Concentration of quantum integrable eigenfunctions on a convex surface of revolution
aa r X i v : . [ m a t h . SP ] A ug CONCENTRATION OF QUANTUM INTEGRABLEEIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION
MICHAEL GEIS
Abstract.
Let ( S , g ) be a convex surface of revolution and H ⊂ S the uniquerotationally invariant geodesic. Let ϕ ℓm be the orthonormal basis of joint eigen-functions of ∆ g and ∂ θ , the generator of the rotation action. The main resultis an explicit formula for the weak-* limit of the normalized empirical measures, Σ ℓm = − ℓ || ϕ ℓm || L ( H ) δ mℓ ( c ) on [ − , . The explicit formula shows that, asymptotically,the L norms of restricted eigenfunctions are minimal for the zonal eigenfunction m = 0 , maximal for Gaussian beams m = ± , and exhibit a (1 − c ) − type singu-larity at the endpoints. For a pseudo-differential operator B we also compute thelimits of the normalized measures P ℓm = − ℓ h Bϕ ℓm , ϕ ℓm i δ mℓ . Introduction
This article is concerned with concentration properties of an orthonormal basis ofQuantum completely integrable Laplace eigenfunctions(1.1) − ∆ g ϕ λ j = λ j ϕ λ j on a closed Riemannian manifold ( M, g ) in the λ j → ∞ limit. The concentration of asequence ϕ λ j of eigenfunctions is often measured by studying the limits of matrix ele-ments h Aϕ λ j , ϕ λ j i L ( M ) of pseudodifferential operators, which are known as microlocaldefect measures. One may also study concentration on a submanifold H ⊂ M viathe limits of L p norms of restricted eigenfunctions || ϕ λ j | H || L p ( H ) . We take a newapproach to the study of eigenfunction concentration in the quantum completely in-tegrable setting in the simple case of a convex surface of revolution ( S , g ) where wecan obtain explicit results. We let ∂ θ be the smooth vector field which generates the S symmetry and we study an L orthonormal basis of joint eigenfunctions ϕ ℓm of thecommuting operators − ∆ g and D θ = i ∂ θ : ( − ∆ g ϕ ℓm = λ ℓ ϕ ℓm D θ ϕ ℓm = mϕ ℓm On a convex surface of revolution, there exists an operator b I which commuteswith − ∆ g and D θ which has a joint spectrum with D θ consisting of a lattice ofsimple eigenvalues Spec ( b I , D θ ) = { ( ℓ, m ) ∈ Z | ℓ ≥ | m | ≤ ℓ } Thus b I ϕ ℓm = ℓϕ ℓm , D θ ϕ ℓm = mϕ ℓm . Our purpose is to study the relative rates ofconcentration of the eigenfunctions ϕ ℓm along the equator, H , the unique rotationally Date : August 27, 2020.Partially supported by NSF RTG grant DMS-1502632.Written in progress of a PhD in mathematics at Northwestern University. invariant geodesic, within a single b I eigenspace. To do this we calculate the weak-*limits of the following empirical measure :(1.2) µ ℓ = 1 M ℓ ℓ X m = − ℓ || ϕ ℓm || L ( H ) δ mℓ The constant M ℓ normalizes µ ℓ , making it a probability measure on [ − , . Ifwe view c ∈ [ − , as a continuous version of the ratio m/ℓ , the weak-* limits canbe viewed as the asymptotic distribution of mass across the b I eigenspaces. In thecalculation of the limit of µ ℓ , we need to compute the weak-* limit of another familyof empirical measures(1.3) ν ℓ ( B ) = 1 N ℓ ( B ) ℓ X m = − ℓ h Bϕ ℓm , ϕ ℓm i L ( S ,dV g ) δ mℓ Here B ∈ Ψ is a homogeneous pseudo-differential operator of order zero.1.1. Statement of results.
In order to state the results, we need to briefly describethe underlying geometry. The principal symbols of b I and b I = D θ , I and I = p θ are homogeneous, poisson commuting smooth functions on T ∗ S \ and are calledthe action variables for the geodesic flow. Their Hamiltonian flows are π -periodicso their joint flow Φ t defines a homogeneous, Hamiltonian action of the torus T .The joint flow preserves level sets of both I and p θ and by homogeneity, all of theinformation is contained in the I = 1 level set, which we denote by Σ ⊂ T ∗ S \ .On Σ , | I | ≤ and for c ∈ [ − , , we let T c = I − ( c ) ∩ Σ . For c = ± , these levelsets are diffeomorphic to T and consist of a single orbit of the joint flow. The levels T ± consist of I unit covectors tangent to H with the sign reflecting the orientationrelative to ∂ θ . We let dµ L denote Liouville measure on Σ and dµ c,L = dµ L /dp θ denoteLiouville measures on the regular tori T c . The torus action Φ t commutes with thegeodesic flow G t = exp tH | ξ | g and we can write(1.4) | ξ | g = K ( I , I ) For a smooth function K on R \ , homogeneous of degree 1. We let ( ω , ω ) = ∇ I K ( I , I ) be the so-called frequency vector associated to this action. The ω i arethemselves functions of the action variables I , I . For a homogeneous pseudo B ∈ Ψ ( S ) , we let σ ( B ) denote its principal symbol and set [ σ ( B )( c ) = R T c σ ( B ) dµ c,L .We also let ω ( B ) = R Σ σ ( B ) dµ L be the Liouville state on B . We note that for ( x, ξ ) ∈ T H S ∩ T c we have, p θ ( x, ξ ) = | ξ | g a ( r ) cos φ = K ( c, a ( r ) cos φ where φ is the angle between the covector ξ and H and r is the distance fromthe north pole to H so that H = { r = r } . Let L ( H ) be the length of H . Then a ( r ) = L ( H ) / π . Theorem 1.1.
Let ( S , g ) be a convex surface of revolution where g = dr + a ( r ) dθ in geodesic polar coordinates. Let H ⊂ S be the equator, the unique rotationallyinvariant geodesic. Then in terms of action angle variables we have, ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION3 (a) For every f ∈ C ([ − , , Z − f ( c ) dµ ℓ ( c ) = 1 M ℓ ℓ X m = − ℓ || ϕ ℓm || L ( H ) f (cid:16) mℓ (cid:17) → M Z − f ( c ) ω ( c, q − (2 π ) c K ( c, L ( H ) dc (b) For any f ∈ C ([ − , , Z − f ( c ) dν ℓ ( c ) = 1 N ℓ ( B ) ℓ X m = − ℓ h Bϕ ℓm , ϕ ℓm i L ( S ,g ) f (cid:16) mℓ (cid:17) → ω ( B ) Z − f ( c ) [ σ ( B )( c ) dc The constant appearing in (a) is M = Z − ω ( c, q − (2 π ) c K ( c, L ( H ) dc and normalizes the limit measure to have mass 1 on [ − , . We note that when c = ± , T c collapses to the set of I unit covectors tangent to H . On T ± , ω = ∂K∂I = ∂K∂I = a ( r ) − = (2 π ) L ( H ) and φ = 0 . The left hand of (1.5) therefore blows up at c = ± . When ( S , g can ) is the standard sphere, L ( H ) = 2 π , K ( c,
1) = 1 and ω ( c,
1) = 1 ,hence(1.5) ω ( c, q − (2 π ) c K ( c, L ( H ) = 1 √ − c Remark 1.2.
It would be interesting to know when the above formula holds. It isplausible that this is true on an ellipsoid of revolution where one has explicit formulaefor the frequencies ω i . We would also like to find the weak-* limit of the measure(1.2) when H is any latitude circle. We leave that for future investigations.The measures µ ℓ considered here are closely related to the empirical measuresassociated to a polarized toric Kähler manifold L → M n studied in [15], µ zk = 1Π h k ( z, z ) X α ∈ kP ∩ Z d | s α ( z ) | h k δ αk Here, s α ( z ) are the holomorphic sections of L k . These correspond to lattice pointsinside the k th dialate of a certain Delzant polytope P ⊂ R n . This polytope is theimage of the moment map µ : M → P associated to the torus action on M . In oursetting, M is analogous to the phase space energy surface Σ = { I = 1 } ⊂ T ∗ S with the moment map I : Σ → [ − , . The joint eigenfunctions of b I -eigenvalue ℓ correspond to the lattice points inside the ℓ th dialate of I (Σ) = [ − , and areanalogous to the holomorphic sections s α . In both cases the measures are dialatedback to be supported on the image of the moment map and normalized to have mass1. The submanifold H plays the role of the continuous parameter z ∈ M in theKähler setting. In [15] it is shown that as k → ∞ , a central limit theorem typerescaling of these measures tends to a Gaussian measure centered on µ ( z ) while inour case the measures µ ℓ tend to an absolutely continuous limit which blows up at M. GEIS the end points with a (1 − c ) − type singularity. The blow-up reflects the fact thatthe Gaussian beams m = ± ℓ are concentrated on T ∗ H ∩ Σ in phase space.In addition, we codify the similarity of the operator b I on ( S , g ) to the degreeoperator A = q − ∆ g can + − on the round sphere ( S , g can ) by showing that b I and A are conjugate via a unitary Fourier integral operator that leaves invariant D θ ,at least up to a finite rank operator. Theorem 1.3.
