Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods
aa r X i v : . [ m a t h . A P ] A p r UNIQUE CONTINUATION FOR QUASIMODES ON SURFACESOF REVOLUTION: ROTATIONALLY INVARIANTNEIGHBOURHOODS
HANS CHRISTIANSON
Abstract.
We prove a strong conditional unique continuation estimate forirreducible quasimodes in rotationally invariant neighbourhoods on compactsurfaces of revolution. The estimate states that Laplace quasimodes whichcannot be decomposed as a sum of other quasimodes have L mass boundedbelow by C ǫ λ − − ǫ for any ǫ > C δ λ − δ for some fixed δ > Introduction
We consider a compact periodic surface of revolution X = S x × S θ , equippedwith a metric of the form ds = dx + A ( x ) dθ , where A ∈ C ∞ is a smooth function, A > ǫ > . Our analysis is microlocal, soapplies also to any compact surface of revolution with no boundary, and to certainsurfaces of revolution with boundary under mild assumptions, however we willconcentrate on the toral case for ease of exposition.From such a metric, we get the volume form d Vol = A ( x ) dxdθ, and the Laplace-Beltrami operator acting on 0-forms∆ f = ( ∂ x + A − ∂ θ + A − A ′ ∂ x ) f. We are concerned with quasimodes, which are the building blocks from whicheigenfunctions are made, however we need to define the most basic kind of quasi-modes, which we will call irreducible quasimodes , meaning the quasimodes whichcannot be decomposed as a sum of two or more nontrivial quasimodes. In order tomake our definitions, we recall first that the geodesic flow on T ∗ X is the Hamilton-ian system associated to the principal symbol of the Laplace-Beltrami operator: p ( x, ξ, θ, η ) = ξ + A − ( x ) η . A fixed energy level p = const. consists of all the geodesics of that constant “speed”.For the case of the geodesic Hamiltonian system on T ∗ X , there are two conserved The author is extremely grateful to Andr´as Vasy and Jared Wunsch for their help with themoment map definition of irreducible quasimode, and for their comments and suggestions onearlier versions of this paper. This work is supported in part by NSF grant DMS-0900524.
Figure 1.
The reduced phase space of a toral surface of revolu-tion with many periodic latitudinal geodesics.quantities, the total energy and the angular momentum η . The moment map isthe map sending points of T ∗ X to their associated conserved quantities, that is M ( x, ξ, θ, η ) = (cid:18) ξ + A − ( x ) η η (cid:19) . When the gradient of M has rank 2, then M defines a submersion, so each con-nected component of the preimage is a 2-manifold. Points in T ∗ X where M hasrank 1 or 0 are called critical points, and points ( P, Q ) ∈ R such that { M = ( P, Q ) } contains critical points are called critical values. Critical points correspond to lati-tudinal periodic geodesics, which can also carry quasimode mass, and critical valueshave preimages which may have infinitely many latitudinal periodic geodesics. Thesemiclassical wavefront set is always a closed invariant subset of the energy surface,so our definition of irreducible quasimode will be one which has wavefront massconfined to the closure of one of these two kinds of sets, distinguished by rank of M . Definition 1.1. An irreducible quasimode is a quasimode whose semiclassical wave-front set is contained in the closure of a single connected component in T ∗ X wherethe moment map has constant rank.We also will require a limit on the geodesic complexity by assuming there are onlya finite number of connected regions of latitudinal periodic geodesics. This will notpreclude having infinitely many periodic latitudinal geodesics, but merely havingaccumulation points of connected components of latitudinal geodesics. We thereforewill assume that the moment map has a finite number of critical values, each ofwhich has a preimage of finitely many non-empty connected components. Note thisallows intervals of latitudinal periodic geodesics, but does not allow accumulationof such sets. For an example, see Figure 1.Finally, we will require a certain 0-Gevrey regularity on the manifold, whichin a sense says our manifold is not too far from being analytic. Such a 0-Gevreyassumption nevertheless allows for non-trivial functions which are constant on in-tervals, so this is a very general class of manifolds. Of course this includes analytic NIQUE CONTINUATION 3 manifolds, for which we have a stronger estimate. See Subsection 2.1 for the precisedefinitions.
Theorem 1.
Let X be as above, for a generating curve in the -Gevrey class A ( x ) ∈ G τ ( R ) for some τ < ∞ . Assume the moment map has finitely many criticalvalues, with preimages consisting of finitely many connected components. Suppose u is a (weak) irreducible quasimode satisfying k u k = 1 and ( − ∆ − λ ) u = O ( λ − β ) , for some fixed β > . Let Ω ⊂ X be a rotationally invariant neighbournood, Ω = ( a, b ) x × S θ . Then either (1) k u k L (Ω) = O ( λ −∞ ) , or (2) for any ǫ > , there exists C = C ǫ, Ω ,β > such that (1.1) k u k L (Ω) > Cλ − − ǫ . Remark 1.2.
