RESONANCE-FREE REGIONS FOR DIFFRACTIVE TRAPPING BY CONORMAL POTENTIALS

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1 RESONANCE-FREE REGIONS FOR DIFFRACTIVE TRAPPING BY CONORMAL POTENTIALS ORAN GANNOT AND JARED WUNSCH Abstract. We consider te Scrödinger operator P = 2 g + V on R n equipped wit a metric g tat is Euclidean outside a compact set. Te real-valued potential V is assumed be compactly supported and smoot except at conormal singularities of order 1 α along a compact ypersurface Y. For α > 2 (or even α > 1 if te classical flow is unique), we sow tat if E 0 is a non-trapping energy for te classical flow, ten te operator P as no resonances in a region [E 0 δ, E 0 + δ] i[0, ν 0 log(1/)]. Te constant ν 0 is explicit in terms of α and dynamical quantities. We also sow tat te size of tis resonance-free region is optimal for te class of piecewise-smoot potentials on te line. 1. Introduction 1.1. Main results. Let X = R n, equipped wit a smoot Riemannian metric g suc tat g ij = δ ij outside a compact set. Wit g denoting te nonnegative Laplacian on (X, g), consider te semiclassical Scrödinger operator wit compactly supported potential, P = 2 g + V. We assume tat V I [ 1 α] (Y ) is conormal to a compact ypersurface Y X wit α > 1. Tis notation means te following: V is C away from Y. In local coordinates (x 1, x ) near Y, wit Y given by {x 1 = 0}, V (x 1, x ) = e ix1 ξ 1 v(x, ξ 1 ) dξ 1, v S 1 α (R n x; R ξ 1). Suc a potential V is at least C 1,γ for some γ > 0. Let p = σ (P ) denote te semiclassical principal symbol of P, p = ξ 2 g + V. Te Hamilton vector field H p is continuous, ence always as global solutions. We furter assume tat H p as unique integral curves; tis is always true if α > 2 (were 1

2 2 ORAN GANNOT AND JARED WUNSCH H p is Lipscitz) but in general fails in te range α (1, 2]. In particular, tere is a well-defined flow ρ exp thp (ρ) on T X, wic is tangent to eac energy surface {p = E}. Let E 0 > 0 be a non-trapping energy level for te H p flow, i.e., assume tat x in bot directions along all integral curves of H p in {p = E 0 }. We sow tat for a suitable ν 0 > 0 and, δ > 0 sufficiently small, tere are no resonances of P in te spectral window [E 0 δ, E 0 + δ] i[0, ν 0 log(1/)]. Te quantity ν 0 > 0 as a dynamical caracterization wic we discuss next. Let H E T Y denote te set of yperbolic points for p E; tese are te points in pase space were te flow is transverse to Y. More precisely, if we introduce normal coordinates (x 1, x ) for g wit respect to Y, so tat Y = {x 1 = 0} locally, ten we can write p(x, ξ) E = (ξ 1 ) 2 + K(x)ξ, ξ + V (x) E (ξ 1 ) 2 r(x, ξ, E) for a positive definite matrix K(x). In tese coordinates, (x, ξ ) H E r(0, x, ξ, E) > 0. We also remark for later use tat te glancing set G E T Y is defined in coordinates by te equation r(0, x, ξ, E) = 0. For x Y, let π : T x X T x Y denote te canonical projection, wic in local coordinates is just te map (0, x, ξ 1, ξ ) (x, ξ ). Note tat π is two-to-one over H E and one-to-one over G E. Given E R, we introduce te affine lengt of te longest H p trajectory connecting two yperbolic points: diam E (Y ) = sup{ t : tere exists ρ π 1 (H E ) wit exp thp (ρ) π 1 (H E )}. (1.1) Since Y is compact, diam E (Y ) is finite for a non-trapping energy E. For I R we also define diam I (Y ) = sup{diam E (Y ) : E I}. Tis is again finite if I = [E 0 δ, E 0 + δ] for E 0 non-trapping and δ > 0 sufficiently small. Teorem 1. Let (X, g) and V I [ α 1] (Y ) be as above. If E 0 > 0 is non-trapping, ten tere exists δ 0 > 0 wit te following property. Given δ (0, δ 0 ) and α 0 < ν 0 < diam [E0 δ,e 0 +δ](y ),