Let ( S , g ) be a convex surface of revolution and A = q − ∆ g can + − be the degree operator on the round sphere. There exists a homogeneous unitaryFourier integral operator W : L ( S , g can ) → L ( S , g ) such that [ W, D θ ] = 0 and W ∗ b I W = A + R where R is a finite rank operator.Consequently, if Y ℓm denotes the standard orthonormal basis of L ( S , g can ) such that AY ℓm = ℓY ℓm , D θ Y ℓm = mY ℓm , then for ℓ large enough, there are constants c ℓm with | c ℓm | = 1 so that (1.6) W Y ℓm = c ℓm ϕ ℓm In [6], Lerman proves that there is only one homogeneous hamiltonian action ofthe torus T on T ∗ S \ up to symplectomorphism. In particular, letting p ( x, ξ ) = | ξ | g can ( x ) be the principal symbol of A , p θ and p generate such an action, so there is ahomogeneous symplectomorphism χ on T ∗ S \ which pulls back the functions p θ , I to p θ , p . Theorem 1.2 is essentially an operator theoretic version of this statement. Acknowledgements.
I would like to thank Emmett Wyman for many helpful con-versations regarding the symbol calculus of FIOs, as well as my advisor Steve Zelditchfor his continual patience and guidance.1.2.
Outline of the computation of weak-* limits.
We compute the weak-*limits of the measures (1.3), (1.2) by expressing their un-normalized versions as atrace and using the symbol calculus of Fourier integral operators to compute theleading order contribution as ℓ → ∞ . We refer to [3],[1] for background on Fourierintegral operators and the symbol calculus. Let Π ℓ : L ( S , dV g ) → L ( S , dV g ) denote the orthogonal projection onto the b I = ℓ eigenspace. Suppose that A : C ∞ ( S ) → C ∞ ( S ) is an operator which commutes with D θ . Then the kernel of theoperator(1.7) f (cid:18) D θ ℓ (cid:19) A Π ℓ is equal to(1.8) ℓ X m = − ℓ Aϕ ℓm ( x ) ϕ ℓm ( y ) f (cid:16) mℓ (cid:17) And thus we have
ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION5 (1.9) Trace f (cid:18) D θ ℓ (cid:19) A Π ℓ = ℓ X m = − ℓ h Aϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) We use this formula to compute the weak-* limits of both sequences of empiricalmeasures. When A is a pseudo-differential operator, this forumula returns the un-normalized measures (1.3) tested against f . To use this formula for the measures(1.2), we express the L norms on H ⊂ S as a global matrix element as follows:let γ H : C ∞ ( S ) → C ∞ ( H ) denote restriction to H and γ ∗ H denote the L adjointof γ H with respect to the Riemannian volume measure dV g . Thus, for g ∈ C ∞ ( H ) , f ∈ C ∞ ( S ) we have h γ ∗ H g, f i L ( S ,dV g ) = Z H gf | H dS where dS is the induced surface measure. From this it follows that || ϕ ℓm || L ( H,dS ) = h γ ∗ H γ H ϕ ℓm , ϕ ℓm i One problem with this setup is that (1.9) requires the operator A to commute with D θ , and this will not be true for every pseudo B ∈ Ψ ( S ) nor for the operator γ ∗ H γ H .We deal with this by averaging against the torus action generated by D θ and b I . For t = ( t , t ) ∈ T , let(1.10) U ( t ) = exp i [ t D θ + t b I ] In section 3 we review that this is a torus action on L ( S , dV g ) by unitary Fourierintegral operators. For any operator A : C ∞ ( S ) → C ∞ ( S ) we set(1.11) ¯ A = (2 π ) − Z T U ( t ) ∗ AU ( t ) d t The average ¯ A commutes with both D θ and b I since(1.12) [ D θ , ¯ A ] = (2 π ) − Z T − ∂ t [ U ( t ) ∗ AU ( t )] d t = 0 And similarly for b I . We also note that h Aϕ ℓm , ϕ ℓm i L ( S ) ,dV g = h ¯ Aϕ ℓm , ϕ ℓm i L ( S ,dV g ) This means replacing A with ¯ A in the trace will not change the right hand side of(1.9). When A ∈ Ψ ( S ) , Egorov’s theorem tells us that ¯ A ∈ Ψ ( S ) as well, and σ ( ¯ A ) = (2 π ) − Z T Φ ∗ t σ ( A ) d t where Φ t is the joint flow generated by I = p θ and I . In section 4, we analyzethe averaged restriction operator(1.13) ¯ V = (2 π ) − Z T U ∗ ( t )( γ ∗ H γ H ) U ( t ) d t M. GEIS
And show that, after applying microlocal cutoffs to γ ∗ H γ H , it splits into the sum ofa pseudo-differential operator and a Fourier integral operator. The canonical relationof the non-pseudo-differential part of ¯ V is related to the notion of a mirror reflectionmap on covectors based on H (See section 4 for details). Both summands can be madeto commute with U ( t ) . The strategy of using the operator ¯ V to study restricted L norms (and more generally restricted Ψ DO matrix elements) has been used in [14]and we closely follow their analysis here. As mentioned, for this analysis to work weneed to microlocally cut off γ ∗ H γ H away from both N ∗ H and T ∗ H . Literally speakingwe fix ε > and instead work with the operator(1.14) ( γ ∗ H γ H ) ≥ ε = (1 − b χ ε/ )( γ ∗ H γ H )(1 − b χ ε ) Where ( I − b χ ε ) is a homogeneous pseudo-differential operator with wave front setoutside conic neighborhoods of both N ∗ H and T ∗ H . The cutoff away from the normaldirections is technical and related to the choice to use the homogeneous calculus,while the cutoff away from the tangential directions is necessary since otherwise thecanonical relation of ¯ V would be singular. We show in section 5 that we can usethe cutoff operator ( γ ∗ H γ H ) ≥ ε to compute the weak-* limits of (1.2) by letting ε → afterwards.2. Quantum toric integrability for convex surfaces of revolution
Let ( S , g ) be a surface of revolution. We denote the two fixed points of the S action by N and S . Fix a meridian geodesic γ which joins N to S and let ( r, θ ) denote geodesic polar coordinates from N , i.e. so that the curve r ( r, is the arclength parametrized geodesic γ . In these coordinates the metric takes the form g = dr + a ( r ) dθ for some smooth function a : [0 , L ] → R such that a k (0) = a k ( L ) = 0 and a ′ (0) = 1 , a ′ ( L ) = 1 . Here L is the distance between the poles. A convex surface ofrevolution is one such that a ( r ) has exactly one non-degenerate critical point which isa maximum, a ′′ ( r ) < . The latitude circle H = { ( r = r ) } is the unique rotationallyinvariant geodesic.Recall that we say the Laplacian − ∆ g of a Riemannian manifold ( M n , g ) is quan-tum completely integrable if there exists n first order homogeneous pseudo-differentialoperators P , . . . , P n ∈ Ψ ( M ) satisfying: • [ P i , P j = 0] • p − ∆ g = K ( P , . . . , P n ) for some polyhomogeneous function K ∈ C ∞ ( R n \ • If p j = σ ( P j ) are the principal symbols, the regular values of the associatedmoment map P = ( p , . . . , p n ) : T ∗ M \ → R n \ form an open, dense subsetof T ∗ M .For background on quantum integrable Laplacians, see chapter 11 of [4]. If ( S , g ) any surface of revolution, and D θ = i ∂ θ is the self-adjoint differential operator asso-ciated to the generator of the S action, it is clear by writing ∆ g in polar coordinatesthat [∆ g , D θ ] = 0 . Hence every surface of revolution is quantum completely inte-grable by taking P = p − ∆ g and P = D θ . The third condition is satisfied, forinstance, if a ( r ) is assumed to be Morse. In the special case of a convex surface ofrevolution, Colin de Verdière in [5] has shown that the Laplacian is quantum toric ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION7 completely integrable. This means that there exists b I , b I first order, homogeneous,commuting pseudo-differential operators satisfying the above conditions of quantumcomplete integrability, but with the additional property that(2.1) exp 2 πi b I j = IdIn particular, one can take b I = D θ and b I to be self-adjoint and elliptic. Note thatcondition (2.1) implies that the joint spectrum of b I , b I is a subset of Z . In fact it isshown in [5] that it consists of all simple eigenvalues and(2.2) Spec ( b I , b I ) = { ( m, ℓ ) ∈ Z | | m | ≤ ℓ ; ℓ > } We fix a particular orthonormal basis of joint eigenfunctions { ϕ ℓm } satisfying b I ϕ ℓm = ℓϕ ℓm and D θ ϕ ℓm = mϕ ℓm .2.1. The moment map and classical toric integrability.
