSICALFIOSExample:Semi-classical pseudo-differential operators.Take Xi = X2 = Rn andI = AM = graph of the identity = {(r, s, , s)} C M × M-where M = T*Rn. ThenA= 02 0I = (cr, r, s, -s) C T*(R" × R").On A one has tAaT+x = E sidr - sidc; = 0, so one take phase function pA = 0.To find a generating function, one use the fibrationT:Rn × Rn × Rn→R"×Rn,(a,,s)-(c,y).Then by definition,I = [(r, y, s, n1, n2, 0, r, y, n1, n2))If wetakeyEC(Rn)tobethefunctionp(r, y,s) =(ri - yi)si,thenAp= [(c, y,s, s, -s, r-y))and it is easy to seeA=TApMoreover, the set C, is defined by the equations = r - yi = O, i.e.Cp = [(r, r, s)),and the map P is given explicitly byPp:Co →A, (r,a,s) (r,r,s, -s).So i*p = 0 = papA. In other words, satisfies the normalizing condition (11).What are the semi-classical Fourier integral operators associated to T? By def-inition,Fm(r)consist of those operatorsthatmaps f(r)drtohm一号一号et(m-w)-a(r, ,s,h)f(r)dede) dya.They are the semi-classical pseudo-differential operators (of semi-classical order m)we learned earlier in this course! (We used special symbols of the form a(r,s,h)or a(y, s, h) or a(“, s,h), but we could use general symbols a(r, y,s,h). One canshow that any general symbol corresponds to a unique left/right/Weyl symbol. c.fA. Martinez, page 37. )In general, if Xi = X2 = X is a smooth manifold, then the construction abovegives us semi-classical pseudo-differential operators on X. In other words, m(X) =Fm(AM)
8 LECTURE 29-30 FIO – SEMICLASSICAL FIOS ¶ Example: Semi-classical pseudo-differential operators. Take X1 = X2 = R n and Γ = ∆M = graph of the identity = {(x, s, x, s)} ⊂ M × M−, where M = T ∗R n . Then Λ = σ2 ◦ Γ = {(x, x, s, −s)} ⊂ T ∗ (R n × R n ). On Λ one has ι ∗ ΛαT ∗X = Psidxi − sidxi = 0, so one take phase function ϕΛ = 0. To find a generating function, one use the fibration π : R n × R n × R n → R n × R n , (x, y, s) 7→ (x, y). Then by definition, Γπ = {(x, y, s, η1, η2, 0, x, y, η1, η2)}. If we take ϕ ∈ C ∞(R n ) to be the function ϕ(x, y, s) = X(xi − yi)si , then Λϕ = {(x, y, s, s, −s, x − y)} and it is easy to see Λ = Γπ ◦ Λϕ. Moreover, the set Cϕ is defined by the equations ∂ϕ ∂si = xi − yi = 0, i.e. Cϕ = {(x, x, s)}, and the map pϕ is given explicitly by pϕ : Cϕ → Λ, (x, x, s) 7→ (x, x, s, −s). So ι ∗ϕ = 0 = p ∗ ϕϕΛ. In other words, ϕ satisfies the normalizing condition (11). What are the semi-classical Fourier integral operators associated to Γ? By definition, F m(Γ) consist of those operators that maps f(x)dx 1 2 to ~ m− n 2 − n 2 Z Rn×Rn e i ~ (x−y)·s a(x, y, s, ~)f(x)dxdξ dy 1 2 . They are the semi-classical pseudo-differential operators (of semi-classical order m) we learned earlier in this course! (We used special symbols of the form a(x, s, ~) or a(y, s, ~) or a( x+y 2 , s, ~), but we could use general symbols a(x, y, s, ~). One can show that any general symbol corresponds to a unique left/right/Weyl symbol. c.f A. Martinez, page 37. ) In general, if X1 = X2 = X is a smooth manifold, then the construction above gives us semi-classical pseudo-differential operators on X. In other words, Ψm(X) = F m(∆M)