The proof will show that a more or less straightforward commuta-tor/contradiction argument gives a lower bound of λ − − β . The difficulty comes intrying to beat this lower bound.In the analytic category, we have a significant improvement. Of course in thecase of an analytic manifold, there can be no infinitely degenerate critical elements,nor can there be any accumulation points of sets of latitudinal periodic geodesics,so we do not need to make the assumption about finite geodesic complexity. Corollary 1.3.
Let X be as above, and assume X is analytic. Suppose u is a(weak) irreducible quasimode satisfying k u k = 1 and ( − ∆ − λ ) u = O (1) . Then for any open rotationally invariant neighbourhood Ω ⊂ X , either (1) k u k L (Ω) = O ( λ −∞ ) , or (2) there exists a fixed δ > and a constant C = C Ω > such that k u k L (Ω) > Cλ − δ . Remark 1.4.
The assumption that Ω ⊂ X is a rotationally invariant neighbour-hood of the form Ω = ( a, b ) x × S θ is necessary for this level of generality. To seethis, consider the case where X has part of a 2-sphere embedded in it. Then thereare many periodic geodesics close to the latitudinal one. But these geodesics canbe rotated in θ without changing the angular momentum. Each one of these iselliptic and can carry a Gaussian beam type quasimode. Hence one can createan irreducible quasimode as a superposition of these Gaussian beams. The result-ing “band” of quasimodes need not have nontrivial mass except in a rotationallyinvariant neighbourhood. See Figure 2.2. Preliminaries
In this section we review some of the definitions and preliminary computationsnecessary for Theorem 1, as well as recall the spectral estimates we will be using.
HANS CHRISTIANSON θ Figure 2.
A surface of revolution with a piece of S embedded.Also sketched are two “isoenergetic” periodic geodesics which are θ rotations of each other. One can construct pathological quasi-modes which are continuous, compactly supported superpositionsof isoenergetic quasimodes associated to such geodesics.2.1. The -Gevrey class of functions. For this paper, we use the following 0-Gevrey classes of functions with respect to order of vanishing, introduced in [Chr13].
Definition 2.1.
For 0 τ < ∞ , let G τ ( R ) be the set of all smooth functions f : R → R such that, for each x ∈ R , there exists a neighbourhood U ∋ x and aconstant C such that, for all 0 s k , | ∂ kx f ( x ) − ∂ kx f ( x ) | C ( k !) C | x − x | − τ ( k − s ) | ∂ sx f ( x ) − ∂ sx f ( x ) | , x → x in U. This definition says that the order of vanishing of derivatives of a function isonly polynomially worse than that of lower derivatives. Every analytic function isin one of the 0-Gevrey classes G τ for some τ < ∞ , but many more functions are aswell. For example, the function f ( x ) = ( exp( − /x p ) , for x > , , for x G p +1 , but f ( x ) = ( exp( − exp(1 /x )) , for x > , , for x τ .2.2. Conjugation to a flat problem.
We observe that we can conjugate ∆ byan isometry of metric spaces and separate variables so that spectral analysis of ∆is equivalent to a one-variable semiclassical problem with potential. That is, let T : L ( X, d
Vol) → L ( X, dxdθ ) be the isometry given by
T u ( x, θ ) = A / ( x ) u ( x, θ ) . Then e ∆ = T ∆ T − is essentially self-adjoint on L ( X, dxdθ ). A simple calculationgives − e ∆ f = ( − ∂ x − A − ( x ) ∂ θ + V ( x )) f, NIQUE CONTINUATION 5 where the potential V ( x ) = 12 A ′′ A − −
14 ( A ′ ) A − . If we now separate variables and write ψ ( x, θ ) = P k ϕ k ( x ) e ikθ , we see that( − e ∆ − λ ) ψ = X k e ikθ P k ϕ k ( x ) , where P k ϕ k ( x ) = (cid:18) − d dx + k A − ( x ) + V ( x ) − λ (cid:19) ϕ k ( x ) . Setting h = | k | − and rescaling, we have the semiclassical operator(2.1) P ( z, h ) ϕ ( x ) = ( − h d dx + V ( x ) − z ) ϕ ( x ) , where the potential is V ( x ) = A − ( x ) + h V ( x )and the spectral parameter is z = h λ . In Section 3 we will at first let h = λ − be our semiclassical parameter for the whole quasimode, but then switch to h = | k | − to estimate the parts of the quasimode microsupported where the criticalelements are located. The relevant microlocal estimates near critical elements aresummarized in the following Subsection.2.3. Spectral estimates for weakly unstable critical sets.
In this subsectionwe summarize the spectral estimates we will use for weakly unstable critical ele-ments obtained in [Chr07, Chr10, Chr11, CW11, CM13, Chr13].
Definition 2.2.