Let I j = σ ( b I j ) be theprincipal symbols. The associated moment map P = ( I , I ) : T ∗ S \ → R \ hasimage equal to the closed conic wedge B = { ( x, y ) | | x | ≤ y ; y > } The set of critical points, Z , of P consists of covectors lying tangent to the equator.If ( ρ, η ) are the dual coordinates to ( r, θ ) on the fibers of T ∗ S , Z = { ( r , θ, , η ) | η = 0 } = T ∗ H \ P maps Z to the boundary ∂ B , so the interior of B consists entirely of regularvalues. Consider a regular level set of the form T c = P − (1 , c ) , for c ∈ ( − , . Byhomogeneity, all other regular levels are dialates of these. For each c , T c is connectedand diffeomorphic to a torus T ∼ = R / π Z × R / π Z . The singular levels correspond to c = ± and are equal to the set of covectors T ± = { ( r , θ, , ± } . One consequenceof quantum toric integrability is of course classical toric integrability. That is, letting H I j denote the hamilton vector fields of I j , equation (2.1) implies that both H I j generate π -periodic flows. Since { I , I } = 0 , we let for t = ( t , t ) ∈ T ,(2.3) Φ t : T × T ∗ S \ → T ∗ S \ t ( x, ξ ) = exp t H I ◦ exp t H I ( x, ξ ) The joint flow Φ t thus defines a homogeneous, Hamiltonian action of T on T ∗ S \ which commutes with the geodesic flow G t = exp tH | ξ | g . It preserves the level sets ofthe moment map and each torus T c consists of a single orbit of the joint flow.2.2. The standard torus action on T ∗ S . In [6], Lerman shows that up to sym-plectic equivalence, there is only one homogeneous Hamiltonian action of T on T ∗ S \ . The simplest example of a convex surface of revolution is the standardsphere ( S , g can ) . For the standard sphere we can take b I = A = q − ∆ g can + − ,the so-called degree operator. The associated torus action on T ∗ S is generated by | ξ | g can and p θ . If I = p θ and I are the action variables associated to a convex surfaceof revolution, there is a homogeneous symplectomorphism M. GEIS χ : T ∗ S \ → T ∗ S \ such that χ ∗ p θ = p θ and χ ∗ I = | ξ | g can . Theorem 1.2 is the statement that thesymplectic equivalence of the torus action on a convex surface of revolution to thatof the round sphere can be quantized. That is, the generators of the standard torusunitary torus action D θ and A on the round sphere are unitarily conjugate via ahomogeneous Fourier integral operator to the quantized action operators b I j on anyconvex surface of revolution.3. The Quantum torus action
In this section we briefly review the fact that the commuting operators b I = D θ and b I on a convex surface of revolution ( S , g ) together generate an action of T on L ( S , dV g ) by unitary Fourier integral operators. (See for instance p. 245 of [4]).For t = ( t , t ) ∈ T we set(3.1) U ( t ) = exp i [ t D θ + t b I ] Proposition 3.1.
The operator U ( t , t ) is a homogeneous Fourier integral operatorbelonging to the class I − ( T × S × S ; C U ) . Its canonical relation is given by thespace-time graph of the joint flow C U = { ( t , p θ ( x, ξ ) , t , I ( x, ξ ) , y, η, x, ξ ) | ( y, η ) = Φ ( t ,t ) ( x, ξ ) ; ( x, ξ ) ∈ T ∗ S \ } The half density part of the symbol σ ( U ) pulls back along the parametrizing map ι : ( t , t , x, ξ ) ( t , p θ ( x, ξ ) , t , I ( x, ξ ) , Φ ( t ,t ) ( x, ξ ) , x, ξ ) to the half density | dt ∧ dt | ⊗ | dx ∧ dξ | on T × T ∗ S .Proof. Since exp it D θ just acts by pulling back a function along the flow of the vectorfield ∂ θ , one can check in coordinates that this is a Fourier integral operator in theclass I − ( S × S , S ; C ) where C = { t , p θ ( x, ξ ) , y, η, x, ξ ) | ( y, η ) = exp t H p θ ( x, ξ ); ( x, ξ ) ∈ T ∗ S \ } The half density symbol pulls back along the parametrizing map ι : ( t , x, ξ ) ( t , p θ ( x, ξ ) , exp t H p θ ( x, ξ ) , x, ξ ) to | dt | ⊗ | dx ∧ dξ | . Now I is a first order, self-adjoint, elliptic pseudo-differentialoperator with integer spectrum, so by [10] we have that exp it b I ∈ I − ( S × S × S ; C ′ ) where C ′ = { t , I ( x, ξ ) , y, η, x, ξ ) | ( y, η ) = exp t H I ( x, ξ ); ( x, ξ ) ∈ T ∗ S \ } Now the composition of C with C ′ is transverse since they are essentially canonicalgraphs. By standard transverse composition of FIOs the orders add and we get thedescription of U ( t ) stated in the proposition. (cid:3) ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION9 Restricted L norms as matrix elements In order to calculate the weak-* limit of (1.2) using trace formulae, we need torelate the restricted L norms of the joint eigenfunctions to matrix elements. Let γ H : C ∞ ( S ) → C ∞ ( H ) be the operator which restricts functions to H . Then if γ ∗ H is the L adjoint, wehave || ϕ ℓm || L ( H ) = h γ ∗ H γ H ϕ ℓm , ϕ ℓm i L ( S ,dV g ) and since ϕ ℓm are joint eigenfunctions of b I j , we can replace γ ∗ H γ H with the average ¯ V = (2 π ) − Z T U ( t ) ∗ ( γ ∗ H γ H ) U ( t ) d t without changing the above matrix elements. The problem is the operator ¯ V hasa singular canonical relation. To fix this, we replace γ ∗ H γ H with a microlocally cut offoperator ( γ ∗ H γ H ) ≥ ε described below. After doing this, ¯ V ε = (2 π ) − Z T U ( t ) ∗ ( γ ∗ H γ H ) ≥ ε U ( t ) d t becomes a genuine Fourier integral operator and we calculate its order and symbolicdata.4.1. The cutoff restriction operator on the sphere.
Let T ∗ H S = { ( x, ξ ) ∈ T ∗ S | x ∈ H } Denote the set of covectors with footprint on H . Since γ H is just pullback along theinclusion map, it is a Fourier integral operator associated with the pullback canonicalrelation C = { ( x, ξ | T H , x, ξ ) | ( x, ξ ) ∈ T ∗ H S \ } ⊂ T ∗ H × T ∗ S The left factor contains elements of the zero section whenever ξ ∈ N ∗ H , so it is nota homogeneous Fourier integral operator in the sense of [3]. Because of this defect,the wave front set of γ ∗ H γ H is(4.1) W F ′ ( γ ∗ H γ H ) = C H ∪ N ∗ H × T ∗ M ∪ T ∗ M × N ∗ H Where C H ⊂ T ∗ M \ × T ∗ M \ is the homogeneous canonical relation C H = { ( x, ξ, x, ξ ′ ) | ( x, ξ ) , ( x, ξ ′ ) ∈ T ∗ H S \ ξ | T x H = ξ ′ | T x H } Note that since ∂ θ is tangent to H , ( x, ξ ) | T H = ( x, ξ ′ ) | T H is equivalent to I ( x, ξ ) = I ( x, ξ ′ ) . In order to get rid of the last two components of wave front set, we insertmicrolocal cutoff operators as in [14]. In this setting we can take them to be functionsof the action operators b I j . Let φ ε and ψ ε be smooth cutoff functions on R such that(4.2) φ ε ( x ) = ( for | x | ≤ ε/ for | x | > ε (4.3) ψ ε ( x ) = ( for | x | > − ε/ for | x | < − ε Then we set b χ nε = φ ε ( b I b I ) and b χ tε = ψ ε ( b I b I ) . Finally set b χ ε = b χ nε + b χ tε . Note that theoperator ( I − b χ ε ) has no wave front set in a conic ε/ neighborhood of both N ∗ H and T ∗ H . We now define(4.4) ( γ ∗ H γ H ) ≥ ε = ( I − ˆ χ ε/ ) γ ∗ H γ H ( I − ˆ χ ε ) (4.5) ( γ ∗ H γ H ) ≤ ε = ˆ χ ε/ γ ∗ H γ H ˆ χ ε Proposition 4.1.
We have the decomposition (4.6) γ ∗ H γ H = ( γ ∗ H γ H ) ≥ ε + ( γ ∗ H γ H ) ≤ ε + K ε where h K ε ϕ λ j , ϕ λ j i L ( S ,dV g ) = O ε ( λ −∞ j ) and ϕ λ j are any orthonormal basis of eigen-functions of − ∆ g . For the proof of this, see section 9.1.1 in [14]. We also quote the following descrip-tion of the cutoff restriction operator:
Proposition 4.2.