3 tere exists 0 > 0 suc tat P as no resonance z wit DIFFRACTIVE TRAPPING 3 z [E 0 δ, E 0 + δ] i[0, ν 0 log(1/)] for (0, 0 ). If diam [E0 δ,e 0 +δ](y ) = 0, ten te conclusion is valid for any ν 0 (0, ). Wen V is smoot (so tat α can be taken arbitrarily large), te fact tat Teorem 1 olds for arbitrary ν 0 (0, ) is originally due to Martinez [11]. Te size of te resonance-free region in Teorem 1 is already optimal for te class of compactly supported piecewise-smoot potentials on R. Consider te operator P = (D x ) 2 + V, were V satisfies te following properties: (1) Tere exists L > 0 suc tat supp V [0, L]. (2) Te restriction of V to [0, L] is smoot. If V vanises to order k at x = 0 and to order l at x = L, ten V I [ 1 min(k,l)] ({0, L}). For use in Teorem 2 below define te quantity φ = (1/2π) arg(v (k) (0 + ) V (l) (L )), wic arises as te pase sift in a Bor Sommerfeld type formula. Consider a spectral interval [a, b], were a > sup V. Certainly any energy E [a, b] is nontrapping for te H p flow. Define (alf) te action and period by S(E) = L 0 (E V (s)) 1/2 ds, T (E) = L 0 1 ds. (1.2) 2(E V (s)) 1/2 Of course tis is a sligt abuse of terminology, since te endpoints are not turning points for te classical dynamics. Observe tat in te notation of (1.1). T (E) = diam E ({0, L}) Noting tat S(E) is increasing on any interval [a, b] as above, define [α, β] = S([a, b]) and ten set N() = {n Z : π(n + (l k)/4 + φ) 1 [α, β]}. Note tat N() 1 β α /π. We ten ave te following semiclassical analogue of [14, Teorem 6] on te existence and asymptotics of resonances. Teorem 2. Let k and l denote te orders of vanising of V at x = 0 and x = L, respectively. For eac n N() let E n = S 1 (π(n + (l k)/4 + φ)).

4 4 ORAN GANNOT AND JARED WUNSCH Tere exists 0 > 0 suc tat for eac (0, 0 ) and n N() tere is a unique resonance z n satisfying i(l + k) z n = E n 2T (E n ) log(1/) + i 2T (E n ) + O( 2 log(1/) 2 ). ( log V (k) (0 + ) V (l) (L ) (1/2)(l + k + 4) log(4e n ) ) (1.3) Furtermore, if M > 0 is sufficiently large, ten {z n : n N()} are all te resonances in [a, b] + i[ M log(1/), 0]. Because E S(E) = T (E) and T (E) is uniformly positive for E [a, b] it follows tat z n z m C n m for some C > 0 and eac n, m N(). Tus te z n are all distinct, wic gives 1 β α /π as an asymptotic formula for te number of resonances in [a, b]+[ im log(1/), 0] Context and previous work. Wen diffraction of singularities is te only trapping, an increasing body of work suggests tat resonances may occur at Im z C log(1/) for various values of C > 0, but no closer to te real axis. Tis is farter into te lower alf-plane tan te resonances occurring in cases of elliptic trapping (were tey rapidly approac te real axis as 0) or even for yperbolic trapped sets, were tere is an O() resonance-free region (see [13] for references on te subject of classical dynamical trapping and resonances). In te diffractive case, te regularization of te trapped wave wit eac successive diffraction is by contrast responsible for te faster rate of decay as measured by resonance widt. Most of te literature substantiating tis euristic is in te omogeneous rater tan te semiclassical setting; tere te analog of resonances at E 0 iν 0 log(1/) are resonances close to a curve Im λ C log Re λ (1.4) as λ. Tat diffraction of singularities can in fact create strings of resonances along suc log curves was demonstrated in te omogeneous setting by Zworski [14] (see also earlier work by Regge [12]). In te setting of diffraction by analytic corners in te plane, Burq [3] likewise sowed tat resonances lie asymptotically on families of curves (1.4) for various values of C > 0. In te setting of manifolds wit conic singularities, were similar diffractive propagation occurs, a number of recent teorems ave explored te same teme. Baskin Wunsc [1] sowed on te one and tat some nontrivial region of te form Im λ > ν 0 log Re λ, λ > R

5 DIFFRACTIVE TRAPPING 5 contains no resonances (subject to some genericity conditions on te relationsip among te conic singularities) tis is analogous to te gap teorem obtained ere. Galkowski [7] ten found te largest ν 0 wic could be obtained by te Vainberg parametrix metod employed in [1]: ν 0 = (n 1)/(2L 0 ), wit n te dimension and L 0 te maximal distance between cone points. Work of Hillairet Wunsc using a trace formula of Ford Wunsc [6] sowed tat tis constant was in general optimal by proving existence of resonances wit Im λ ν 0 log Re λ, wile [9] refined te description of te resonances on and near tis curve. Closely related work of Datcev Kang Kessler [4] studied te distribution of resonances on surfaces of revolution wit a cone point and a funnel (as well as oter types of infinite ends). Te autors find examples were te Laplacian admits resonances wit Im λ C log Re λ as Re λ, altoug te classical flow is non-trapping; in suc cases te metric is continuous wit a conormal singularity. Te results ere are, to te best of our knowledge, te first results on resonances generated by diffractive trapping for semiclassical Scrödinger operators. Te crucial new tecnical ingredient is te propagation of singularities results recently obtained by te autors in [8], wic include estimates on te size of te wave of diffractively reflected singularities. Tese singularities bounce back off of even a mild singularity of V in violation of te most naive application of te principle of geometric optics, wic would say tat wavefront set travels along classical trajectories. Te autors intend in future work to complement te results ere, wic sow a resonance-free region in all dimensions and existence in one-dimension, by sowing te existence of resonances just below te gap obtained in Teorem 1 in all dimensions, at least in settings were te dynamics is tractable. We tus conjecture, based on te evidence of Teorem 2, tat te ν 0 obtained in Teorem 1 is optimal in general (at least generically). Acknowledgements. Te autors are grateful to Nicolas Burq and Jeff Galkowski for elpful discussions. OG was partially supported by NSF grant DMS ; JW was partially supported by NSF grant DMS Resonances 2.1. Complex scaling. We define resonances of P by te metod of complex scaling following [5, Sections 4.5, 6.2.1]. Fix R 0 > 0 suc tat outside of B(0, R 0 ), V = 0, g ij = δ ij,

