RESONANCES AND LOWER RESOLVENT BOUNDS

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1 RESONANCES AND LOWER RESOLVENT BOUNDS KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI Abstract. We show how the presence of resonances close to the real axis implies exponential lower bounds on the norm of the cut-off resolvent on the real axis. In this note we establish exponential lower bounds on the scattering resolvent on the real line. We show that these lower bounds can be understood in terms of resonances close to the real axis. To fix the concepts, consider a semiclassical Schrödinger operator on R n : P (h) = h 2 + V (x), x R n, V C c (R n ; R), (.) supp V B(0, R 0 ); V (0) = V 0 > 0, V (0) = 0, V (0) > 0; x V (x) 0 on {V V 0 }, x V (x) < 0 on {V = V 0 } \ {0}. Take R > R 0 and define the cutoffs (.2) χ = l B(0,R0 ), ψ = l B(0,R+)\B(0,R ). (.3) Theorem 3 in 4 shows that for any R > R 0 there exists a constant c > 0 independent of h and E 0 (h) = V 0 + O(h) such that χ(p (h) E 0 (h) ± i0) χ L 2 L2 exp(c/h), (.4) ψ(p (h) E 0 (h) ± i0) χ L 2 L2 exp(c/h). (.5) A very general exponential upper bound corresponding to (.4) was first proved by Burq [Bu], with generalizations by Vodev [Vo], and more recently by Datchev [Da]. The lower bound is immediate from much easier arguments involving quasimodes. The non-trapping upper bound (for R large enough) ψ(p (h) E 0 (h) ± i0) ψ L 2 L 2 C 0 h, (.6) was again given by Burq [Bu] (with a log /h loss) and Vodev [Vo] see [Da] for a neat new proof. It is (.5) which seems to be the novel aspect. It shows that having a one sided cutoff to the exterior of the interaction region cannot prevent exponential blow up of the resolvent. The method also applies to the case of Riemannan manifolds, (M, g), considered recently by Rodnianski Tao [RoTa] see Fig.. In that case the support of V is

2 2 KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI Figure. Examples of manifolds for which the estimate (.7) holds: on the left a surface of revolution with two Euclidean ends to which Theorem 4 applies directly; on the right a surface with one end to which a modification of the same method applies (see [Ga]). The same examples work in any dimension. replaced in (.3) by the set where the metric is different from the Euclidean metric, and we obtain a sequence of λ k such that See Theorem 4 in 4 for details. ψ( g λ k ± i0) χ L 2 (M) L 2 (M) e c λ k. (.7) The reason behind these estimates is the presence of resonances close to real axis. The resolvent R h (z) = (P (h) z) : L 2 (R n ) H 2 (R n ), z [0, + ), has a meromoprhic continuation to the Riemann surface of z for n odd, and to the Riemann surface of log z for n even, as a family of operators L 2 comp(r n ) H 2 loc (Rn ), see for example [DyZw] and references given there. Resonances, defined as the poles of this continuation, replace discrete spectral data for problems on non-compact domains. For the situations considered here, in particular for the case (.), Rellich s theorem (see [DyZw]) shows that there are no resonances on the positive real axis, that is, the operators (P (h) E ± i0) are well-defined for E > 0. Theorem 2 in 3 gives general lower bounds based on existence of resonances with certain properties. It is then applied in Theorems 3 and 4 in 4 to obtain examples, in particular of Riemannian manifolds with Euclidean ends. The simple proofs here are based on previous work on scattering resonances, in particular those by Bony Michel [BoMi], Gérard Martinez [GéMa], Helffer Sjöstrand