Let (
P, Q ) be a critical value of the moment map. Then thereare points in M − ( P, Q ) where the moment map has rank 1 (or 0, but these pointsare easy to handle (see below)). For these points, there are latitudinal periodicgeodesics. If the principal part of the potential, A − ( x ), for the reduced Hamil-tonian ξ + A − ( x ) has an “honest” minimum at x in the sense that if [ a, b ] isthe maximal closed interval containing x with A − ( x ) = A − ( x ) on it, then( A − ) ′ < x < a in some small neighbourhood, and ( A − ) ′ > x > b insome other small neighbourhood, then we say this critical element is weakly stable .In all other cases, we say the critical element is weakly unstable .In the following subsections, we review the microlocal estimates from [Chr13]for weakly unstable critical elements. Taken together, they imply the followingtheorem. Theorem 2.
Let Λ be a weakly unstable critical element in the reduced phase space T ∗ S x , and assume u has h -wavefront set sufficiently close Λ . Then for any ǫ > ,there exists C = C ǫ such that k u k Ch − − ǫ k (( hD ) + V ( x ) − z ) u k , for any z ∈ R . HANS CHRISTIANSON
Unstable nondegenerate critical elements.
A nondegenerate unstable criticalelement exists where the principal part of the potential V ( x ) = A − ( x ) has anondegenerate maximum. To say that x = 0 is a nondegenerate maximum meansthat x = 0 is a critical point of V ( x ) satisfying V ′ (0) = 0, V ′′ (0) < Lemma 2.3.
Suppose x = 0 is a nondegenerate local maximum of the principalpart of the potential V , V (0) = 1 . For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact support in {| ( x, ξ ) | ǫ } . Then there exists C ǫ > such that (2.2) k P ( z, h ) ϕ w u k > C ǫ h log(1 /h ) k ϕ w u k , z ∈ [1 − ǫ, ǫ ] . Remark 2.4.
This estimate is known to be sharp, in the sense that the logarithmicloss cannot be improved (see, for example, [CdVP94a]).2.3.2.
Unstable finitely degenerate critical elements.
In this subsection, we consideran isolated critical point at an unstable but finitely degenerate maximum. Thatis, we now assume that x = 0 is a degenerate maximum for the function V ( x ) = A − ( x ) of order m >
2. If we again assume V (0) = 1, then this means that near x = 0, V ( x ) ∼ − x m . Critical points of this form were first studied in [CW11].This Lemma and the proof are given in [CW11, Lemma 2.3]. Lemma 2.5.
For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact supportin {| ( x, ξ ) | ǫ } . Then there exists C ǫ > such that (2.3) k P ( z, h ) ϕ w u k > C ǫ h m/ ( m +1) k ϕ w u k , z ∈ [1 − ǫ, ǫ ] . Remark 2.6.
This estimate is known to be sharp, in the sense that the exponent2 m/ ( m + 1) cannot be improved (see [CW11]).2.3.3. Finitely degenerate inflection transmission critical elements.
We next studythe case when the principal part of the potential has an inflection point of finitelydegenerate type. That is, let us assume the point x = 1 is a finitely degenerateinflection point, so that locally near x = 1, the potential V ( x ) = A − ( x ) takes theform V ( x ) ∼ C − − c ( x − m +1 , m > C > c >
0. Of course the constants are arbitrary (chosen to agreewith those in [CM13]), and c could be negative without changing much of theanalysis. This Lemma and the proof are in [CM13]. Lemma 2.7.
For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact supportin {| ( x − , ξ ) | ǫ } . Then there exists C ǫ > such that (2.4) k P ( z, h ) ϕ w u k > C ǫ h (4 m +2) / (2 m +3) k ϕ w u k , z ∈ [ C − − ǫ, C − + ǫ ] . Remark 2.8.
This estimate is also known to be sharp in the sense that the expo-nent (4 m + 2) / (2 m + 3) cannot be improved (see [CM13]). NIQUE CONTINUATION 7
Unstable infinitely degenerate and cylindrical critical elements.
In this sub-section, we study the case where the principal part of the potential V ( x ) = A − ( x )+ h V ( x ) has an infinitely degenerate maximum, say, at the point x = 0. Let V ( x ) = A − ( x ). As usual, we again assume that V (0) = 1, so that V ( x ) = 1 − O ( x ∞ )in a neighbourhood of x = 0. Of course this is not very precise, as V could beconstant in a neighbourhood of x = 0 and still satisfy this. So let us first assumethat V (0) = 1, and V ′ ( x ) vanishes to infinite order at x = 0, however, ± V ′ ( x ) < ± x >
0. That is, the critical point at x = 0 is infinitely degenerate but isolated. Lemma 2.9.
For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact supportin {| ( x, ξ ) | ǫ } . Then for any η > , there exists C ǫ,η > such that (2.5) k P ( z, h ) ϕ w u k > C ǫ,η h η k ϕ w u k , z ∈ [1 − ǫ, ǫ ] . For our next result, we consider the case where there is a whole interval ata local maximum value. That is, we assume the principal part of the effectivepotential V ( x ) has a maximum V ( x ) ≡ x ∈ [ − a, a ], and that ± V ′ ( x ) < ± x > a in some neighbourhood. Lemma 2.10.