6 6 ORAN GANNOT AND JARED WUNSCH and let R 1 > R 0. We define te complex scaled operator P θ using a contour Γ θ = f θ (R n ), were f θ (x) = x + i x F (x), and F is a smoot convex function satisfying F = 0 near B(0, R 1 ), F = tan θ 2 x 2 for x 2R 1. Here we take any fixed θ (0, π/2). In particular, te semiclassical principal symbol of p θ is given by { (1 + i 2 F (x)) 2 ξ, ξ, x > R 0, p θ (x, ξ) = ξ 2 g + V (x) x < R 1. Referring to te proof of [5, Proposition 6.10], we record te following important observation: for an interval I, exp thre pθ (ρ) {Im p θ = 0} for all t I = exp thre pθ (ρ) = exp thp (ρ) for all t I Te time-dependent Scrödinger equation. Define M = R t X, Y M = R t Y. (2.1) As a preliminary step, we pass to te time-dependent semiclassical Scrödinger operator on M given by Q = D t + P. If τ denotes te momentum dual to t, ten te principal symbol of Q (wic does not depend on t) is q(x, τ, ξ) = τ + p(x, ξ). Note tat τ is conserved under te H q flow, and t evolves at unit speed. We will also work wit te complex-scaled operator Q θ = D t + P θ, writing q θ = τ + p θ for its principal symbol. Since Im q θ (τ, x, ξ) = Im p θ (x, ξ) for all τ, observe tat (2.1) also olds wit q replacing p. Given E R, consider te joint energy surface {q = 0, τ = E}. In general, tere may be non-trivial beavior of solutions to Qw = 0 in te caracteristic set of (τ, ξ) 2 q at fiber-infinity in T M, since Q is not elliptic in te non-semiclassical sense. However, we explicitly avoid suc issues by restricting to τ = E. Tus (t, x, E, ξ) {q = 0, τ = E} (x, ξ) {p = E}, wit similar observations for p θ and q θ. We now consider propagation of singularities for te operator Q near T Y M M. Define te lift of H E by Ĥ E = {(t, x, E, ξ ) : (x, ξ ) H E } T Y M,

7 DIFFRACTIVE TRAPPING 7 wit te analogous definition for ĜE. We also write ˆπ : TmM TmY M for te projection wenever m Y M. Adapting te results of [8], we ave te following teorem on propagation of singularities for Q, stated locally: Teorem 3. Let w = w() be -tempered in H 1,loc (M) suc tat Qw = O( ) L 2 loc. (1) If q ĤE, let µ ± {q = 0, τ = E} be te preimages of q under ˆπ wit opposite normal momenta. If µ + WF s (w) for some s R, ten tere exists ε > 0 suc tat exp thq (µ + ) WF s (w), or exp thq (µ ) WF s α (w), or bot, for all t (0, ε). (2) If q ĜE, let µ {q = 0, τ = E} be te unique preimage of q under ˆπ wit vanising normal momentum. If µ WF s (w) for some s R, ten tere exists ε > 0 suc tat exp thq (µ) WF s (w) for all t (0, ε). Te (minor) modifications to [8] needed to prove Teorem 3 are outlined in Appendix A. Note tat singularities propagate straigt troug ˆπ 1 (ĜE). Combined wit ordinary propagation of singularities for Q θ away from T Y M M, we obtain te following: Corollary 4. Let w = w() be -tempered in H 1,loc (M) suc tat Q θw = O( ) L 2 loc. Let s R and T 0. If µ {q θ = 0, τ = E} is suc tat exp thre qθ (µ) is disjoint from ˆπ 1 (ĤE) for eac t [0, T ], ten exp T HRe qθ (µ) / WF s (v) = µ / WF s (v). Proof. Suppose conversely tat µ WF s (v), and define T 0 = sup{t 0 : exp t H Re qθ (µ) WF s (v) for all t [0, t]}. Note tat µ = exp T0 H Re qθ (µ) WF s (v) since wavefront set is closed. We claim tat T 0 T, wic tus completes te proof. Indeed, if T 0 < T, ten by ypotesis µ / T Y M M or µ ˆπ 1 (ĜE). In te first case Q θ is smoot in a neigborood of µ, and since Im q θ 0 we can apply propagation of singularities forward along te H Re qθ flow to deduce a contradiction. In te second case q = q θ in a neigborood of µ, so we can apply te second part of Teorem 3 to deduce a contradiction. We make te following dynamical definitions, recalling te definition of R 1 from Section 2.1.