3 RESONANCES AND LOWER RESOLVENT BOUNDS Figure 2. The potential V used in Theorem. [HeSj2], Tang-Zworski [TaZw] and Nakamura Stefanov Zworski [NaStZw]. To make the basic idea accessible, we present in 2 an elementary and self-contained one dimensional example which captures the basic reason for (.5); the argument of 2 does not directly use resonances though it could be used to show their existence. Acknowledgements. We would like to thank André Martinez and András Vasy for helpful discussions of resolvent estimates. We are also grateful for the support by a National Science Foundation postdoctoral fellowship (KD), a Clay Research Fellowship (SD) and by a National Science Foundation grant DMS-2047 (MZ). 2. An explicit example The following one dimensional example shows the reasons for (.5) in an explicit setting: Theorem. Let V Cc (R) be a nonnegative potential satisfying the following conditions (see Figure 2): V (x) = V ( x), V (x) = x 2 + for x [, ]; V (x) = 4 x for x [2, 3.5]; V (x) < for x > 3; supp V [ 5, 5]. (2.) Put R 0 = 4, fix R > 5, and define χ, ψ by (.3). Then there exists c > 0 and families E 0 (h) = + O(h), u(h), f(h) Cc (R) such that (P (h) E 0 (h))u = f, f = ψf; χu L 2 (R) =, f L 2 (R) e c/h. Note that (2.2) implies (henceforth suppressing the dependence on h) (2.2) χr h (E 0 ± i0)ψ L 2 L 2 ec/h, (2.3) ψr h (E 0 ± i0)χ L 2 L 2 ec/h. (2.4) Indeed, since u C c and f = ψf, we have χu = χr h (E 0 ± i0)ψf; this shows (2.3). The bound (2.4) follows since R h (E 0 ± i0) = R h (E 0 i0).

4 4 KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI The key component of the proof of Theorem is the existence of quasimodes for the operator P E 0, namely functions that satisfy (P E 0 )v = O(e c/h ): Lemma 2.. There exist h-dependent families E 0 = E 0 (h) R, and v = v(r; h) C ([ R, R]), such that E 0 = + O(h) and for some C, c > 0, (P E 0 )v = 0, v L 2 ( 3.5,3.5) C, v H h ({R < x <R}) Ce c/h. Here Hh denotes the semiclassical Sobolev space where in the standard definition D x is replaced by hd x see for instance [Zw, 7., 8.3]. To derive Theorem from Lemma 2., we take χ 0 C c ( R, R) such that χ 0 = on [ (R ), R ] and put u := αχ 0 v, f = (P E 0 )u = α[p, χ 0 ]v, here the constant α = α(h) is chosen so that χu L 2 = and we have α C. We furthermore see that supp f {R < x < R} and f L 2 e c/h (the constant C can be absorbed into the exponential by replacing c by a smaller constant and taking h small enough). The rest of this section contains the proof of Lemma 2.. We take R(h) R, R(h) = R+O(h), to be chosen at the end of this section in (2.5), and let v be an eigenfunction of P on [ R(h), R(h)] with Dirichlet boundary conditions with eigenvalue E 0 close to the ground state +h of the quantum harmonic oscillator h 2 2 x+x 2 +. The existence of such eigenvalue is given by the following Lemma 2.2. For h small enough and given R(h) [R, R + ], there exists E 0 R and v C ([ R(h), R(h)]) such that Proof. Define (P E 0 )v = 0, v( R(h)) = v( R(h)) = 0; v L 2 ( R(h), R(h)) =, v H h ( R(h), R(h)) C, E 0 = + h + O(e 0h ). v (x) := h /4 e x2 2h, x [, ]. Note that, since P = h 2 2 x + x 2 + on [, ], we have Now, take χ C c (P ( + h))v = 0, x [, ]; v L 2 ( /2,/2) C, v H h ({/2< x <}) Ce 0h. (, ) such that χ = on [ /2, /2]. Then χv L 2 C, (P ( + h)) χv L 2 Ce 0h, and χv satisfies the Dirichlet boundary conditions at ± R(h) (since it vanishes there). Now, P ( + h) is self-adjoint on L 2 ([ R(h), R(h)]) when Dirichlet boundary conditions are imposed. Since the norm of its inverse is at least C e 0h, we see that this