For each ε > , ( γ ∗ H γ H ) ≥ ε is a Fourier integral operator in theclass I ( M, M ; C H ) where C H is the homogeneous canonical relation (4.7) C H = { ( x, ξ, x, ξ ′ ) ∈ T ∗ H S \ × T ∗ H S \ | I ( x, ξ ) = I ( x, ξ ′ ) } In polar coordinates ( r, θ, ρ, η ) on T ∗ S , the set C H is parametrized by the map ι C H : ( θ, η, ρ, ρ ′ ) ( r , θ, ρ, η, r , θ, ρ ′ , η ) The half density part of the symbol of ( γ ∗ H γ H ) ≥ ε pulls back under ι C H to the halfdensity (4.8) (1 − χ ε/ )( r , θ, ρ, η )(1 − χ ε )( r , θ, ρ ′ , η ) | dθ ∧ dη ∧ dρ ∧ dρ ′ | This follows from Lemma 18 in [14] setting Op H ( a ) = Id, because the geodesicpolar coordinates ( r, θ ) are Fermi normal coordinates along H .4.2. The I reflection map and the set b C H . Here we include more geometricpreliminaries to the description of the averaged restriction operator(4.9) ¯ V ε = (2 π ) − Z T U ∗ ( t )( γ ∗ H γ H ) ≥ ε U ( t ) d t which is found in the next subsection. We begin by describing the so-called I reflection map along H . ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION11
Proposition 4.3.
Suppose ( x, ξ ) ∈ T ∗ H S . If ( x, ξ ) / ∈ T ∗ H , there are is exactly onecovector ( x, ξ ′ ) ∈ T ∗ H S such that I ( x, ξ ) = I ( x, ξ ′ ) , ( x, ξ ) = ( x, ξ ′ ) and ξ | T H = ξ ′ | T H .We refer to the map r H : ( x, ξ ) ( x, ξ ′ ) As the I -reflection map.Proof. We’ll show that on the set { I = c } , I is an invertible function of the length q ( x, ξ ) = | ξ | g ( x ) . Thus, if I ( x, ξ ) = I ( x, ξ ′ ) and I ( x, ξ ) = I ( x, ξ ′ ) , then | ξ | g ( x ) = | ξ ′ | g ( x ) and this means that ( x, ξ ′ ) = ( r , θ, ± q | ξ | g ( x ) − c , c ) in polar coordinates.The reflection map then flips the sign of the component dual to r . From [5], we havethe formula(4.10) I ( x, ξ ) = Z r r s | ξ | g ( x ) − p θ ( x, ξ ) a ( r ) dr + p θ Where r and r are the two solutions of a ( r ) = p θ ( x,ξ ) | ξ | g . Now r = r if and only if ( x, ξ ) ∈ T ∗ H thus we have that r = r and ∂∂ | ξ | I ( x, ξ ) = Z r r | ξ | g q | ξ | g ( x ) − c a ( r ) dr > This shows that I is an increasing function of | ξ | g on { I = c } ⊂ T ∗ S \ . (cid:3) From section 4.1, we know that for each ε > , the operator ( γ ∗ H γ H ) ≥ ε is a Fourierintegral operator with canonical relation C H = { ( x, ξ, x, ξ ′ ) | ( x, ξ ) , ( x, ξ ′ ) ∈ T ∗ H S ; ξ | T H = ξ ′ | T H } In the study of ¯ V ε , a related set appears. Define(4.11) b C H = { ( x, ξ, x, ξ ′ ) | x ∈ H ; I ( x, ξ ) = I ( x, ξ ′ ); I ( x, ξ ) = I ( x, ξ ′ ) } It is clear from proposition 4.3, b C H has the following simple description Proposition 4.4.
The set b C H is an immersed submanifold of dimension 3 which canbe written as the union of the two embedded submanifolds b C H = ∆ T ∗ H S [ graph r H | T ∗ H S These intersect along the set ∆ T ∗ H where b C H fails to be embedded. Description of the averaged restriction operator ¯ V ε . The purpose of thissection is to describe the averaged restriction operator(4.12) ¯ V ε = (2 π ) − Z T U ( t ) ∗ ( γ ∗ H γ H ) ≥ ε U ( t ) d t As a Fourier integral operator and calculate its symbolic data. In order to statethe proposition, we set some notation. For any set U ⊂ T ∗ S × T ∗ S , we define itsflow-out Fl ( U ) by Fl ( U ) = [ t ∈ T Φ t × Φ t ( U ) = { (Φ t ( x, ξ ) , Φ t ( y, η )) | ( x, ξ, y, η ) ∈ U } In the calculation of the symbol of ¯ V ε , there are two important submersions. Define i D , i R : T × T ∗ H S → T ∗ S × T ∗ S by(4.13) i D ( t , x, ξ ) = (Φ t ( x, ξ ) , Φ t ( x, ξ )) (4.14) i R ( t , x, ξ ) = (Φ t ( x, ξ ) , Φ t ( r H ( x, ξ ))) The image of these maps are the diagonal and reflection flow-outs, Fl (∆ T ∗ H S ) ,Fl ( graph r H | T ∗ H S ) Proposition 4.5.
Both maps i D and i R are smooth submersions. Over any point ( y, η, y ′ , η ′ ) ∈ T ∗ S × T ∗ S in the image of either map, the fiber can be identified withthe set (4.15) { ( x, ξ ) ∈ T ∗ H S | P ( x, ξ ) = P ( y, η ) } For ( y, η ) / ∈ T ∗ H S , the fiber is identified with two distinct copies of H correspondingto the choice of the northern or southern pointing covector lying on the torus P ( y, η ) .Proof. Fix a point ( y, η, y, η ) in the image of i D . Then Φ t ( x, ξ ) = ( y, η ) for some t ∈ T and ( x, ξ ) ∈ T ∗ H S . The covector ( x, ξ ) lies on the level set P − ( y, η ) and byproposition 4.3 there are two covectors in this set lying over x . Since the flow of H I translates around the equator, for each covector ( x, ξ ) in the set (4.15), there is aunique time t so that Φ t ( x, ξ ) = ( y, η ) . In this way the fiber is identified with twocopies of H (cid:3) These maps induce half densities on the flow-outs Fl (∆ T ∗ H S ) and Fl ( graph r H | T ∗ H S ) as follows. We let µ be the half density on T × T ∗ H S which is equal to 1 on theproduct basis ∂ t ⊗ { ∂ θ , ∂ ρ , ∂ η } . Then the exact sequence → ker di R → T ( T × T ∗ H S ) → T ( Fl ( graph r H | T ∗ H S )) → implies that µ = | dθ | ⊗ µ / | dθ | , where, under the identification of the fiberof i with two copies of H , | dθ | is the volume density such that R H | dθ | = 2 π andthe quotient half density µ / | dθ | assigns the value to the basis ( d Φ t v i , d Φ t dr H v i ) where v i ∈ { H I , ∂ θ , ∂ ρ , ∂ η } . The same is true for the flowout of the diagonal replacing i R with i D . In this case the quotient density µ assigns to the basis ( d Φ t v i , d Φ t v i ) . Proposition 4.6.
The operator ¯ V ε = (2 π ) − Z T U ∗ ( t )( γ ∗ H γ H ) ≥ ε U ( t ) d t is a Fourier integral operator in the class I ( S × S ; C ¯ V ) . Its canonical relation is C ¯ V = Fl ( b C H ) = Fl (∆ T ∗ H S ) [ Fl ( graph r H | T ∗ H S ) The half density symbol of ¯ V ε is equal to ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION13 σ ( ¯ V ε )(Φ t ( x, ξ ) , Φ t ( x, ξ ′ )) = 1 π (1 − χ ε )( x, ξ ) ω ( x, ξ ) q − I ( x,ξ ) | ξ | g a ( r ) µ | dθ | where µ / | dθ | is the half density induced by the fibrations of proposition 4.5. In order to analyze ¯ V ε , we will view it as a composition of pullbacks and pushfor-wards applied to the Fourier integral operator(4.16) V ε ( t , t ′ ) = U ( t ) ∗ ( γ ∗ H γ H ) ≥ ε U ( t ′ ) We begin by describing this operator.
Proposition 4.7.
The operator V ε ( t , t ′ ) is a Fourier integral operator in the class I − ( T × T × S , S ; C V ) (4.17) C V = { ( t , P ( x, ξ ) , t ′ , P ( x, ξ ′ ) , Φ t ( x, ξ ) , Φ t ′ ( x, ξ ′ ) | ( x, ξ, x, ξ ′ ) ∈ C H } The map ι V : T × T × C H → T ∗ ( T × T × S × S ) given by ι V : ( t , t ′ , x, ξ, x, ξ ′ ) = ( t , P ( x, ξ ) , t ′ , P ( x, ξ ′ ) , Φ t ( x, ξ ) , Φ t ′ ( x, ξ ′ )) is a Lagrangian embedding whose image is C V . The half density part of the principalsymbol pulls back along ι to | d t ∧ d t ′ | ⊗ σ (( γ ∗ H γ H ) ≥ ε ) Proof.