8 8 ORAN GANNOT AND JARED WUNSCH Definition 2.1. Let E R and ρ {p = E} \ π 1 (H E ). We say tat ρ IC E if tere exists T 0 0 suc tat exp thp (µ) is disjoint from π 1 (H E ) for eac t [0, T 0 ] and x(exp T0 H p (ρ)) 2R 1. If ρ {p = E} \ π 1 (H E ), ten we say tat ρ Γ ± E exp thp (ρ) π 1 (H E ). if tere exists t > 0 suc tat Tus, IC E consists of points tat are incoming from infinity witout interaction wit te yperbolic region, wile Γ ± E represent points itting te yperbolic region in backward-/forward-time (using notation in loose analogy wit tat of unstable/stable manifolds). Note tat { x 2R 1 } {Im p θ 0}. (2.2) Te next observation follows immediately from Corollary 4. Lemma 5. Let w = w() be -tempered in H 1,loc (M) suc tat Q θw = O( ) L 2 loc, and let E R. If (x, ξ) IC E and we set µ = (t, x, E, ξ) {q θ = 0, τ = E}, ten µ / WF s (w) for all s R. Proof. Let T 0 > 0 be as in Definition 2.1. It cannot be tat exp thre qθ (µ) {Im q θ = 0} for all t [0, T 0 ], since using Im q θ (τ, x, ξ) = Im p θ (x, ξ) and (2.1), we would conclude tat exp thre qθ (µ) = exp thq (µ) for all t [0, T 0 ], (2.3) and in particular exp T0 H q (µ) {Im q θ = 0}; tis contradicts te definition of T 0 according to (2.2). Tus we can define T = inf{t [0, T 0 ] : exp T HRe qθ (µ) {Im q θ 0}}. Note tat µ = exp T HRe qθ (µ) / T Y M M since Im q θ = 0 in a neigborood of T Y M Y. By semiclassical ellipticity in te smoot setting and te definition of T, tere exists δ > 0 suc tat µ = exp (T +δ)hre qθ (µ) / WF (w), and ence by forward propagation in te smoot setting we ave µ / WF (w) as well. On te oter and, by te definition of T we must ave tat (2.3) olds for all t [0, T ]. Since (x, ξ) IC E, it follows tat te backward H Re qθ flow from µ to µ is disjoint from ˆπ 1 (ĤE), and ence µ / WF s (w) for all s R by Corollary 4. Now we make te assumption tat E is a non-trapping energy level. Lemma 6. Let w = w() be -tempered in H 1,loc (M) suc tat Q θw = O( ) L 2 loc, and let E > 0 be non-trapping. If tere exists µ WF s (w) {q θ = 0, τ = E}

9 DIFFRACTIVE TRAPPING 9 for some s R, ten tere exists r R and µ = (t, x, E, ξ) WF r (w) {q θ = 0} suc tat (x, ξ) Γ E. Proof. First assume µ / ˆπ 1 (ĤE). Arguing precisely as in Lemma 5, te backwards flow exp thre qθ (µ) must encounter a yperbolic point µ + for some t > 0. Oterwise, if µ ˆπ 1 (ĤE) to begin wit, simply set µ + = µ. Now let µ project to te same yperbolic point as µ +, but wit opposite normal momentum. We know from te first part of Teorem 3 tat tere exists µ wit te requisite properties, obtained by flowing backwards along H q from eiter µ + (taking r = s) or µ (taking r = s α) for a sort time ε > 0, noting tat q = Re q θ = q θ near µ ±. Note in Lemma 6 we assume tat µ is in bot WF s (w) and te caracteristic set {q θ = 0}, since te inclusion of former in te latter is only guaranteed for a certain range of s owing to te singularity of V (see [8, Proposition 7.5]). Lemma 7. Let E > 0 be non-trapping and ρ Γ E. If diam E(Y ) > 0 and T > diam E (Y ), ten If diam E (Y ) = 0, ten ρ IC E. exp T Hp (ρ) IC E. Proof. Since ρ Γ E, tere exists t 0 > 0 suc tat ρ 0 = exp t0 H p (ρ) π 1 (H E ). Clearly if diam E (Y ) = 0, ten ρ IC E. Oterwise, if 0 < diam E (Y ) < T, set ρ = exp T Hp (ρ 0 ) = exp (t0 T )H p (ρ). Ten exp thp (ρ ) is disjoint from π 1 (H E ) for all t 0, and since E is non-trapping te proof is finised. We record a key lemma, wic is a refinement of Lemma 6. Lemma 8. Let w = w() be -tempered in H 1,loc (M) suc tat Q θw = O( ) L 2 loc, and let E > 0 be nontrapping. If (x, ξ) Γ E and µ = (0, x, E, ξ) WF s (w) {q θ = 0} for some s R, ten for all δ > 0 tere exists T 0 diam E (Y ) + δ and suc tat (x 0, ξ 0 ) Γ E. µ = ( T 0, x 0, E, ξ 0 ) WF s α (w) {q θ = 0}