5 RESONANCES AND LOWER RESOLVENT BOUNDS 5 operator has an eigenvalue which is O(e 0h ); we denote the corresponding eigenvalue of P by E 0 and the corresponding L 2 normalized eigenfunction by v. Finally, to establish a bound on the Hh norm of v it suffices to multiply the equation (P E 0)v = 0 by v and integrate by parts. Lemma 2. follows once we establish the following exponential bound on v: v H h ({3.5 x R(h)}) Ce c/h. (2.5) We will show (2.5) for positive x; the case of negative x is handled similarly (since V is even, v can be taken to be even as well). The main idea is the following: if v is not exponentially large in /h near, say, x = 2 relative to its size on [3.5, R(h)], then one expects v to give an approximate Dirichlet eigenfunction to the operator P on [2, R(h)] with eigenvalue E 0. However, then E 0 has to satisfy a quantization condition determined by the behavior of V on [2, R(h)]; since E 0 = + h + o(h), one can choose R(h) to ensure that the quantization condition is not satisfied and thus obtain a contradiction. For f, f 2 C (R) we define the (semiclassical) Wronskian by and note that W (f, f 2 ) = f h x f 2 f 2 h x f, h x W (f, f 2 ) = f 2 (P E 0 )f f (P E 0 )f 2. The interval [2, R] can be split into three regions where the behavior of v is different, based on the sign of V (x) : the elliptic or classically forbidden region [2, 3), where v will grow exponentially in h as x decreases, the neighborhood of the turning point x = 3, and the hyperbolic region (3, R), where the equation (P (h) E 0 )v = 0 has two solutions which are bounded as h 0. We start with the hyperbolic region, considering the phase function Φ(x) := x 4 E 0 E0 V (y) dy. Note that Φ is well-defined on x [4 E 0, R + ], since E 0 V (y) = y (4 E 0 ) for y [4 E 0, 3.5]; in fact, we have Φ(x) = 2 3 (x (4 E 0)) 3/2 for x [4 E 0, 3.5]. (2.6) Define now the following WKB solutions: v ± (x) := (E 0 V (x)) /4 e ± iφ(x) h, x [3.5, R + ], then we have uniformly in x [3.5, R + ], (P E 0 )v ± (x) = O(h 2 ), W (v +, v )(x) = 2i + O(h). (2.7)

6 6 KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI Denote then Since (P E 0 )v = 0, we have v(x) = (v (x), v 2 (x)) := (W (v, v + )(x), W (v, v )(x)), v(x) = v 2(x) v + (x) v (x) v (x), W (v +, v )(x) (2.8) h x v(x) = v 2(x) h x v + (x) v (x) h x v (x). W (v +, v )(x) (2.9) h x W (v, v ± ) = v (P E 0 )v ±. From (2.7) and (2.8) we see that for x [3.5, R + ], Therefore, This and (2.8), (2.9) show that x v(x) Ch v(x). v(x) = v(3.5)( + O(h)), x [3.5, R(h)]. (2.0) v H h (3.5, R(h)) C v(3.5). (2.) The final component of the proof is the following solution in the region [2, 3.5] which describes the transformation from the hyperbolic to the elliptic region via the turning point, and is exponentially decaying in the elliptic region. Since V (x) = 4 x in this region, the solution is given by an Airy function, and its properties are as follows: Lemma 2.3. There exists a solution w(x) to the equation (P E 0 )w = 0 for x [2, 3.5] such that w H h (2,2.5) Ce c/h for some constants C, c > 0 and ( ) ( ) ( ) w(x) = e iπ v+ (x) 4 e iπ v (x) 4 + O(h), x [3.25, 3.5]. (2.2) h x w(x) h x v + (x) h x v (x) Proof. The solution w is given by w(x) = 2i πh /6 Ai(h 2/3 (4 E 0 x)), and its properties follow from the following asymptotic formulæ for the Airy function Ai as y + : ( Ai(y) = y /4 2 π exp 2 )( 3 y3/2 + O(y 3/2 )), ( 2 3 y3/2 + π ) 4 Ai( y) = y /4 π ( sin ) + O(y 3/2 ), and similar formulæ for its derivatives, see for example [HöI II, (7.6.20), (7.6.2)].