Viewing both U ∗ ( t ) , U ( t ′ ) as operators U, U ∗ : C ∞ ( S ) → C ∞ ( T × S ) thenthe composition we are talking about is really V ε ( t , t ′ ) = Id ⊗ U ∗ ( t ) ◦ Id ⊗ ( γ ∗ H γ H ) ≥ ε ◦ U ( t ′ ) The compositions are all transverse provided that C H and C U intersect transverselyin the sense that the maps π i : C H → T ∗ S are transverse to the projections ρ i : C U → T ∗ S onto either factor. This follows from the fact that C U is essentially acanonical graph. It implies the orders add to give the stated order and one can checkeasily that the composite canonical relation and symbol is what was stated in theproposition. (cid:3) Now we describe the pullback under the time diagonal map. Let ∆ : T × S × S → T × T × S × S be the map ∆ : ( t , x, y ) ( t , t , x, y ) . Proposition 4.8.
The kernel of the operator V ε ( t ) = U ∗ ( t )( γ ∗ H γ H ) ≥ ε U ( t ) is in theclass I − ( T × S × S ; ∆ ∗ C V ) Where ∆ ∗ C V is the pullback of C V , the image of theLagrangian embedding i ∆ ∗ C V : T × C H → T ∗ ( T × S × S ) given by (4.18) ι ∆ ∗ C V : ( t , x, ξ, x, ξ ) ( t , P ( x, ξ ) − P ( x, ξ ′ ) , Φ t ( x, ξ ) , Φ t ( x, ξ ′ )) The half density symbol of V ε ( t ) pulls back under ι ∆ ∗ C V to | d t | ⊗ σ (( γ ∗ H γ H ) ≥ ε . Proof.
Recall that the pullback of Lagrangian distributions is well-defined under atransversality condition. Namely, V ε ( t ) = ∆ ∗ V ( t , t ′ ) is a Lagrangian distribution aslong as the maps π | C V → T × T × S × S and ∆ are transverse, which is easilyverified. Letting N ∗ ∆ ⊂ T ∗ ( T × S × S ) × T ∗ ( T × T × S × S ) be the co-normalbundle to the graph of ∆ and π : N ∗ ∆ → T ∗ ( T × T × S × S ) , projection ontothe factor on the right, this implies that the pullback diagram F C V N ∗ ∆ T ∗ ( T × T × S × S ) ιπ is transverse. The left projection of F into T ∗ ( T × S × S ) is then the set(4.19) ∆ ∗ C V = { t , P ( x, ξ ) − P ( x, ξ ′ ) , Φ t ( x, ξ ) , Φ t ( x, ξ ′ ) } Which inherits a canonical half density determined by the symbol of V ε ( t , t ′ ) on C V , the canonical half density on N ∗ ∆ ∼ = T × T ∗ S × T ∗ S and the symplectic halfdensity on T ∗ ( T × T × S × S ) . This is the symbol of V ε ( t ) . (cid:3) Next, let π : T × S × S → S × S be the projection onto the rightmost factors, π ( t , x, y ) = ( x, y ) . Let let π ∗ : C ∞ ( T × S × S ) → C ∞ ( S × S ) be the pushforwardmap defined on smooth functions by π ∗ u ( t , x, y ) = (2 π ) − Z T u ( t , x, y ) d t Lemma 4.9.
Let N ∗ π ⊂ T ∗ ( T × S × S ) × T ∗ ( S × S ) denote the co-normal bundleto the graph of π and ρ L : N ∗ π → T ∗ ( T × S × S ) denote the left projection. Thepushforward diagram F ∆ ∗ C V N ∗ π T ∗ ( T × S × S ) ιρ L is clean away from the singular set i ∆ ∗ C V ( T × T ∗ H ) ⊂ ∆ ∗ C V .Proof. Recall that above diagram is clean if the fiber product F is a submanifold of ∆ ∗ C V × N ∗ π and the linearization T F T (∆ ∗ C V ) T ( N ∗ π ) T ( T ∗ ( T × S × S )) dιdρ L is also a fiber product. Note that the fiber F is the set F = { ( t , , Φ t ( x, ξ ) , Φ t ( x, ξ ′ ) , t , , Φ t ( x, ξ ) , Φ t ( x, ξ ′ ) , Φ t ( x, ξ ) , Φ t ( x, ξ ′ ) | ( x, ξ, x, ξ ′ ) ∈ b C H } The natural parametrization i F : T × b C H → F is an embedding on the smoothparts of b C H . The image i F ( T × T ∗ H ) of the non-smooth part corresponds to the ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION15 singular set i ∆ ∗ C V ( T × T ∗ H ) . Hence we see that F is a submanifold of dimension 5away from this set. To prove that the diagram is clean, we have to verify that T F is given by the kernel of the map τ : T (∆ ∗ C V × N ∗ π ) → T ( T ∗ ( T × S × S )) givenby τ ( u, v, w ) = v − u . Suppose that u = di ∆ ∗ C V ( α, v, v ′ ) ∈ dρ L T ( N ∗ π ) . Then we have ( v, v ′ ) ∈ C H with dP v − dP v ′ = 0 . But this implies that ( v, v ′ ) ∈ T ( b C H ) and thetangent vector ( u, u, w ) ∈ ker τ ⊂ T (∆ ∗ C V × N ∗ π ) is actually equal to di F ( α, v, v ′ ) ,i.e. it is tangent to F . (cid:3) Now since the pushforward diagram is clean, the right projection ρ R : F → T ∗ ( S × S ) is a smooth submersion whose image ρ R ( F ) = C ¯ V = Fl (∆ T ∗ H S ) [ Fl ( graph r H | T ∗ H S ) is a Lagrangian submanifold of T ∗ S × T ∗ S . We now describe how the half densitieson N ∗ π and ∆ ∗ C V determine a half density on the image ρ R ( F ) = C ¯ V . More precisely,at each point p ∈ F , the clean diagram determines an element µ ⊗ ν ∈ | ker d ( ρ R ) p | ⊗| T ρ R ( p ) C ¯ V | . The half density at the point q ∈ C ¯ V is then given by integrating thedensity over the fiber of ρ R over q:(4.20) Z ρ − R ( q ) µ ! ν First consider the sequence of maps → T p F → T i F ( p ) (∆ ∗ C V × N ∗ π ) → im τ → Where τ is the map above. Because the diagram is clean, this sequence is exact. Wesuppose that p = i F ( t , x, ξ, x, ξ ′ ) . We will make use of several different bases whichwe pause to notate here. First, let B = ( H I , ∂ θ , ∂ ρ , ∂ η ) ∈ T ( T ∗ S ) . We will write di N ∗ π ( ∂ t ⊗ B ) denote the basis on T ( N ∗ π ) obtained by pushing forward the productbasis on T × T ∗ S × T ∗ S determined by ∂ t and B . We also let B ′ denote the basis ( ∂ θ , ∂ θ ) , ( ∂ η , ∂ η ) , ( ∂ ρ , , (0 , ∂ ρ ) ∈ T C H and similarly, di ∆ ∗ C V ( ∂ t ⊗ B ′ ) denote the basison T (∆ ∗ C V ) obtained by pushing forward the product basis on T × C H .Now, since both smooth branches of b C H are graphs over T ∗ H S , we have a naturalhalf density µ ∈ | T ( T × b C H ) | which pulls back to | d t | ⊗ | dθ ∧ dη ∧ dρ | on T × T ∗ H S . We let B be a basis of T p F such that µ ( B ) = 1 . We complete this toa basis of T (∆ ∗ C V × N ∗ π ) by adding the 10 vectors ⊗ di N ∗ π ( ∂ t ⊗ B ) in addition tothe vector (0 , d P ∂ ρ , , d Φ t ∂ ρ , ) . We claim that the change of basis matrix betweenthis completed basis and the product basis di ∆ ∗ C V ( ∂ t ⊗ B ′ ) ⊗ , ⊗ di N ∗ π ( ∂ t ⊗ B ) hasdeterminant equal to ± . Lemma 4.10.
Let | Ω | denote the symplectic half density on T ∗ S . Then Ω ( B ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂I ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) Proof.