10 10 ORAN GANNOT AND JARED WUNSCH Proof. Define t 0 sup { t 0: exp t H Re qθ (µ) WF s (w) for all t [0, t] }. We claim tat t 0 always exists and t 0 diam E (Y ). To see tis, first note tat for t 0, exp t H Re qθ (µ) WF s (w) for all t [0, t] = exp t H Re qθ (µ) = exp t H q (µ) for all t [0, t]. Indeed, if te conclusion fails, ten by (2.3) tere must be t [0, t] suc tat exp t H Re qθ (µ) Im{q θ 0}. But in te neigborood of suc a point q θ is smoot, and ence by semiclassical ellipticity exp t H Re qθ (µ) / WF s (w), wic is a contradiction. Furtermore, by Lemmas 5 and 7, if T > diam E (Y ), ten wic sows tat t < T. exp T Hq (µ) / WF s (w), Now set µ + = exp t0 H Re qθ (µ); we ave µ + WF s (w) since wavefront set is closed. Also by definition of t 0, tere is a sequence ε n > 0 wit ε n 0 suc tat exp εnh Re qθ (µ + ) / WF s (w), By ordinary propagation of singularities of Q θ away from T Y M Y we must ave µ + T Y M M, and by Teorem 3, ˆπ(µ + ) must be a yperbolic point. Let µ denote te point in ˆπ 1 (ˆπ(µ + )) wit opposite normal momentum. Ten Teorem 3 sows tat µ exp δhre qθ (µ ) WF s α (w) for δ > 0 arbitrarily small. Now T 0 = t(µ ) = t 0 + δ diam E (Y ) + δ as desired. We are now ready to prove Teorem Resonance widts. We begin by working quite generally, witout prejudice as to te operator P. Suppose tat u = u() is -tempered in H,loc 1 (X). Given a family z = z() C, set Im z() ν() = log(). Now form te functions w(t, x) = e izt/ u(x), and note tat a sufficient condition to guarantee tat w = w() is -tempered in H 1,loc (M) is ν() = O(1). Furtermore, if (P z)u = O( ) L 2 loc, ten Qw = O( ) L 2 loc. We need a simple lemma comparing te wavefront of w wit its restriction to a fixed time slice {t = t 0 }. For our purposes it will suffice to consider wavefront set away

11 DIFFRACTIVE TRAPPING 11 from fiber-infinity. In addition, to te assumption ν() = O(1), we also assume tat z() = E + o(1) for some E R. Note tat WF s (w) T M {τ = E} for all s, since (D t + z)w = 0 and D t + z is elliptic wen τ E. Of course tis leaves open te possibility tat w as wavefront set at fiber-infinity in a direction wit τ = 0. We need to be sligtly more precise: let χ 0 Cc (R) ave support near E and χ 1 Cc (R) ave support near t 0. Ten for eac K X compact, after integrating by parts. For eac fixed t 0 R, introduce te set (1 χ 0 (D t ))χ 1 (t)w L 2 (R K) = O( ) (2.4) W s (t 0 ) = {(x, ξ) T X : (t 0, x, E, ξ) WF s (w)}. We ave te following lifting lemma: Lemma 9. If z = z() C satisfies z() = E + o(1) and ν() ν 0 for some E R and ν 0 > 0, ten for eac s, t 0 R and δ > 0 W s δ (t 0 ) WF s ( ν()t 0 u) T X W s (t 0 ). Proof. First observe tat WF s (w(t 0, )) = WF s ( ν()t 0 u) for eac fixed t 0 R. Now suppose tat (x 0, ξ 0 ) T X satisfies (t 0, x 0, E, ξ 0 ) / WF s (w). For appropriate cutoffs φ 0, φ 1 C c (X), and χ 1 C c (R) supported near ξ 0, x 0, and t 0, respectively, (2.4) implies tat φ 0 (D x )φ 1 (x)χ 1 (t)w = O( s ) L 2 (M), wic immediately implies tat (x 0, ξ 0 ) / WF s (w(t 0, )). (x 0, ξ 0 ) / WF s (w(t 0, )) T X, so in te notation above, e izt 0/ φ 0 (D x )φ 1 (x)u = O( s ) L 2 (X). Conversely, suppose tat Now if χ 1 as above as sufficiently small support depending on δ > 0, ten we can arrange tat φ 0 (D x )φ 1 (x)χ 1 (t)w = O( s δ ) L 2 (M). Again using (2.4), tis implies tat (x 0, ξ 0 ) / W s δ (t 0 ) for eac δ > 0. Now we place ourselves in te setting of Teorem 1. If tere is a resonance in te set [E 0 δ, E 0 + δ] i[0, ν 0 log(1/)] for some ν 0 > 0, ten we can find a sequence k 0, complex numbers z( k ) satisfying Re z( k ) = E + o(1), Im z( k ) ν 0 k log(1/ k ),