7 RESONANCES AND LOWER RESOLVENT BOUNDS 7 We are now ready to finish the proof of (2.5). Since (P E 0 )v = (P E 0 )w = 0 on [2, 3.5], the Wronskian W (v, w) is constant on this interval. Using the estimates v H h (2,2.5) C, w H h (2,2.5) Ce c/h, we see that W (v, w) Ce c/h. Now, computing the same Wronskian at x = 3.5 and using (2.2), we get W (v, w) = e iπ 4 v (3.5) e iπ 4 v2 (3.5) + O(h) v(3.5), It remains to prove that, for a certain choice of R(h) independent of E 0, we have e iπ 4 v (3.5) e iπ 4 v2 (3.5) C v(3.5). (2.3) Indeed, in this case v(3.5) Ce c/h, which together with (2.) gives (2.5). Using (2.0), we rewrite (2.3) as follows: e iπ 4 v ( R(h)) e iπ 4 v2 ( R(h)) C v( R(h)). (2.4) Since v satisfies the Dirichlet boundary condition at R(h), we have W (v, v ± )( R(h)) = E /4 0 e ± iφ( R(h)) h h x v( R(h)), so that v ( R(h)), v 2 ( R(h)) C v( R(h)) and e iπ 4 v 2 ( R(h)) e iπ 4 v ( R(h)) = exp ( i h ( 2Φ( R(h)) + πh/2 ) ). To prove (2.4), we choose R(h) = R + O(h), R(h) R, so that min j Z Φ( R(h)) + (j + /4)πh πh 4 ; (2.5) this can be done independently of E 0, since E 0 = + h + O(e 0h ) and thus Φ( R(h)) = 5 3 h + h V (y) dy + + h( R(h) 5) + O(e 0h ). 3. A general argument Suppose that P (h) is an operator satisfying the general assumptions of [NaStZw], that is a black box self-adjoint operator, close to the Laplacian and having analytic coefficients near infinity and with a barrier at energy V 0. (A barrier separates the interaction region from infinity see (3.3) and Fig. 3). We assume that R n \ B(0, R 0 ) is contained in the outside of the black box and the trapped set at energy V 0. We also assume the Hilbert space on which the operator acts, H = H R0 L 2 (R n \ B(0, R 0 ), is equipped with an involution u ū equal to complex conjugation on L 2 (R n \ B(0, R 0 )) and satisfying zu = zū, z C. The abstract reality assumption on P (h) reads P (h)u = P (h)u.

8 8 KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI (a) (b) (c) (d) Figure 3. Examples of one-dimensional potentials satisfying: (a) condition (.2) and hence (3.3) and (4.); (b) conditions (3.3) and (4.), but not (.2); (c) condition (3.3), but not (4.); (d) neither (3.3) nor (4.). The dashed line corresponds to V 0. In particular, examples (a) and (b) satisfy the assumptions of Theorems 2 and 3. An example to keep in mind is given by the operator P (h) = h 2 g + V : H 2 (R n ) L 2 (R n ), where the measure on L 2 is obtained from the Riemannian metric and H := L 2 (R n ). The potential V (x) and the metric coefficients g ij (x) are smooth, extend analytically to Ω := {z C n : z R, Im z δ z }, and g ij (z) δ ij 0, V (z) 0, z, z Ω. (3.) The trapped set, K E, at energy E > 0 is then defined by K E := {(x, ξ) R n R n : p(x, ξ) = E, e thp (x, ξ), t ± }, (3.2) where H p := n j= ξ j p xj xj p ξj and p(x, ξ) := n i,j= gij (x)ξ j ξ i + V (x). The assumption on the interaction region in this case means that π(k V0 ) B(0, R 0 ). The barrier assumption means that p (V 0 ) = K V0 Σ V0, Σ V0 K V0 =, Σ V0 is closed. (3.3) The more general black box setting allows obstacle problems and other geometric situations. However, the barrier assumption cannot be satisfied for connected manifolds without having a nontrivial potential V. We denote by Res(P (h)) the set of resonances of P (h) (in an h-independent neighborhood of {Re z > 0, Im z > 0}).