For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact supportin {| x | a + ǫ, | ξ | ǫ } . Then for any η > , there exists C ǫ,η > such that (2.6) k P ( z, h ) ϕ w u k > C ǫ,η h η k ϕ w u k , z ∈ [1 − ǫ, ǫ ] . Infinitely degenerate and cylindrical inflection transmission critical elements.
In this subsection, we assume the effective potential has a critical element of in-finitely degenerate or cylindrical inflection transmission type. This is very similarto Subsection 2.3.4, but now the potential is assumed to be monotonic in a neigh-bourhood of the critical value.We begin with the case where the potential has an isolated infinitely degeneratecritical point of inflection transmission type. As in the previous subsection, wewrite V ( x ) = A − ( x ) + h V ( x ) and denote V ( x ) = A − ( x ) to be the principalpart of the potential. Let us assume the point x = 1 is an infinitely degenerateinflection point, so that locally near x = 1, the potential takes the form V ( x ) ∼ C − − ( x − ∞ , where C >
1. Of course the constant is arbitrary (chosen to again agree withthose in [CM13]). Let us assume that our potential satisfies V ′ ( x ) x = 1,with V ′ ( x ) < x = 1 in some neighbourhood so that the critical point x = 1 isisolated. Lemma 2.11.
For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact supportin {| ( x − , ξ ) | ǫ } . Then for any η > , there exists C = C ǫ,η > such that (2.7) k P ( z, h ) ϕ w u k > C ǫ h η k ϕ w u k , z ∈ [ C − − ǫ, C − + ǫ ] . On the other hand, if V ′ ( x ) ≡ x − ∈ [ − a, a ] with V ′ ( x ) < x − < − a and x − > a , we do not expect anything better than Lemma 2.11.The next lemma says that this is exactly what we do get. To fix an energy level,assume V ≡ C − on [ − a, a ]. HANS CHRISTIANSON
Lemma 2.12.
For ǫ > sufficiently small, let ϕ ∈ S ( T ∗ R ) have compact supportin {| x − | a + ǫ, /, | ξ | ǫ } . Then for any η > , there exists C = C ǫ,η > suchthat (2.8) k P ( z, h ) ϕ w u k > C ǫ h η k ϕ w u k , z ∈ [ C − − ǫ, C − + ǫ ] . Proof of Theorem 1 and Corollary 1.3
Recall the conjugated Laplacian is − e ∆ = − ∂ x − A − ( x ) ∂ θ + V ( x ) , where V ( x ) has been computed above. We will do some analysis and reductionsnow before separating variables. If we are considering quasimodes( − e ∆ − λ ) u = E ( λ ) k u k , where E ( λ ) = O ( λ − β )for some β >
0, then we begin by rescaling. Set h = λ − so that( − h ∂ x − h A − ( x ) ∂ θ + h V ( x ) − u = e E ( h ) k u k , where e E ( h ) = h E ( h − ) = O ( h β ). With ξ, η the dual variables to x, θ as usual,the semiclassical symbol of this operator is p = ξ + A − ( x ) η + h V ( x ) − , and the semiclassical principal symbol is p = ξ + A − ( x ) η − . It is worthwhile to point out that at this point our semiclassical parameter is h = λ − . After separating variables later in the proof, we will let h = | k | − , where k isthe angular momentum parameter. However, in the regime where we so take h , | k | and λ will be comparable, so it is merely a choice of convenience.It is important to keep in mind for the remainder of this paper what the variousparameters represent. Here, the variable η represents hD θ . As we will eventuallybe decomposing in Fourier modes in the θ direction, this means that the variable η takes values in h Z .We next record that a standard h -parametrix argument tells us that any quasi-mode is concentrated on the energy surface where { p = 0 } . The proof is standard. Lemma 3.1.