12 12 ORAN GANNOT AND JARED WUNSCH for some E E 0 < δ, and eigenfunctions u( k ) L 2 (X) of te complex-scaled operator, (P θ z( k ))u( k ) = 0. Trougout, we suppress te index k. Normalize u by u L 2 = 1. In order to eventually apply Teorem 3, we verify tat te H 1 (in fact, te H2 ) norms of u are also uniformly bounded in. Tis follows from standard semiclassical analysis by observing tat P θ V is smoot, tat uniformly on X, and tat V L (X). σ (P θ V )(x, ξ) δ ξ 2 C 0 Observe tat WF s (u) for eac s > 0, since u L 2 = 1. Taking s = α, we conclude tat WF α (u) { ξ 2 (p θ E) = 0} Indeed, away from Y tis is ordinary semiclassical elliptic regularity (ence applies wit any s), wereas near T Y X, were p θ = p, we can apply [8, Proposition 7.5] (wose proof applies nearly verbatim even wen z is not real valued). Of course since p θ is elliptic at fiber-infinity, tis can be rewritten as WF α (u) {p θ = E}. Next, we let w = e izt/ u wit u = u() our family of eigenfunctions as above, and observe tat Lemma 9 applies to te family w = w(). Proof of Teorem 1. As noted above, tere exists (x 0, ξ 0 ) WF α (u) {p θ = E}. Let µ 0 = (0, x 0, E, ξ 0 ) {q θ = 0, τ = E} and note tat µ 0 WF α (w) by Lemma 9. Applying Lemmas 6 and 9 we conclude tere exists ρ WF r (u) Γ E for some r R. Tis is already a contradiction if diam E (Y ) = 0, since by Lemmas 5, 7, and 9 we ave ρ IC E and ence ρ / WF r (u). Oterwise, we derive a lower bound for ν 0 as follows. Since WF 0 (u) =, we may define s 0 = inf{s : WF s (u) Γ E }. Pick any s > s 0, so tere is (x, ξ) WF s (u) Γ E, and apply Lemma 8 to µ = (0, x, E, ξ), wic by Lemma 9 indeed satisfies µ WF s (w) {q θ = 0}. Tus for eac δ > 0 we can find T < diam E (Y ) + δ suc tat µ = ( T, x 1, E, ξ) WF s α (w) {q θ = 0}

13 DIFFRACTIVE TRAPPING 13 and (x 1, ξ 1 ) Γ E. Furtermore, by Lemma 9, (x 1, ξ 1 ) WF s α+ν 0T +δ (u), and ence (x 1, ξ 1 ) WF s α+ν 0T +δ (u) Γ E. By definition of s 0, we must ave s α + ν 0 T + δ > s 0. But s > s 0 and δ > 0 are bot arbitrary, wic implies tat ν 0 α/ diam Y (E). Tis completes te proof of te Teorem Existence of resonances in one dimension 3.1. WKB solutions. Trougout tis section we adopt te notation of Teorem 2 and te paragrap preceding it. Since supp V [0, L], a complex number z C is a resonance of P = (D x ) 2 + V precisely if it satisfies (P z)u = 0, D x u(0) + z 1/2 u(0) = 0, (3.1) D x u(l) z 1/2 u(l) = 0. Fix an interval [a, b] wit a > sup V. Given M > 0, set Ω M () = [a, b] + i[ M log(1/), 0]. Te fact tat I = [0, L] is a classically allowed region implies te following: Lemma 10. If u solves te equation (P z)u = 0 wit initial conditions u(0) = O(1), D x u(0) = O(1), ten u and x u are polynomially bounded on I = [0, L] uniformly in z Ω M () Proof. Tis follows from a semiclassical version of te energy estimates in [10, Section 23.2]; cf. [5, Lemma E.60]. At most polynomial growt in arises from te fact tat te imaginary part of z can be of size O( log(1/)). Next we consider approximate WKB solutions to (P z)u = 0. Following [2], for tis problem it is convenient to consider approximate solutions in exponential form. Define ψ 0 = (z V ) 1/2, and ten set ψ +,0 = ψ,0 = ψ 0. For i 1, define ψ ±,i recursively by ψ ±,k (x) = ± i 2ψ 0 (x) xψ ±,k 1 (x) 1 k 1 ψ ±,j (x)ψ ±,k j (x). (3.2) 2ψ 0 (x) j=1

14 14 ORAN GANNOT AND JARED WUNSCH Eac ψ ±,k is smoot on I and depends olomorpically on z [a, b] + i[ C 0, 0]. Let ψ ± be a function admitting an asymptotic expansion ψ ± j ψ ±,j j=0 and depending olomorpically on z. We ten set ϕ ± (x) = x 0 ψ ± (s) ds, u ± = e ±iϕ ±/. If z Ω M (), ten u ± are polynomially bounded on I, and ence te recursion relation (3.2) guarantees tat (P z)u ± = O( ) C (I), z Ω M () for any fixed M > 0. Te usefulness of tis exponential form comes from te following observation (cf. [2, Appendix 2]). Lemma 11. For eac j, ψ ±,j (x) = ( ) j ±i ψ (j) 0 (x) + F j (ψ 0 (x), ψ 2ψ 0 (x) 0(x),..., ψ (j 1) 0 (x)) for a smoot function F j (t 0, t 1,..., t j 1 ) suc tat F j (t 0, 0,..., 0) = 0 for all t 0 R. Proof. Tis follows by induction from te recursion relations (3.2). We ten obtain te following corollary: if V vanises to order k 1 at a point x 0 I, ten ψ ±,j (x 0 ) = 0 for j = 1,..., k 1, and ψ ±,k (x 0 ) = i ±k (2z 1/2 ) k 1 V (k) (x 0 ). (3.3) If x 0 is an endpoint of I = [0, L], ten te same formulas old in te sense of one-sided limits. Indeed, V (x) = c 0 (x x 0 ) k + O( x x 0 k+1 ), c 0 = V (k) 0 (x 0 )/k!, wic sows tat ψ 0 (x) = z 1/2 (c 0 /2)z 1/2 (x x 0 ) k + O( x x 0 k+1 ). Te formula (3.3) follows immediately from tis expression.