9 RESONANCES AND LOWER RESOLVENT BOUNDS 9 Theorem 2. Let P (h) satisfy the general assumptions above. Suppose that z 0 = z 0 (h) Res(P (h)), Re z 0 = V 0 + O(h), z 0 is simple, and Im z 0 = O(h ), d(z 0 (h), Res(P (h)) \ {z 0 (h)}) > h N, (3.4) for some N. Suppose that χ and ψ are given by (.3) with R > R 0. Then there exist C 0 > 0 and h 0 such that for 0 < h < h 0, and χ(p (h) Re z 0 i0) χ H H ψ(p (h) Re z 0 i0) χ H H C 0 Im z 0, (3.5) C 0 Im z0 h. (3.6) Remark. As stated in the introduction, it is (3.6) that seems to be the novel aspect. The presence of the square root is (morally) consistent with the results of [BuZw, Lemma A.2] and of [DaVa, Theorem 2]. Proof. We only present the proof of (3.6). Using the involution u ū we define (u v)f := u f, v, where we use the inner product on the black box Hilbert space. Since z 0 is simple, we have ψ(p (h) z) χ = ψu χu z z 0 + ψr z0 (z, h)χ, where u is the corresponding normalized resonant state and R z0 (z, h) is holomorphic in [Re z 0 h N, Re z 0 + h N ] + i( h N, ). From [NaStZw, (5.)] and [BoMi, (.2)] (or [NoZw, (8.8)]) we see that χu H = + O(h ), u = χu + O(h ) L 2 loc. (3.7) Using the maximum principle as in [TaZw, Lemma 2] and the estimates on the resolvent in [TaZw, Lemma ] we see that χr z0 (Re z 0, h)ψ H H = O(h M ), (3.8) for some M. Hence to obtain (3.6) we need to estimate from below. To estimate ψu H from below we write Im z 0 ψu χu H H = ψu H χu H, (3.9) Im z 0 0 = Im (P (h) z 0 )u, l B(0,R+t) u H = Im P (h)u, l B(0,R+t) u H Im z 0 l B(0,R+t) u 2 H.

10 0 KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI Since P (h) is self-adjoint on H and acts as a symmetric second order operator on Cc (R n \ B(0, R 0 )), we obtain Im z 0 l B(0,R+t) u 2 H = Im ūn(x, hd)uds(x), B(0,R+t) where N(x, hd x ) is a first order semiclassical differential operator. For example, if P (h) = h 2 + V, then N(x, hd x ) = i(x/ x ) hd x. Since from (3.7), l B(0,R+t) u 2 H = + O t(h ), we see that 2 Im z h u N(x, hd x )u ds(x)dt B(0,R+t) 3 = h u N(x, hd x )u dx B(0,R+ 2 3 )\B(0,R+ 3 ) C h u 2 dx C h ψu 2 H, B(0,R+)\B(0,R) where we used the equation (P z 0 )u = 0 and elliptic estimates to control the first order term term N(x, hd x )u. This shows that ψu H Im z 0 /Ch, which combined with (3.8),(3.9) and (3.7) completes the proof of (3.6). 4. A metric example We start by using Theorem 2 to obtain a generalization of Theorem to higher dimensions and to more general potentials: Theorem 3. Consider a Schrödinger operator P (h) = h 2 + V on L 2 (R n ) where V satisfies (3.) and (3.3). Suppose also that K V0 = {(x 0, 0)}, V (x 0 ) = 0, V (x 0 ) > 0. (4.) Then there exists z 0 satisfying the assumptions of Theorem 2 with d(z 0, Res(P (h)) \ {z 0 }) > h/c, Im z 0 < e c 0/h. (4.2) In particular in the notation of (3.6), for 0 < h < h 0 and some c > 0. ψ(p (h) Re z 0 ± i0) χ L 2 L 2 exp c h, (4.3) Proof. The existence of z 0 follows from Corollary in [NaStZw, 5]. The reference operator P (h) there can be chosen as P (h) = h 2 + V (x), where V (x) = V (x) in a small neighbourhood of x 0 where x 0 is the only critical point and V (x) > V (x 0 ) + ε, ε > 0, outside of that neighbourhood. Since the eigenvalue of P (h) corresponding to