Since ( r, θ, ρ, η ) are canonical coordinates if we write H I in terms of the basis ∂ r , ∂ θ , ∂ ρ , ∂ η , the coefficient of ∂ r is ∂I ∂ρ . Hence the change of basis from this symplecticbasis to B has determinant | ∂I ∂ρ | (cid:3) Now let σ ∈ | T (∆ ∗ C V × N ∗ π ) | denote the tensor product of the natural halfdensity on N ∗ π and the symbol of V ε ( t ) on ∆ ∗ C V . Then in light of the lemma, σ onthe completed basis above is equal to(4.21) (1 − χ ε )( x, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ∂I ∂ρ ( x, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) This means that the exact sequence, together with our reference half density µ determines the half density ν on im τ which assigns the value (4.21) to the 11 vectors di N ∗ π ( t ⊗B ) , (0 , − d P ∂ ρ , , − d Φ t ∂ ρ ) . We complete this to a basis of T ( T ∗ ( T × S × S )) by adding the vector (0 , ∂ τ , , . Then the symplectic half density on this basis isequal to | ∂I /∂ρ | . Hence, using the exact sequence → im τ → T ( T ∗ ( T × S × S )) → coker τ → We get the negative half density on coker τ which assigns the value (1 − χ ε )( x, ξ ) | ∂I /∂ρ | − to the residue class of (0 , ∂ τ , , .To finish, we use the exact sequence associated thesubmersion ρ R : → ker d ( ρ R ) p → T p F → T ρ R ( p ) C V → Note that this is the exact sequence determined by either i D or i R of proposition4.5 depending on whether ( x, ξ, x, ξ ′ ) is the diagonal or reflection branch of b C H . Nowcoker τ is symplectic dual to ker dρ R . This allows us to identify the minus half densityon coker τ with the half density (1 − χ ε )( x, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ∂I ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) − | dθ | The symbol of ¯ V ε on the diagonal branch is therefore equal to σ ( ¯ V ε )(Φ t ( x, ξ ) , Φ t ( x, ξ )) = (2 π ) − Z i − D (Φ t ( x,ξ ) , Φ t ( x,ξ )) (1 − χ ε )( y, η ) (cid:12)(cid:12)(cid:12)(cid:12) ∂I ∂ρ ( y, η ) (cid:12)(cid:12)(cid:12)(cid:12) − | dθ | ! µ | dθ | and on the reflection branch we have σ ( ¯ V ε )(Φ t ( x, ξ ) , Φ t ( r H ( x, ξ ))) = (2 π ) − Z i − R (Φ t ( x,ξ ) , Φ t ( x,ξ )) (1 − χ ε )( y, η ) (cid:12)(cid:12)(cid:12)(cid:12) ∂I ∂ρ ( y, η ) (cid:12)(cid:12)(cid:12)(cid:12) − | dθ | ! µ | dθ | The proof is then completed by the following proposition:
Proposition 4.11.
For ( x, ξ ) ∈ T ∗ H S in the support of the cutoff − χ ε ( x, ξ ) , wehave (4.22) ∂I ∂ρ ( x, ξ ) = q − I ( x,ξ ) | ξ | g a ( r ) ω ( x, ξ ) where ω is the second component of the frequency vector ω = ∂K∂I . ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION17
Proof.
We have I = G ( | ξ | g , p θ ) . Since p θ does not depend on ρ , ∂I ∂ρ = ∂I ∂ | ξ | g ∂ | ξ | g ∂ρ Now for ( x, ξ ) ∈ T ∗ H S , we have | ξ | g = q ρ + p θ a ( r ) . So ∂I ∂ | ξ | g = ω − ( x, ξ ) and ∂ | ξ | g ∂ρ = q | ξ | g − p θ a ( r ) | ξ | g (cid:3) Since the symbol of the cutoff, χ ε and all of the quanities appearing in (4.22) arefunctions of I and I , they are constant on the fibers of i D and i R . Hence the integralsappearing above can be simplified to σ ( ¯ V ε )(Φ t ( x, ξ ) , Φ t ( x, ξ )) = 1 π (1 − χ ε )( x, ξ ) ω ( x, ξ ) q − I ( x,ξ ) | ξ | g a ( r ) µ | dθ | σ ( ¯ V ε )(Φ t ( x, ξ ) , Φ t ( r H ( x, ξ ))) = 1 π (1 − χ ε )( x, ξ ) ω ( x, ξ ) q − I ( x,ξ ) | ξ | g a ( r ) µ | dθ | This completes the proof of proposition 4.6. We now want to show that ¯ V ε can bewritten as the sum of a pseudo-differential operator and a Fourier integral operator. Proposition 4.12.
We have a decomposition ¯ V ε = P ε + F ε where P ε is an order zeropseudo-differential operator with scalar symbol equal to σ ( P ε )( y, η ) = 1 π (1 − χ ε )( y, η ) ω ( y, η ) q − p θ ( y,η ) | η | y a ( r ) | dy ∧ dη | F ε ∈ I ( S × S ; Fl ( graph r H | T ∗ H S )) . The symbol of F ε is the half density σ ( F ε )(Φ t ( x, ξ ) , Φ t ( r H ( x, ξ ))) = 1 π (1 − χ ε )( x, ξ ) ω ( x, ξ ) q − I ( x,ξ ) | ξ | g a ( r ) µ | dθ | where µ / | dθ | is the half density on the flow-out of the reflection graph determinedin proposition 4.5.Proof. Note that the two flow-out sets Fl (∆ T ∗ H S ) S Fl ( graph r H | T ∗ H S are disjointwhen ( x, ξ ) is restricted to the support of a the cutoff − χ ε . Since V ε only haswave front set in the flow-outs of this region, we can let Ψ ∈ C ∞ c ( T ∗ S × T ∗ S ) be asmooth cutoff function such that ψ = 1 in a neighborhood of the diagonal flow-outand has support disjoint from the reflection flow-out. Then we have ¯ V ε = b ψ ¯ V ε + ( I − b ψ ) ¯ V ε The diagonal flow-out is inside ∆ T ∗ S so the first term is a pseudo-differentialoperator. The symbol is unchanged due to the fact that ψ and − ψ are equal to 1 on neighborhoods of the diagonal, reflected flow-outs. On the diagonal branch of theflow-out, we also have the natural symplectic half density | dy ∧ dη ∧ dy ∧ dη | . It iseasy to check that (see lemma 4.10) µ | dθ | = (cid:12)(cid:12)(cid:12)(cid:12) ∂I ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) − | dy ∧ dη ∧ dy ∧ dη | This accounts for the difference between the symbol of P ε stated here and thesymbol of ¯ V ε on the diagonal branch. (cid:3) Calculation of weak-* limits: Proof of theorem 1.1
In this section we compute the weak-* limits of the measures (1.3), (1.2) by ex-panding their un-normalized versions in ℓ . Recall that we let Π ℓ : L ( S , dV g ) → L ( S , dV g ) denote the orthogonal projection onto the b I = ℓ eigenspace. And if A : C ∞ ( S ) → C ∞ ( S ) is an operator which commutes with D θ . Then we have(5.1) Trace f (cid:18) D θ ℓ (cid:19) A Π ℓ = ℓ X m = − ℓ h Aϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) We will use the symbol calculus to expand the left hand side of (5.1) in powers of ℓ . To begin with, we need a description of the operator f ( D θ /ℓ ) . Proposition 5.1.
Let f ∈ C ∞ c ( R ) . The operator f (cid:0) D θ ℓ (cid:1) is a semi-classical pseudo-differential operator in the class Ψ −∞ ℓ − ( S ) with principal symbol equal to f ( p θ ( y, η )) .Proof. Note that by Fourier inversion, we can write(5.2) f (cid:18) D θ ℓ (cid:19) = 12 π Z R b f ( t ) e i tℓ D θ dt Becauase the flow of D θ is just linear translation in the polar coordinates ( r, θ, ρ, η ) ,we can write (exp i tℓ D θ )( r, θ, r ′ , θ ′ ) = (2 π ) − Z R e i [( r − r ′ ) ρ +( θ − θ ′ ) η ] e i tℓ η dρ dη Now change variables ρ ′ = ρ/ℓ , η = η/ℓ . Then (exp i tℓ D θ )( r, θ, r ′ , θ ′ ) = ℓ (2 π ) Z R e iℓ [( r − r ′ ) ρ +( θ − θ ′ ) η ] e itη ′ dρ ′ dη ′ Inserting this expression into (5.2) and integrating in t finishes the proof. (cid:3) We also need a description of Π ℓ as a semi-classical Fourier integral operator. Fordetails, see for instance theorem 1 of [9]. Although this is written for the cluster pro-jection of a Zoll Laplacian, the same argument applies to the operator b I consideredhere. ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION19
Proposition 5.2.