Suppose u satisfies ( − h ∂ x − h A − ( x ) ∂ θ + h V ( x ) − u = e E ( h ) k u k , where e E ( h ) = h E ( h − ) = O ( h β ) , and Γ ∈ S satisfies Γ ≡ in a small fixedneighbourhood of { p = 0 } . Then (1 − Γ w ) u = O ( h β ) . Hence we will restrict our attention to the characteristic surface where { p = 0 } .Using our moment map idea, we know that η is invariant under the classical flow.Hence if η is very large, our operator will be elliptic, while if η is very small, theparameter ξ will be bounded away from zero, and hence we will have uniform NIQUE CONTINUATION 9 propagation estimates. Let us make this more precise. Let A = min( A ( x )) and A = max( A ( x )), and let 1 = ψ ( η ) + ψ ( η ) + ψ ( η )be a partition of unity satisfying ψ ≡ {| η | A } with support in {| η | A } ; ψ ≡ {| η | > A } with support in {| η | > A } . Then, on supp ψ , we have η A − ( x ) η A − , and on supp ψ , we have η A − ( x ) > η A − > . Now for our quasimode u , write u = u + u + u + u := ψ w Γ w u + ψ w Γ w u + ψ w Γ w u + (1 − Γ w ) u. Since hD θ commutes with − e ∆ and we can choose Γ = Γ( p ) so that [ p w , Γ w ] = O ( h ), each of these u j are also quasimodes of the same order as u (but of coursemay have small or even trivial L mass).3.0.6. Estimation of u . Observe that on the support of ψ , since η is invariant, wehave | ξ | > / − O ( h ), which means the propagation speed in the x -direction isbounded below. We claim this implies k u k L x,θ c k u k L ([ a,b ] x × S θ ) for some c >
0. In other words, u is uniformly distributed in the sense that themass cannot be vanishing in h on any set.The claim follows by propagation of singularities. The standard propagationof singularities result applies whenever the classical flow propagates singularitiesfrom one region to another in phase space. Since we are analyzing the region where ξ = 0, we have uniform propagation in the x direction. A general statement is givenin the following Lemma (a refinement of H¨ormander’s original result [H¨or71]). Fora proof in this context, see, for example, [Chr07, Lemma 6.1] and [BZ04, Lemma4.1]. Lemma 3.2.
Suppose V ⋐ T ∗ X , p is a symbol, T > , A an operator, and V ⋐ T ∗ X a neighbourhood of γ satisfying ∀ ρ ∈ { p − (0) } \ V, ∃ < t < T and ǫ = ± such that exp( ǫsH p )( ρ ) ⊂ { p − (0) } \ V for < s < t, and exp( ǫtH p )( ρ ) ∈ V ;(3.1) and A is microlocally elliptic in V × V . If B ∈ Ψ , ( X, Ω X ) and WF h ( B ) ⊂ T ∗ X \ V , then k Bu k C (cid:0) h − k P u k + k Au k (cid:1) + O ( h ∞ ) k u k . (3.2) Fix two non-empty intervals in the x direction, ( a, b ) and ( c, d ) and assume u = u is L normalized. Now using that P u = O ( h β ) k u k , we have k u k L (( c,d ) × S Ch − k P u k + C k u k L (( a,b ) × S ) Ch β k u k L ( S × S ) + C k u k L (( a,b ) × S ) , for some C >
0. For h > u has mass boundedbelow independent of h in any x neighbourhood ( c, d ), then k u k L (( a,b ) × S ) > c ′ > h . Rescaling in terms of u if u is not normalized, we recover k u k L (( a,b ) × S ) > c ′ k u k . Since the interval ( a, b ) is arbitrary, we have shown that the L -mass on anyrotationally invariant neighbourhood is positive independent of h . Thus (1.1) holdswith a lower bound independent of h = λ − .3.0.7. Estimation of u . On the other hand, on the support of ψ , we have theprincipal symbol satisfies | p | > , so we claim that an elliptic argument shows k u k L = O ( h ∞ ) k u k L . That is, since | p | > on support of ψ , there is an h -parametrix for P there:there exists Q such that QP ψ w = ψ w + O ( h ∞ ) , and further Q has bounded L norm. Hence k u k = k QP u k + O ( h ∞ ) k u k C k P u k + O ( h ∞ ) k u k = O ( h β ) k u k . This implies u = O ( h ∞ ).3.0.8. Estimation of u . In order to consider the final part u , which is microsup-ported where all the critical points are, we will employ one further reduction. Since u is microsupported in a region where | η | is bounded between two constants, say, a | η | a , and η = hk for some integer k , a priori the number of angularmomenta k in the wavefront set of u is comparable to h − . We can do better thanthat. Using the semiclassical calculus, we will next show that there exists k ∈ Z such that for any ǫ >
0, we have u = X | k − k | h − ǫ e ikθ ϕ k ( x ) + O ( h ∞ ) k u k . That is, we claim that the Fourier decomposition of u can actually only have O ( h − ǫ ) non-trivial modes. To prove this claim, fix k ∈ Z and any ǫ >