11 RESONANCES AND LOWER RESOLVENT BOUNDS the minimum V 0 = V (x 0 ) is separated from other eigenvalues by h/c (see for instance [HeSj] and references given there) the same corollary shows the separation from other resonances. Remarks.. The condition (.2) implies that (3.3) and (4.) hold (since H p (x ξ) > 0 on {p = V 0 } except at x = ξ = 0), but the converse is not true see Figure When V is analytic and satisfies certain well-in-the-island hypotheses, Theorem 3 follows from the work of Helffer Sjöstrand [HeSj2] and under these stronger assumptions Theorem 2 can then be proved in the same way using the earlier results of Gérard Martinez [GéMa] in place of the results of [NaStZw]. 3. For P (h) = h 2 + V, and for E s satisfying (4.) (with V 0 = E), a result of Nakamura [Na, Proposition 4.] and [NaStZw, Corollary, 5] show that χ(p (h) E i0) χ L 2 L 2 Ch q, E z j (h) h q, where q and z j (h) are the resonances of P (h). Since the density of Re z j (h) satisfies a Weyl law, this means that the bound is O(h q ), outside of a set of measure O(h q n ), q > n. The example in Theorem 3 can be used directly to obtain examples of resolvent growth for asymptotically conic metrics of the type studied by Rodnianski Tao [RoTa]. Theorem 4. Let (M, g) be the following Riemannian manifold: M = R x S n θ, g = dx 2 + V (x) dθ 2, n >, where dθ 2 is the round metric on the sphere of radius and V (x) C (R; (0, )) is a function satisfying the assumptions of Theorem 3 and Put V (x) = x 2, x R 0. χ(x) = l x <R0, ψ(x) = l R < x <R+, R > R 0. Then there exists a sequence λ k such that for some constant c > 0. ψ( g λ k ± i0) χ L 2 (M) L 2 (M) exp(c λ k ), (4.4) Proof. In the (x, θ) coordinates, the Laplacian g has the form g = x 2 (n )V (x) x + V (x) S. 2V (x) Here S is the Laplacian on S n. For k 0, let Y k (θ) be (any) spherical harmonic of order k, i.e. a smooth function on S n such that ( S k(k + n 2))Y k = 0, Y k L 2 (S n ) =,