For A ∈ Ψ a homogeneous order zero pseudo-differential operator, A Π ℓ is a semi-classical Fourier integral operator of order associated to the canonicalrelation C Π = { ( x, ξ, y, η ) ∈ Σ × Σ | ∃ t ∈ [0 , π ) exp tH I ( x, ξ ) = ( y, η ) } Where
Σ = { I = 1 } . Along the parametrizing map ι Π : S × Σ → T ∗ S × T ∗ S ι Π : ( t, x, ξ ) ( x, ξ, exp tH I ( x, ξ )) The half density symbol pulls back to ι ∗ Π σ ( A Π ℓ ) = ℓ e − iℓt | dt | ⊗ σ ( A ) | dµ L | Where dµ L is Liouville measure on the energy surface Σ and σ ( A ) is the scalarsymbol of A with respect to the canonical symplectic half density on N ∗ ∆ . Weak-* limit of ν ℓ ( B ) . Let B ∈ Ψ ( S ) and ¯ B be the average (1.11) of B with respect to the torus action U ( t ) . Then the un-normalized version of ν ℓ ( B ) tested against f ∈ C ∞ c ( − , is ℓ X m = − ℓ h Bϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) = Trace f (cid:18) D θ ℓ (cid:19) ¯ B Π ℓ The right hand side is the trace of a semi-classical Fourier integral operator andby standard symbol calculus it has the leading order asymptotics ℓ X m = − ℓ h Bϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) = ℓ Z Σ f ( p θ ) σ ( ¯ B ) dµ L + O (1) Similarly, the normalizing coefficient N ℓ is N ℓ = Trace ¯ B Π ℓ = ℓ Z Σ σ ( ¯ B ) dµ L + O (1) Finally, since σ ( ¯ B ) is just the average of σ ( B ) with respect to the torus action Φ t ,we have R Σ σ ( ¯ B ) dµ L = R Σ σ ( B ) dµ L = ω ( B ) . We also write Z Σ f ( p θ ) σ ( ¯ B ) dµ L = Z − f ( c ) Z T c σ ( ¯ B ) dµ L,c dc = Z − f ( c ) [ σ ( B )( c ) dc This completes the proof of theorem 1.1 (b) when f is compactly supported. As for µ ℓ , the full statement follows from the fact that [ σ ( B )( c ) is an L function on [ − , .5.2. Weak-* limit of µ ℓ . To begin with, we need to relate the un-normalized versionof (1.2) to a trace formula.
Proposition 5.3.
Let f ∈ C ∞ c ( − , . For each ε > , (5.3) ℓ X m = − ℓ || ϕ ℓm || L ( H ) f (cid:16) mℓ (cid:17) = Trace f (cid:18) D θ ℓ (cid:19) ¯ V ε Π ℓ + R ( ε, ℓ ) where lim sup ℓ →∞ | R ( ε, ℓ ) | ℓ = O ( ε ) Proof.
Note that by proposition 4.1, we have(5.4) ℓ X m = − ℓ || ϕ ℓm || L ( H ) f (cid:16) mℓ (cid:17) = ℓ X m = − ℓ h ( γ ∗ H γ H ) ≥ ε ϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) + ℓ X m = − ℓ h ( γ ∗ H γ H ) ≤ ε ϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) + ℓ X m = − ℓ h K ε ϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) The first term on the right hand side is just the trace appearing in the proposition.Further, since |h K ε ϕ ℓm , ϕ ℓm i| = O ε ( ℓ −∞ ) , we just need to show that(5.5) lim sup 1 ℓ (cid:12)(cid:12)(cid:12)(cid:12) ℓ X m = − ℓ h ( γ ∗ H γ H ) ≤ ε ϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) = O ( ε ) As in the discussion on page 37 of [14], we can bound the sum ℓ (cid:12)(cid:12)(cid:12)(cid:12) ℓ X m = − ℓ h ( γ ∗ H γ H ) ≤ ε ϕ ℓm , ϕ ℓm i f (cid:16) mℓ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) By a sum of terms of the form ℓ ℓ X m = − ℓ || γ H b χ jε ϕ ℓm || L ( H ) where b χ jε is either the tangential or the normal cutoff operator. In both cases, thesymbol of the operator appearing is supported inside a set of volume O ( ε ) inside Σ .By the pointwise Weyl law, lim sup ℓ →∞ ℓ ℓ X − ℓ | b χ jε ϕ ℓm ( x ) | = O ( ε ) and integrating this along H preserves this bound. (cid:3) Proposition 5.4.
For each ε > ,Trace f (cid:18) D θ ℓ (cid:19) ¯ V ε Π ℓ = 4 πℓ Z − f ( c )(1 − χ ε )( c ) ω ( c, q − c K ( c, a ( r ) dc + O ε (1) Proof.
By proposition 4.12, we have ¯ V ε = P ε + F ε . From propositions 5.1,5.2, and4.12, the contribution of the P ε term in the trace is equal to ℓ (cid:18)Z Σ f ( p θ ) σ ( P ε ) dµ L (cid:19) + O ε (1) ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION21
Since the symbol of P ε is a function of I and I , it is constant on each torus T c and the leading term is equal thus equal to (2 π ) ℓ Z − f ( c ) σ ( P ε )( c, dc which is the stated term in the proposition. To finish the proof, we need to showthat the contribution to the trace from the F ε piece is of size O ε (1) . For this, notethat f (cid:0) D θ ℓ (cid:1) F ε Π ℓ is a semi-classical Fourier integral operator of order associated tothe canonical relation C R Π = { ( x, ξ, y, η ) | ( x, ξ ) = Φ t ( r H ( x ′ , ξ ′ )) and (Φ t ( x ′ , ξ ′ ) , y, η ) ∈ C Π } The trace is controlled by the symbol on the intersection C R Π ∩ ∆ T ∗ S . This isequal to the set { (Φ t ( x ′ , ξ ′ ) , Φ t ( r H ( x ′ , ξ ′ )) ∈ C Π | t ∈ T , ( x ′ , ξ ′ ) ∈ T ∗ H S } And this is equivalent to the statement that ( x ′ , ξ ′ ) and r H ( x ′ , ξ ′ ) lie along the same I bicharacteristic. But if ( x ′ , ξ ′ ) / ∈ T ∗ H , this would mean that the projection of the I bicharacteristic to S has a self-intersection, which is impossible. Thus it must bethat ( x ′ , ξ ′ ) = r H ( x ′ , ξ ′ ) ∈ T ∗ H . Due to the cutoff χ ε , the symbol of F ε vanishes onthe aforementioned set. Hence the order ℓ term in the trace vanishes as claimed. (cid:3) Proposition 5.5.
The normalizing factor M ℓ = P ℓm = − ℓ || ϕ ℓm || L ( H ) satisfies lim ℓ →∞ M ℓ ℓ = 4 π Z − ω ( c, q − c K ( c, a ( r ) dc Proof.
In the same fashion as the proof of proposition 5.3, we can write M ℓ = Trace ¯ V ε Π ℓ + R ′ ( ε, ℓ ) Trace ¯ V ε Π ℓ = ℓ Z − (1 − χ ε )( c ) ω ( c, q − c K ( c, a ( r ) dc + O ε (1) where lim sup ℓ →∞ | R ′ ( ε, ℓ ) | /ℓ = O ( ε ) . Since Z − (1 − χ ε )( c ) ω ( c, q − c K ( c, a ( r ) dc → Z − ω ( c, q − c K ( c, a ( r ) dc as ε → , the statement follows. (cid:3) Now in light of propositions 5.1,5.2,and 5.3, for f ∈ C ∞ c ( − , , h µ ℓ , f i = 1 M ℓ ℓ X m = − ℓ || ϕ ℓm || L ( H ) f (cid:16) mℓ (cid:17) = 4 π ℓM ℓ Z − f ( c )(1 − χ ε )( c ) ω ( c, q − c K ( c, a ( r ) dc + R ′′ ( ε, ℓ ) where lim sup | R ′′ ( ε, ℓ ) | = O ( ε ) . Taking ℓ → ∞ and then ε → finishes the proof oftheorem 1.1 (a) when f is compactly supported. We can freely upgrade this statementto f ∈ C ([ − , because ω ( c, q − c K ( c, a ( r ) is an L function of c on [ − , .6. Unitary conjugation to the round sphere
In this section we prove theorem 1.2. That is, we construct a unitary Fourierintegral operator W : L ( S , g ) → L ( S , g can ) such that ( W ˆ I W ∗ = AW D θ W ∗ = D θ Where A = q − ∆ g c an + − is the degree operator on the round sphere. Webegin by describing the outline of the proof. First, using the canonical transformation χ : T ∗ S \ → T ∗ S \ of section 2.2 which satisfies χ ∗ I = | ξ | g c an , χ ∗ p θ = p θ , wecan find a unitary Fourier integral operator W so that [ W , D θ ] = 0 and(6.1) W ˆ I W ∗ = A + R − where R − is a pseudo-differential operator of order − . We then use the averagingargument of Guillemin (See [7]) to show that there exists a unitary pseudo-differentialoperator F of order zero such that(6.2) F ( A + R − ) F ∗ = A + R − where [ A, R − ] = 0 and [ F, D θ ] = 0 . This is contained in propositions 6.1, 6.2, and6.3. Then W = F W is a unitary Fourier integral operator which commutes with D θ and conjugates ˆ I to A + R − , where R − is an order − pseudo commuting with A .Using the fact that A + R − and A have the same spectrum, we easily see that R is a finite rank operator. Proposition 6.1.
There exists a unitary Fourier integral operator W such that W ˆ I W ∗ = A + R − where R − ∈ Ψ − is self-adjoint and [ W , D θ ] = 0 . In thiscase we also have [ R − , D θ ] = 0 Proof.