15 DIFFRACTIVE TRAPPING Outgoing condition. Observe tat u ± (0) = 1 and D x u ± (0) = ±ψ ± (0), so if we form te function v = u z1/2 ψ (0) z 1/2 + ψ + (0) u +, ten D x v(0) = z 1/2 v(0), and v is polynomially bounded on I for z Ω M (). Lemma 12. Let u solve te equation (P z)u = 0 wit initial data u(0) = v(0) and D x u(0) = z 1/2 u(0). Ten uniformly for z Ω M (). u = v + O( ) C (I) Proof. Tis follows from te identity u = W (u, u + )u W (u, u )u +, W (u, u + ) were W (f, g) = f ( x g) ( x f) g is te semiclassical Wronskian. Indeed, te fact tat u and u ± are polynomially bounded on I for z Ω M () implies tat all Wronskians appearing in te formula above are constant modulo O( ). It is ten a straigtfoward computation of te Wronskians at x = 0 using te specified initial conditions. Fix M > 0. In view of (3.1) and Lemma 12, resonances z Ω M () are caracterized as solutions to an equation of te form ( ) ( ) ψ+ (L) z 1/2 ψ (0) z 1/2 e iϕ +(L)/ e iϕ (L)/ = O( ). (3.4) ψ + (L) + z 1/2 ψ (0) + z 1/2 Bot sides of tis equation are olomorpic in z Ω M (). We replace tis wit a simpler expression by inserting te asymptotics of ψ ±, making sure to only incur errors tat are olomorpic in z Ω M (); we will continue to use ordinary Landau notation to denote tese errors. First, note tat according to (3.3), ψ + (L) z 1/2 = i l l (2z 1/2 ) l 1 V (l) (L )(1 + O()) ψ (0) z 1/2 = i k k (2z 1/2 ) k 1 V (k) (0 + )(1 + O()), ψ + (L) + z 1/2 = 2z 1/2 + O( l ), ψ (0) + z 1/2 = 2z 1/2 + O( k ). We also multiply (3.4) troug by e iϕ (L)/ = O( N ) (for some N N). Define te pase function ϕ(x) = x 0 (z V (s)) 1/2 ds, (3.5) and observe tat ϕ + (L) + ϕ (L) = 2ϕ(L) + O( 2 ) since ψ +,1 = ψ,1. In particular, e i(ϕ +(L)+ϕ (L))/ = e 2iϕ(L)/ (1 + O()).

16 16 ORAN GANNOT AND JARED WUNSCH Inserting tis information into (3.4), we obtain an equivalent equation i l k l+k (2z 1/2 ) l k 4 (V (k) (0 + ) V (l) (L ))e 2iϕ(L)/ 1 = O(). Tis equation already implies te necessity of Teorem 2 (i.e., te fact tat resonances in Ω M () may only be of te form (1.3)) by considering te modulus and argument of tis equation. To sow existence of tese resonances, consider te function F (w, E, ) = i l k l+k (2E 1/2 ) l k 4 (V (k) (0 + ) V (l) (L ))e 2i(S(E)+wT (E))/ 1. Here E [a, b] is treated as a parameter, and F (w, E, ) is olomorpic in w C. We ave used te notation S(E) and T (E) from Teorem 2. Note tat if z = E + w Ω M (), ten ϕ(l) = S(E) + wt (E) + O( w 2 ). If w and w 2 / are bot small, ten z is resonance if and only if w satisfies an equation of te form F (w, E, ) = O( + w + w 2 /) (3.6) wit bot sides olomorpic in w. Now if n N() and E n are as in Teorem 2, ten F (w n, E n, ) = 0 for te coice w n = i (l + k) log(1/) 2T (E n ) + i ( log V (k) (0 + )V (l) (L ) (1/2)(l + k + 4) log(4e n ) ). 2T (E n ) Furtermore, w F (w n, E n, ) = (2i/)T (E n ), and given ε > 0 tere exists C > 0 suc tat n N() and w w n ε = 2 wf (w, E n, ) C 2. Tus by Taylor s teorem, for any A > 0 we can find 0, C 0 > 0 suc tat F (w, E n, ) A log(1/) 2 for all (0, 0 ) and n N() wenever w w n = C 0 2 log(1/) 2. On te oter and, tere exists B > 0 independent of C 0 suc tat rigt and side of (3.6) is bounded by B log(1/) 2 wenever w w n = C 0 2 log(1/) 2 and (0, 0 ) uniformly in n N(), srinking 0 > 0 depending on C 0 if necessary. Tus we first fix A > B, coose 0, C 0 > 0 as above, and ten apply Roucé s teorem. It follows tat for eac (0, 0 ) and n N() tere exists a unique resonance z n Ω M () satisfying z n = E n + w n + O( 2 log(1/) 2 ), tus completing te proof of Teorem 2.