LECTURE29-30FIO-SEMICLASSICALFIOS9 Example: The semi-classical Fourier transform.Let Xi = X, = Rn. Let T be the graph of the symplectomorphismJ: R" ×Rn →R" ×R", (c,y)→(-y,a),i.e.T = [(α, y, -y,r)) c T*R" × T*R".ThenA = ((r, y, -y, -r) c T*(R" × R").A is exact since taT*x = -Eyidr, -ridy = -d(r - y). We just choose thephase functionp=-r·y.We don't need a fibration to find a generating function, since A is already ahorizontal Lagrangian, with (normalized!) generating functionp(a,y) = -r·y.(So in this example C,= X = Z. What is the map p?)Let μ = e-ryddy I'(X,A). What is the corresponding semi-classicalFourier integral operator? By definition Fμ maps any f(r)dr (with f e Co(Rn)for simplicity)to[f(r)e-tydr) dys,which is the semi-classical Fourier transform Fh!What about the inverse (semi-classical) Fourier transform? Well, repeating theprevious precess one can see that F-l is a semi-classical Fourier integral opera-tor associated to the graph of the symplectomorphism (which is the inverse of theprevious one)J-1 : IRn × Rn → Rn × R", (c,)-→(y, -r)
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 9 ¶ Example: The semi-classical Fourier transform. Let X1 = X2 = R n . Let Γ be the graph of the symplectomorphism J : R n × R n → R n × R n , (x, y) 7→ (−y, x), i.e. Γ = {(x, y, −y, x)} ⊂ T ∗R n × T ∗R n . Then Λ = {(x, y, −y, −x)} ⊂ T ∗ (R n × R n ). Λ is exact since ι ∗ ΛαT ∗X = − Pyidxi − Pxidyi = −d(x · y). We just choose the phase function ϕΛ = −x · y. We don’t need a fibration to find a generating function, since Λ is already a horizontal Lagrangian, with (normalized!) generating function ϕ(x, y) = −x · y. (So in this example Cϕ = X = Z. What is the map pϕ?) Let µ = e − i ~ x·ydx 1 2 dy 1 2 ∈ I 0 (X,Λ). What is the corresponding semi-classical Fourier integral operator? By definition Fµ maps any f(x)dx 1 2 (with f ∈ C ∞ 0 (R n ) for simplicity) to Z Rn f(x)e − i ~ x·y dx dy 1 2 , which is the semi-classical Fourier transform F~! What about the inverse (semi-classical) Fourier transform? Well, repeating the previous precess one can see that F −1 ~ is a semi-classical Fourier integral operator associated to the graph of the symplectomorphism (which is the inverse of the previous one) J −1 : R n × R n → R n × R n , (x, y) 7→ (y, −x)
10LECTURE29-30FIO-SEMICLASSICALFIOS4.THECOMPOSITIONOFh-FIOS The composition of phase functions.LetXi,X2and X3besmoothmanifolds,M,=T*X,.LetI,:M,→Mi+1be exact Lagrangian submanifolds, with phase function pr.Suppose Fr and I2 aretransversally composable.Recall that this implies that the mapa: F= [(m1,m2, m3) / (mi,mi+1) ET)) → Mi × M3, (mi,m2, m3) -→ (m1,m3)is a constant rank map which maps onto F2 oFi.As before we assume Q is properand