0, andchoose a k ∈ Z satisfying | k − k | > h − ǫ . We will show that we can decompose u into (at least) two pieces with disjoint mi-crosupport, one near hk and one near hk . Evidently, these two pieces correspondto different angular momenta η , so have wavefront sets associated to different level NIQUE CONTINUATION 11 sets of the moment map. Of course, level sets sufficiently close (in an h -dependentset) may contribute to a single irreducible quasimode, but the point is to quantifyhow far away from a single level set one needs to go before leaving the microsupportof an irreducible quasimode.In order to make this rigorous, let η j = hk j for j = 0 ,
1, and choose χ ( r ) ∈ C ∞ c ( R )satisfying χ ( r ) ≡ | r | , with support in {| r | } . For j = 0 ,
1, let χ j ( η, h ) = χ (cid:18) η − η j h − ǫ/ (cid:19) . As semiclassical symbols, the χ j are in a harmless h / − ǫ/ calculus, and moreoverthey only depend on η (not on θ ) and commute with − e ∆. On the support of eachof the χ j , we have (cid:12)(cid:12)(cid:12)(cid:12) η − η j h − ǫ/ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) hk − hk j h − ǫ/ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k − k j h − ǫ/ (cid:12)(cid:12)(cid:12)(cid:12) . This implies | k − k j | h − ǫ/ , so as h → χ and χ have disjoint supports. This means the functions χ u and χ u have disjoint h -wavefront sets, so they are almost orthgonal: h χ u , χ u i = O ( h ∞ ) . Hence if each of these functions has nontrivial L mass, then u was not an irreduciblequasimode.Finally, we analyse the function u , but spread over at most O ( h − ǫ ) Fouriermodes. Throughout the remainder of this section, let λ be large and fixed. Let usconsider a single Fourier mode confined to a single angular momentum k .The case of u is the most interesting case, as the microsupport of u containsall of the critical elements. Now recalling again the separated equation (2.1) withthe potential V ( x ) = A − ( x ) + h V ( x ) , let A and A again be the min/max respectively of A ( x ). Our spectral parameternow is z = h λ . We are localized where12 A ( λ − k ) A , or 12 A − z A − . This of course implies that λ and k are comparable. Let ( a, b ) ⊂ S be a non-emptyinterval. We need to show that if u is a weak irreducible quasimode,(( hD ) + V ( x ) − z ) u = O ( h β ) k u k , with k u k = 1, then either k u k L ( a,b ) = O ( h ∞ ) = O ( λ −∞ ), or k u k L ( a,b ) > C ǫ h ǫ for any ǫ >
0. Let us assume that u is nontrivial so that k u k L ( a,b ) > ch N for some N .There are a number of subcases to consider here. We observe that, according toLemma 3.2, we can always microlocalize further to a set close to the energy level A α α ′ a ′ b ′ β Figure 3.
The function A − ( x ) and the weakly unstable criticalpoint β .of interest. That is, for P ( z, h ) = ( hD ) + V ( x ) − z , if P ( z, h ) u = O ( h β ), thenif ψ ( r ) ∈ C ∞ c ( R ) satisfies ψ ≡ r near 0, we have for any δ > ψ w (( ξ + V ( x ) − z ) /δ ) u = u + O ( h β ) . For the rest of this section, we write ψ w for this energy cutoff. Case 1:
Next, assume z is in a small neighbourhood of a critical energy level, and assume A ′ ( x ) = 0 somewhere on ( a, b ). Then let ( a ′ , b ′ ) be a non-empty interval with( a ′ , b ′ ) ⋐ { A ′ = 0 } ∩ ( a, b ) , and let ( α ′ , β ) ⊃ ( a ′ , b ′ ) be the maximal connected interval with A ′ ( x ) = 0 on( α ′ , β ). Now A ′ has constant sign on ( α ′ , β ), so at least one of α ′ or β is part of aweakly unstable critical element (see Figure 3).Without loss in generality, assume A ′ < α ′ , β ) so that at least β lies ina weakly unstable critical element. That is, the principal part of the potential A − ( x ) increases as x → β − , and takes the value, say A − ( β ) = A . Let ( α, β )be the maximal open interval containing ( α ′ , β ) where A − ( x ) < A on ( α, β ).As A − ( x ) < A − ( α ) for x ∈ ( α, β ) and A − ( α ) = A , we have ( A − ( x )) ′ < x ∈ ( α, β ) sufficiently close to α . That means that either α is part of a weaklyunstable critical element, or A ′ ( α ) = 0. We break the analysis into the two separatesubsubcases, beginning with A ′ ( α ) = 0. Case 1a: If A ′ ( α ) = 0, then the weakly unstable/stable manifolds associated to( A − ) ′ ( β ) = 0 are homoclinic to each other (see Figure 4), and in particular, prop-agation of singularities can be used to control the mass along this whole trajectory,as long as we stay away from the right hand endpoint β . That is, propagation ofsingularities implies for any η > h , k ψ w u k L ( α,β − η ) C η ( h − k (( hD ) + V − z ) ψ w u k + k ψ w u k L ( a ′ ,b ′ ) C η h β k u k + k ψ w u k L ( a ′ ,b ′ ) . Hence by taking h > k ψ w u k L ( α,β − η ) frombelow in terms of k u k .Let [ β, κ ] be the maximal connected interval containing β on which A ′ = 0 (weallow κ = β if the critical point is isolated). Let ˜ χ ≡ β, κ ] with supportin a small neighbourhood thereof, and let χ ≡ χ with support in a NIQUE CONTINUATION 13 βα Figure 4. If A ′ ( α ) = 0, the unstable manifold from β flows intothe stable manifold at β (homoclinicity). The interval indicates aregion with propagation speed uniformly bounded below.slightly smaller set so that (1 − ˜ χ ) > (1 − χ ) and (1 − ˜ χ ) > c | χ ′ | . Then writing P ( z, h ) = ( hD ) + V − z , we have from Theorem 2 (for any ǫ > k u k k χu k + k (1 − χ ) u k C ǫ h − − ǫ k P ( z, h ) χu k + k (1 − ˜ χ ) u k C ǫ h − − ǫ ( k χP ( z, h ) u k + k [ P ( z, h ) , χ ] u k ) + k (1 − ˜ χ ) u k C ′ ǫ ( h β − ǫ k u k + h − − ǫ k (1 − ˜ χ ) u k ) + k (1 − ˜ χ ) u k Rearranging and taking h > ǫ < β , we get(3.3) k (1 − ˜ χ ) u k > C ǫ h ǫ k u k . Now either the wavefront set of u is contained in the closure of the lift of ( α, β )or it isn’t. In the latter case there is nothing to prove. In the former case, weconclude that u = O ( h ∞ ) on any open subset whose closure does not meet the set[ α, β ]. We appeal to propagation of singularities one more time. Since A ′ ( α ) = 0,propagation of singularities applies in a neighbourhood of α , so that (shrinking η > c > k u k L ( α,β − η ) > c k u k L ( α − η,β − η ) . Since we have assumed u = O ( h ∞ ) on ( α − η, β + η ) c , this estimate, together with(3.3) and (3.2) allows us to conclude k u k L ( α,β − η ) > C k (1 − ˜ χ ) u k > C ǫ h ǫ k u k . Case 1b:
We now consider the possibility that A ′ ( α ) = 0 as well as A ′ ( β ) = 0(see Figure 5). In this case, propagation of singularities fails at both endpoints of( α, β ), so we can only conclude that for any η > h , k u k L ( α + η,β − η ) C η ( h − k (( hD ) + V − z ) u k + k u k L ( a ′ ,b ′ ) . Hence now it suffices to prove that for some η > h ,we have the estimate k u k L ( α + η,β − η ) > C ǫ h ǫ k u k for any ǫ > β, κ ] be the maximal connected interval containing β on which A ′ = 0, andlet [ ω, α ] be the maximal connected interval containing α on which A ′ = 0. Let˜ χ ≡ β, κ ] ∪ [ ω, α ] with support in small neighbourhoods thereof, and let χ ≡ βα Figure 5. If A ′ ( α ) = 0, the unstable manifold from β flows intothe stable manifold at α and vice versa. The interval indicates aregion with propagation speed uniformly bounded below.on supp ˜ χ with support in a slightly smaller set so that (1 − ˜ χ ) > (1 − χ ) and(1 − ˜ χ ) > c | χ ′ | . Since both [ ω, α ] and [ β, κ ] are weakly unstable, we can applyTheorem 2 and the same argument as above to finish this case. Case 2:
Finally, we assume ( a, b ) ⊂ { A ′ = 0 } . Again, if A − ≡ A on ( a, b ) and z = A , we can use propagation of singularities to control k u k L ( a,b ) from below byits mass on the connected component in { p = z } containing ( a, b ) (as in the case of u above). Hence we are interested in the case where z is in a small neighbourhoodof A .If u = O ( h ∞ ) on ( a, b ) there is nothing to prove, so assume not. Then if [ α, β ] ⊃ ( a, b ) is the maximal connected interval where A − ( x ) ≡ A , the wavefront set of u is contained in a small neighbourhood of [ α, β ], so that for δ > k u k L ([ α − δ,β + δ ] c ) = O δ ( h ∞ ) . That means that, either k u k L ([ a,b ]) > c > , k u k L ([ α − δ,a ]) > c > , or k u k L ([ b,β + δ ]) > c > . If the first estimate is true, we’re done, so assume without loss in generality that k u k L ([ b,β + δ ]) > c >
0. Assume for contradiction that there exists ǫ > k u k L ( a,b ) Ch ǫ . Let χ ∈ C ∞ c be a smooth function such that χ ≡ b, β + δ ] with support in ( a, β + 2 δ ). Write ˜ u = χu . If [ α, β ] is a weakly stablecritical element, modify A − ( x ) on the support of 1 − χ so that [ α, β ] is weaklyunstable. That is, if ( A − ( x )) ′ < x < α in some neighbourhood, replace A with a locally defined function ˜ A satisfying ˜ A ≡ A on supp χ but ( ˜ A − ( x )) ′ > x < α in some neighbourhood. If [ α, β ] is weakly unstable, then let ˜ A ≡ A (seeFigure 6. We apply Theorem 2 once again (for any ǫ > k χu k C ǫ h − − ǫ k (( hD ) + ˜ A − + h V − z ) χu k = C ǫ h − − ǫ k (( hD ) + A − + h V − z ) χu k C ǫ h − − ǫ ( k P ( z, h ) u k + k [ P ( z, h ) , χ ] u k ) C ′ ǫ ( h ǫ − ǫ k u k + h − − ǫ k u k L ( a,b ) ) + O ( h ∞ ) , where the O ( h ∞ ) error comes from the part of the commutator [ P ( z, h ) , χ ] sup-ported outside a neighbourhood of [ α, β ] (the other part contributing the integralover ( a, b )). But our contradiction assumption implies that the right hand side is o (1) as h → ǫ < ǫ . As k χu k > c >
0, this is a contradiction.