12 2 KIRIL DATCHEV, SEMYON DYATLOV, AND MACIEJ ZWORSKI see for example [HöI II, 7.2] for the spectrum of S. Then for u(x) C (R) and λ R, we have ( g (u(x)y k (θ)) = x 2 + (n )V (x) ) x + k(k + n 2)V (x) u(x)y k (θ). 2V (x) Put h k := ( k(k + n 2) ) /2 so that where h 2 k( g λ)(u(x)y k (θ)) = (P (h k ) h 2 kλ)u(x)y k (θ), P (h) := h 2 x 2 + (n )V (x) h 2 x + V (x). 2V (x) Let R(λ) := ( g λ) for λ [0, + ). It follows that R(λ) = k N h 2 k(p (h k ) h 2 kλ) Π k : L 2 (M) L 2 (M), L 2 (M) L 2 (R, V (x) n 2 dx) L 2 (S n ), (4.5) where Π k : L 2 (S n ) L 2 (S n )) is the orthogonal projection onto the space of spherical harmonics of order k. The operator R(λ) : Cc (M) C (M) continues meromorphically to Im λ 0, and (P (h k ) h 2 k λ) : Cc (R) C (R) continues meromorphically for each k. Hence (4.5) is valid for Im λ 0, with the operator acting on Cc (M) Cc (R) Cc (S n ). Hence, ψ( g λ ± i0) χ L 2 (M) L 2 (M) = ψr(λ ± i0)χ L 2 (M) L 2 (M) = sup h 2 k ψ(p (h k ) h 2 kλ ± i0) χ L 2 x L 2, x k N where L 2 x := L 2 (R, V (x) n 2 dx). We now apply Theorem 3 to P (h k ) and put λ k = Re z 0 (h k )/h 2 k. The estimate (4.4) follows from (4.3). Theorem 3 applies to the operator P (h) despite the presence of a first order term, as this term is of order O(h) in the semiclassical calculus and thus does not affect the classical Hamiltonian flow H p, and the results of [NaStZw] and Theorem 2 apply to a wide class of semiclassical differential operators including P (h). References [BoMi] Jean-François Bony and Laurent Michel, Microlocalization of resonant states and estimates of the residue of the scattering amplitude, Comm. Math. Phys. 246(2004), [Bu] Nicolas Burq, Lower bounds for shape resonances widths of long range Schrödinger operators. Amer. J. Math., 24(2002), [BuZw] Nicolas Burq and Maciej Zworski, Geometric control in the presence of a black box. J. Amer. Math. Soc. 7(2004), [Da] Kiril Datchev, Quantitative limiting absorption principle in the semiclassical limit. arxiv:309. 2, to appear in Geom. Funct. Anal.

13 RESONANCES AND LOWER RESOLVENT BOUNDS 3 [DaVa] Kiril Datchev and András Vasy, Semiclassical resolvent estimates at trapped sets, Ann. de l Inst. Fourier, 62(202) [DyZw] Semyon Dyatlov and Maciej Zworski, Mathematical theory of scattering resonances, book in preparation [Ga] Oran Gannot, Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis, arxiv:22.907, to appear in Comm. Math. Phys. [GéMa] Christian Gérard and André Martinez, Semiclassical asymptotics for the spectral function of long-range Schrödinger operators. J. Funct. Anal. 84(989), no., [HeSj] Bernard Helffer and Johannes Sjöstrand, Multiple wells in the semi-classical limit I. Comm. in P.D.E, 9(984), [HeSj2] Bernard Helffer and Johannes Sjöstrand, Resonances en limite semiclassique, Bull. Soc. Math. France, 4, no.24 25, 986. [HöI II] Lars Hörmander, The Analysis of Linear Partial Differential Operators, Volumes I and II, Springer, 983. [HöIII IV] Lars Hörmander, The Analysis of Linear Partial Differential Operators, Volumes III and IV, Springer, 985. [Na] Shu Nakamura, Scattering theory for the shape resonance model. I: Non-resonant energies, Ann. Inst. H. Poincaré Phys. Théor. 50(989), 5 3. [NaStZw] Shu Nakamura, Plamen Stefanov and Maciej Zworski, Resonance expansions of propagators in the presence of potential barriers, J. Funct. Anal. 205(2003), [NoZw] Stéphane Nonnenmacher and Maciej Zworski, Quantum decay rates in chaotic scattering, Acta Math., 203(2009), [RoTa] Igor Rodnianski and Terence Tao, Effective limiting absorption principles, and applications, arxiv: [TaZw] Siu-Hung Tang and Maciej Zworski, From quasimodes to resonances, Math. Res. Lett. 5(998), [Vo] Georgi Vodev, Exponential bounds of the resolvent for a class of noncompactly supported perturbations of the Laplacian. Math. Res. Lett, 7(2000), [Zw] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics 38, AMS, address: datchev@math.mit.edu address: dyatlov@math.mit.edu Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA address: zworski@math.berkeley.edu Department of Mathematics, University of California, Berkeley, CA 94720, USA