has connected level sets, so that FoF is an embedded Lagrangian submanifold.Theorem4.1.F,oi is an eract Lagrangian submanifold of Mi×My.Proof. We denote ti : F, → M, x Mi+1 for i = 1, 2 and denote t3 : F2ofi → Mi x M3.Let pi:F→I,and π:F-→M,be theobvious projections.Thenpi(iaT+XixT+x-) =TiaT*Xi -T2aT*X2.Similar expressions holds for ps(2aT-Xax-x-)and a(ga+XixT+x-), which impliesPi(GiQT+X1XT*X-)+ P2(SQT+X2xT+X)=Q*(gaT+X1xT+X)On the other hand, by definitionaT+XxTX+ = dpr.for i = 1, 2. So if we letP= pipri +p2pr2 E C(F),thendp=pidpri+padpr2=a*(gaT*XixT*xg)For any p I2 oFi, let Fp = α-1(p) be the connected compact fiber over p andlet tp:F,→F be the inclusion.Then Qotp:F,→F2 oF is the constant map. So(α 0 tp)*(tgaT+X1xT+x,) = 0.It followsdipp = tjdp = 0.Since F, is connected, go is constant on Fp. In other words, is constant on eachfiber Fp. So one can find a function (pr2or e Co(T2 oFi) so that=*Pr20r1.Thusa'dpr20ri = dp = α*(cgaT*X1xTx-).Since a:F→F2oF is surjective,Q*is injective.It followsdpT20, = 1aT*X1xT*X,口i.e. is a phase function for T2 o Ti
10 LECTURE 29-30 FIO – SEMICLASSICAL FIOS 4. The composition of ~-FIOs ¶ The composition of phase functions. Let X1, X2 and X3 be smooth manifolds, Mi = T ∗Xi . Let Γi : Mi =⇒ Mi+1 be exact Lagrangian submanifolds, with phase function ϕΓi . Suppose Γ1 and Γ2 are transversally composable. Recall that this implies that the map α : F = {(m1, m2, m3) | (mi , mi+1) ∈ Γi)} → M1 × M3,(m1, m2, m3) 7→ (m1, m3) is a constant rank map which maps onto Γ2 ◦ Γ1. As before we assume α is proper and has connected level sets, so that Γ2 ◦Γ1 is an embedded Lagrangian submanifold. Theorem 4.1. Γ2 ◦ Γ1 is an exact Lagrangian submanifold of M1 × M− 3 . Proof. We denote ιi : Γi ,→ Mi×Mi+1 for i = 1, 2 and denote ι3 : Γ2 ◦Γ1 ,→ M1×M3. Let ρi : F → Γi and πi : F → Mi be the obvious projections. Then ρ ∗ 1 (ι ∗ 1αT ∗X1×T ∗X − 2 ) = π ∗ 1αT ∗X1 − π ∗ 2αT ∗X2 . Similar expressions holds for ρ ∗ 2 (ι ∗ 2αT ∗X2×T ∗X − 3 ) and α ∗ (ι ∗ 3αT ∗X1×T ∗X − 3 ), which implies ρ ∗ 1 (ι ∗ 1αT ∗X1×T ∗X − 2 ) + ρ ∗ 2 (ι ∗ 2αT ∗X2×T ∗X − 3 ) = α ∗ (ι ∗ 3αT ∗X1×T ∗X − 3 ). On the other hand, by definition ι ∗ iαT ∗Xi×T ∗X − i+1 = dϕΓi for i = 1, 2. So if we let ϕ = ρ ∗ 1ϕΓ1 + ρ ∗ 2ϕΓ2 ∈ C ∞(F), then dϕ = ρ ∗ 1dϕΓ1 + ρ ∗ 2dϕΓ2 = α ∗ (ι ∗ 3αT ∗X1×T ∗X − 3 ). For any p ∈ Γ2 ◦ Γ1, let Fp = α −1 (p) be the connected compact fiber over p and let ιp : Fp ,→ F be the inclusion. Then α ◦ ιp : Fp → Γ2 ◦ Γ1 is the constant map. So (α ◦ ιp) ∗ (ι ∗ 3αT ∗X1×T ∗X − 3 ) = 0. It follows dι∗ pϕ = ι ∗ pdϕ = 0. Since Fp is connected, ι ∗ pϕ is constant on Fp. In other words, ϕ is constant on each fiber Fp. So one can find a function ϕΓ2◦Γ1 ∈ C ∞(Γ2 ◦ Γ1) so that ϕ = α ∗ϕΓ2◦Γ1 . Thus α ∗ dϕΓ2◦Γ1 = dϕ = α ∗ (ι ∗ 3αT ∗X1×T ∗X − 3 ). Since α : F → Γ2 ◦ Γ1 is surjective, α ∗ is injective. It follows dϕΓ2◦Γ1 = ι ∗ 3αT ∗X1×T ∗X − 3 , i.e. ϕ is a phase function for Γ2 ◦ Γ1.