NIQUE CONTINUATION 15 ˜ A − α a b βχA − Figure 6.
The setup for Case 2. Here if the quasimode is smallin ( a, b ), we cut off to the right of ( a, b ) and modify A − to the leftto be weakly unstable. We then arrive at a contradiction.3.1. Finishing up the proof.
We now put together the estimates of u , u , u , u .Since u = O ( h β ) and u = O ( h ∞ ), for h > u and u must have L mass bounded below independent of h . If u has L massbounded below independent of h we’re done by the propagation of singularitiesargument in Subsection 3.0.6. Hence we need to conclude Theorem 1 assuming u is small and u carries most of the L mass.Fix ( a, b ) as considered in Subsection 3.0.8 and recall we know that for any ǫ > u = X | k − k | h − ǫ e ikθ ϕ k ( x ) + O ( h ∞ ) k u k . We use the notation Ω = ( a, b ) x × S θ as in the statement of Theorem 1. Each ϕ k satisfies either k ϕ k k L ( a,b ) = O ( | k | −∞ )or for any ǫ > k ϕ k k L ( a,b ) > c | k | − − ǫ k ϕ k k L ( S x ) . In the first case, these ϕ k s have disjoint wavefront sets from the ϕ k s in the lattercase, so leaving them in the sum would mean our quasimode was not irreducible.Removing these from the sum and reindexing if necessary, we conclude k u k L (Ω) = X | k − k | k ǫ k ϕ k ( x ) k L ( a,b ) + O ( h ∞ ) k u k > c ′ k − − ǫ X | k − k | k ǫ k ϕ k ( x ) k L ( S x ) − O ( h ∞ ) k u k = c ′′ λ − − ǫ k u k L ( S x × S θ ) − O ( λ −∞ ) k u k . This concludes the proof of Theorem 1. (cid:3)
References [BZ04] Nicolas Burq and Maciej Zworski. Geometric control in the presence of a black box.
J. Amer. Math. Soc. , 17(2):443–471 (electronic), 2004.[CdVP94a] Y. Colin de Verdi`ere and B. Parisse. ´Equilibre instable en r´egime semi-classique. In
S´eminaire sur les ´Equations aux D´eriv´ees Partielles, 1993–1994 , pages Exp. No. VI,11. ´Ecole Polytech., Palaiseau, 1994. [CdVP94b] Yves Colin de Verdi`ere and Bernard Parisse. ´Equilibre instable en r´egime semi-classique. II. Conditions de Bohr-Sommerfeld.
Ann. Inst. H. Poincar´e Phys. Th´eor. ,61(3):347–367, 1994.[Chr07] Hans Christianson. Semiclassical non-concentration near hyperbolic orbits.
J. Funct.Anal. , 246(2):145–195, 2007.[Chr10] Hans Christianson. Corrigendum to “Semiclassical non-concentration near hyperbolicorbits” [J. Funct. Anal. 246 (2) (2007) 145–195].
J. Funct. Anal. , 258(3):1060–1065,2010.[Chr11] Hans Christianson. Quantum monodromy and nonconcentration near a closed semi-hyperbolic orbit.
Trans. Amer. Math. Soc. , 363(7):3373–3438, 2011.[Chr13] Hans Christianson. High-frequency resolvent estimates on asymptotically Euclideanwarped products. preprint , 2013.[CM13] Hans Christianson and Jason Metcalfe. Sharp local smoothing for manifolds withsmooth inflection transmission. preprint , 2013.[CW11] Hans Christianson and Jared Wunsch. Local smoothing for the schr¨odinger equationwith a prescribed loss.
Amer. J. Math. to appear , 2011.[H¨or71] Lars H¨ormander. On the existence and the regularity of solutions of linear pseudo-differential equations.
Enseignement Math. (2) , 17:99–163, 1971.
Department of Mathematics, University of North Carolina
E-mail address ::