17 DIFFRACTIVE TRAPPING 17 Appendix A. Proof of Teorem 3 In [8], propagation of singularities for operators of te form P = 2 + V wit V real-valued was discussed. Altoug Q is not elliptic at fiber-infinity (unlike P ), since we are only considering propagation of singularities in compact subsets of T M viewed as te interior of T M, tere is little difference in te proofs. Tere are two ingredients: (1) Propagation of singularities along generalized broken bicaracteristics (GBBs). (2) Diffractive improvements at yperbolic and glancing points. Te main difference is in te propagation arguments along GBBs. Te preliminary material in [8, Section 4] goes troug uncanged provided p is replaced wit q = τ + ξ 2 g + V. In te neigborood of a point m Y M we can use coordinates (t, x 1, x ), were (x 1, x ) are normal coordinates on Y wit respect to g. We ten replace P wit Q = Q (D x1 ) (D x1 ) in tese local coordinates. Te elliptic, yperbolic, and glancing sets are defined in te obvious way; we denote tese Ê, Ĥ, Ĝ. In te notation of Section 2.2, Ĥ E = Ĥ {τ = E}, Ĝ E = Ĝ {τ = E}. Note tat Ĥ Ĝ may intersect fiber-infinity of T Y M in directions wit τ 0, but of course ĤE ĜE are compact subsets of T Y M. Te study of te elliptic set Ê in [8, Section 5.1] needs only minor modifications provided we stay away from fiber infinity; tis amounts to working wit compactly microlocalized b-pseudodifferential operators only. Te most important point is to replace te Diriclet form associated to P in [8, Lemma 5.3] wit te expression 2 d X Aw 2 g + V Aw 2 + D t Aw Aw d g, M were d X is te differential on X lifted to M = R X by te product structure, and g is te product metric dt 2 + g. Te proof of [8, Lemma 5.3] applies witout cange. Modifying te Diriclet form affects [8, Eq. 5.3]; in te notation tere, te relevant replacement is an estimate of te form Aw H 1 C GQw H 1 + C Gw H 1 + C 0 ( Aw L 2 + D t Aw L 2) + O( ) w H 1,

18 18 ORAN GANNOT AND JARED WUNSCH wit C 0 > 0 independent of A, namely one must add D t Aw L 2 to te rigt and side. On te oter and, if A as compact b-microsupport, ten we can estimate D t Aw L 2 C 1 Aw L 2 + O( ) w L 2, were C 1 > 0 depends only on te size of te b-microsupport of A; tus te important estimate [8, Eq. 5.3] is still valid. Te proofs of [8, Lemmas 5.5, 5.6] go troug, provided V is replaced wit V + τ. Wit tese modifications, b-elliptic regularity ([8, Proposition 5.2]) continues to old, at least away from fiber-infinity. Te analysis at Ĥ is essentially uncanged, provided one only consider propagation of singularities away from fiber-infinity; tis is certainly te case near ĤE. One needs only to account for te additional localization in (t, τ), and in te tird step of te proof of [8, Lemma 5.10] we make te replacement 2 k = 2 k + D t. In tat case, [8, Proposition 5.8] still olds. (A.1) Finally, we come to te analogues of [8, Teorems 2 and 3], wic are proved using ordinary pseudodifferential operators. Again, since in Teorem 3 only points µ in te finite parts of te fiber of T M (rater tan fiber-infinity in te compactification) are considered, te proofs in [8, Section 7] are still valid provided we make te replacement (A.1). References [1] Dean Baskin and Jared Wunsc. Resolvent estimates and local decay of waves on conic manifolds. J. Differential Geom., 95(2): , [2] Micael V Berry. Semiclassically weak reflections above analytic and non-analytic potential barriers. Journal of Pysics A: Matematical and General, 15(12):3693, [3] Nicolas Burq. Pôles de diffusion engendrés par un coin. Astérisque, (242):ii+122, [4] Kiril R Datcev, Daniel D Kang, and Andre P Kessler. Nontrapping surfaces of revolution wit long living resonances. arxiv: , [5] Semyon Dyatlov and Maciej Zworski. Matematical teory of scattering resonances. mat.mit. edu/~dyatlov/res/. [6] G Austin Ford and Jared Wunsc. Te diffractive wave trace on manifolds wit conic singularities. Advances in Matematics, 304: , [7] Jeffrey Galkowski. A quantitative Vainberg metod for black box scattering. Communications in Matematical Pysics, 349(2): , [8] Oran Gannot and Jared Wunsc. Semiclassical diffraction by conormal potential singularities. arxiv: , [9] Luc Hillairet and Jared Wunsc. On resonances generated by conic diffraction. arxiv: , [10] Lars Hörmander. Te Analysis of Linear Partial Differential Operators, volume 3. Springer- Verlag, Berlin, Heidelberg, New York, Tokyo, [11] André Martinez. Resonance free domains for non globally analytic potentials. Annales Henri Poincaré, 3(4): , 2002.

19 DIFFRACTIVE TRAPPING 19 [12] Tullio Regge. Analytic properties of te scattering matrix. Nuovo Cimento (10), 8: , [13] Jared Wunsc. Resolvent estimates wit mild trapping, Journées Équations aux Dérivées Partielles. [14] Maciej Zworski. Distribution of poles for scattering on te real line. J. Funct. Anal., 73: , Department of Matematics, Lunt Hall, Nortwestern University, Evanston, IL 60208, USA address: and