Let U be any unitary Fourier integral operator whose canonical relation isthe graph of χ . Then by Egorov’s theorem,(6.3) U ˆ I U ∗ = A + R Where R ∈ Ψ . Both the left hand side and A are self-adjoint, so R is as well.The subprincipal symbols of both the left hand side and A vanish which implies that σ ( R ) = 0 so R ∈ Ψ − . We write R − from now on to emphasize this. The only thingleft to do is to show that we can modify U in order to make it commute with D θ .We let V ( t ) = exp itD θ and set ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION23 (6.4) W ′ = 12 π Z π V ( t ) U V ( − t ) dtW ′ is a Fourier integral operator with the same canonical relation as U , althoughit may not be unitary. To fix this, replace W ′ with W = [ W ′ ( W ′ ) ∗ ] − W ′ . Then W W ∗ = I and W is still a Fourier integral operator associated to the same canonicalgraph since W ′ ( W ′ ) ∗ is pseudo-differential. W ′ commutes with D θ so W does aswell. Note that if one replaces U by W , (6.3) is still valid since both operators areassociated to the graph of χ . Since b I and A commute with D θ , we automaticallyhave that R − does as well. (cid:3) The following two propositions constitute a slight modifcation of what Guilleminrefers to as the averaging lemma, found in [7]. The goal of the modification is tomake sure the conjugations commute with D θ . Proposition 6.2.
Let R − be as in proposition 6.1. Then there exists a unitarypseudo-differential operator F ∈ Ψ , a self-adjoint operator R − ∈ Ψ − which com-mutes with A and a smoothing operator R −∞ such that F ( A + R − ) F ∗ = A + R − + R −∞ and [ F, D θ ] = 0 Proof.
We let U ( t ) = exp( itA ) be the unitary group generated by A and for a pseudo-differential operator B , define as before, its average with respect to U ( t ) by(6.5) B av = 12 π Z π U ( t ) BU ( − t ) dt Then B av commutes with A and is self-adjoint if B is. We recall the statementof lemma 2.1 in [7]: If R is any self-adjoint operator of order − k , k ∈ N , thereexists a skew-adjoint pseudodifferential operator S of order − k so that [ A, S ] = R − R av + Ψ − k − . This statement is equivalent to the vanishing of the principal symbol of [ A, S ] − ( R − R av ) which is a first order transport equation for σ ( S ) . This can be solvedfor σ ( S ) explicitly on S ∗ S , which can be extended as a degree − k homogeneousfunction to T ∗ S \ . Since it is imaginary, we can choose S to be skew-adjoint. Givensuch an S , let V ( t ) = exp( itD θ ) and set ¯ S = (2 π ) − R π V ( t ) SV ( − t ) dt . Then ¯ S isstill skew-adjoint and commutes with D θ . If we further suppose that R commuteswith D θ then [ A, ¯ S ] = 12 π Z π V ( t )[ A, S ] V ( − t ) dt (6.6) = 12 π Z π V ( t )( R − R av ) V ( − t ) dt + Ψ − k − (6.7) = R − R av + Ψ − k − (6.8)Hence we may assume from the outset that [ S, D θ ] = 0 . This fact allows us tobuild the operator F in stages. If R − is the operator in proposition 6.1, then usingthe above procedure we can choose S − ∈ Ψ − skew-adjoint such that(6.9) [ A, S − ] = R − − ( R − ) av + R − where R − ∈ Ψ − and so that [ S − , D θ ] = 0 . Then setting F = exp S − , a directcalculation shows that(6.10) F ( A + R − ) F ∗ = A + ( R − ) av + R − By construction, F is unitary and commutes with D θ . We can now choose S − skew-adjoint commuting with D θ such that(6.11) [ A, S − ] = R − − ( R − ) av + R − Then, with F = exp S − exp S − we have(6.12) F ( A + R − ) = A + ( R − ) av + ( R − ) av + R − Continuing in this way, we get a sequence of unitary operators F k = exp S − k · · · exp S − so that F k commutes with D θ and(6.13) F k ( A + R − ) F ∗ k = A + ( R − ) av + · · · + ( R − k ) av + R − k − We also note that F k +1 − F k ∈ Ψ − k − . Let F ′ ∼ P ∞ k =1 ( F k +1 − F k ) , R ∼ P ∞ k =1 ( R − k ) av ,and R − = R av . Then we know that R − − R ∈ Ψ −∞ and if we put F = F ′ + F wehave F − F k ∈ Ψ − k . It is then easy to check that(6.14) F ( A + R − ) F ∗ − ( A + R − ) ∈ Ψ −∞ Furthermore, since all of the F k commute with D θ , we can choose F so that it doesas well. As in the proof of proposition 6.1, F may not be unitary. This is fixed in thesame way, by replacing F with ( F F ∗ ) − F . More explicitly, let G = F F ∗ − I . Notethat F = F k + Ψ − k which implies that G is a smoothing operator. By the functionalcalculus, we can find a self-adjoint operator K so that ( I + K ) = ( I + G ) − and if wereplace F by ( I + K ) F , then F is unitary, [ F, D θ ] = 0 , and we still have F − F k ∈ Ψ − k since K is a smoothing operator. (cid:3) Proposition 6.3.
Suppose that R − and R −∞ ∈ Ψ −∞ are as in proposition 6.2 andthat Spec ( A + R − + R −∞ ) = Spec ( A ) = N . Then there exists a unitary operator L and R ∈ Ψ − , self-adjoint, such that [ R , A ] = 0 and (6.15) L ( I + R + R −∞ ) L ∗ = I + R Furthermore, L − I is a smoothing operator and [ L, D θ ] = 0 Proof.
Let V k denote the k th eigenspace of A and V ′ k the k th eigenspace of A + R − + R −∞ . Also let π k and π ′ k denote orthogonal projection onto these subspaces. Finallylet P k = π ′ k restricted to V ′ k . First we show that there is a C > so that for all N ≥ and k sufficiently large(6.16) || ( A + R − ) N ( P k − π ′ k ) || L ≤ C || ( A + R − ) N R −∞ π ′ k || L ONCENTRATION OF QUANTUM INTEGRABLE EIGENFUNCTIONS ON A CONVEX SURFACE OF REVOLUTION25
To do this, we note that the spectrum of A + R − consists of bands of the form λ jk = k + µ jk where | µ jk | = O ( k − ) . Hence for k sufficiently large, the entire band iscontained in a ball of radius around k . Let γ k be a circle of radius centered at k ∈ N . Then for k sufficiently large,(6.17) π k = 12 πi Z γ k ( λ − ( A + R − )) − dλ and(6.18) π ′ k = 12 πi Z γ k ( λ − ( A + R − + R −∞ )) − dλ Hence(6.19) ( A + R − ) N ( π k π ′ k − π ′ k ) = 12 πi Z γ k ( λ − ( A + R − )) − ( A + R − ) N R −∞ π ′ k ( λ − ( A + R − + R −∞ )) − dλ For λ ∈ γ k , the distance between λ and the spectrum of both A + R − and A + R − + R −∞ is bounded below by . Hence the norms of both resolvents are bounded by ,which implies the norm of the left hand side is bounded by || ( A + R − ) N R −∞ π ′ k || L .Now suppose that we choose k ≥ k so that the above estimate holds. Then, repeatingthe argument on p. 255 of [7] we build a sequence of unitary operators L k : V ′ k → V k . Since L k is a function of P k and A commutes with D θ , each L k does as well.Define the unitary operator L by declaring L = L k on V ′ k for k ≥ k sufficientlylarge so that the above estimate holds. To define L on L ≤ k ≤ k V ′ k , let U k denotethe eigenspace of b I of eigenvalue k . and let ϕ km be an orthonormal basis of U k consisting of joint eigenfunctions of D θ . Then W ϕ km is a basis of V ′ k which are alsojoint eigenfunctions of D θ . Define L by taking W ϕ km to the corresponding standardspherical harmonic of joint eigenvalue ( k, m ) . L clearly commutes with D θ as well as A . Also, by construction L ( A + R − + R −∞ ) L ∗ = A + L ( R − + R −∞ ) L ∗ preserveseach V k eigenspace, so commutes with A . This implies that L ( R − + R −∞ ) L ∗ = R commutes with A . Finally the estimate above is used to prove that L − I is asmoothing operator in the same way as in [7]. (cid:3) Proposition 6.4.
Suppose that Spec ( A + R − ) = Spec ( A ) = N where R − ∈ Ψ − isself-adjoint and commutes with A . Then R − is a finite rank operator.Proof. Since R commutes with A , we can choose an orthonormal basis of V k , e kj satisfying R e kj = µ kj e j . Since R ∈ Ψ − , we have | µ kj | = O ( k − ) . The fact thatSpec ( A + R ) = N implies that for k large, R | V k = 0 which shows that R is finiterank. (cid:3) References [1] Victor Guillemin and Shlomo Sternberg,
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