Mathematical study of scattering resonances
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Abstract
We provide an introduction to mathematical theory of scattering resonances and survey some recent results.
1 Introduction
Scattering resonances appear in many branches of mathematics, physics and engineering. They generalize eigenvalues or bound states to systems in which energy can scatter to infinity. A typical state has then a rate of oscillation (just as a bound state does) and a rate of decay. Although this notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green’s functions or scattering matrices. The poles of these meromorphic continuations capture the physical information by identifying the rate of oscillations with the real part of a pole and the rate of decay with its imaginary part. The resonant state, which is the corresponding wave function, then appears in the residue of the meromorphically continued operator. An example from pure mathematics is given by the zeros of the Riemann zeta function: they are the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either 0 or \( \frac{1}{4}\). A standard example from physics is given by shape resonances created when the interaction region is separated from free space by a potential barrier. The decay rate is then exponentially small in a way depending on the width of the barrier.
In this article we survey some foundational and some recent aspects of the subject selected using the perspective and experience of the author. Proofs of many results can be found in the monograph written in collaboration with Semyon Dyatlov [80] to which we provide frequent references.
What we call scattering resonances appear under different names in different fields: in quantum scattering theory they are called quantum resonances or resonance poles. In obstacle scattering, or more generally wave scattering, they go by scattering poles. In general relativity the corresponding complex modes of gravitational waves are known as quasinormal modes. The closely related poles of power spectra of correlations in chaotic dynamics are called Pollicott–Ruelle resonances. We will discuss mathematical results related to each of these settings stressing unifying features.

in the introduction we provide a basic physical motivation from quantum mechanics, discuss the case of the wave equation in one dimension, intuitions behind semiclassical study of resonances and some examples from modern science and engineering;

in Sect. 2 we present scattering by bounded compactly supported potentials in three dimensions and prove meromorphic continuation of the resolvent (Green function), an upper bound on the counting function, existence of resonance free regions and expansions of waves; we also explain the method of complex scaling in the elementary setting of one dimension; finally we discuss recent progress and open problems in potential scattering;

we devote Sect. 3 to a survey of recent results organized around topics introduced in the special setting of Sect. 2: meromorphic continuation for asymptotically hyperbolic spaces, fractal upper bounds in physical and geometric settings, resonance free strips in chaotic scattering and resonance expansions; we also provide some references to recent progress in some of the topics not covered in this survey;

Section 4 surveys the use of microlocal/scattering theory methods in the study of chaotic dynamical systems. Their introduction by Faure–Sjöstrand [84] and Tsujii [261] led to rapid progress which included a microlocal proof of Smale’s conjecture about dynamical zeta functions [78], first proved shortly before by Giulietti–Liverani–Pollicott [109]. We review this and other results, again related to upper bounds, resonance free strips and resonances expansions.
1.1 Motivation from quantum mechanics
In quantum mechanics a particle is described by a wave function \( \psi \) which is normalized in \( L^2 \), \( \Vert \psi \Vert _{L^2 } = 1 \). The probability of finding the particle in a region \( \Omega \) is the given by the integral of \( \psi ( x ) ^2 \) over \( \Omega \). A pure state is typically an eigenstate of a Hamiltonian P and hence the evolved state is given by \( \psi ( t) := e^{ i t P} \psi = e^{  it E_0 } \psi \) where \( P \psi = E_0 \psi \). In particular the probability density does not change when the state is propagated.

the state \( \psi ( 0 ) \) is not in \( L^2 \)—it has physical meaning only in the “interaction region”; that is justified differently in Euclidean and nonEuclidean scattering—see Sects. 2.2, 2.6 and 3.1 respectively.

it is not clear why the evolution of a physical state should have an expansion in terms of resonant states; that is justified using meromorphic continuation and asymptotic control of Green’s function and—see Sect. 2.5;

one needs to justify the passage from the Fourier transform \( t \mapsto E \) to the probability distribution (1.1); that is done using the scattering matrix or spectral measure – see the end of Sect. 2.5.
1.2 Scattering of waves in one dimension
Harmonic inversion methods, the first being the celebrated Prony algorithm [217], can then be used to extract scattering resonances from solutions of the wave equation (as shown in Fig. 7; see for instance [171]). The resulting complex numbers, that is the resonances for the potential in Fig. 4 are shown in Fig. 5.
1.3 Resonances in the semiclassical limit
In general however it is impossible to obtain an explicit description of individual resonances. Hence we need to consider their properties and their distribution in asymptotic regimes. For instance in the case of obstacle scattering that could mean the high energy limit. In the case of the sphere in Fig. 8 that corresponds to letting the angular momentum \( \ell \rightarrow +\infty \). For a general obstacle that means considering resonances as \( \lambda  \rightarrow + \infty \) and \( {\mathrm{Im}}\,\lambda  \ll  \lambda \).
1.4 Other examples from physics and engineering
We present here a few recent examples of scattering resonances appearing in physical systems.
Figure 1 shows resonance peaks for a scanning tunneling microscope experiment where a circular quantum corral of CO molecules is constructed—see [187] and references given there. The resonances are very close to eigenvalues of the Dirichlet Laplacian (rescaled by \( \hbar ^2/m_{\mathrm{eff}} \) where \( m_{\mathrm{eff}} \) is the effective mass of the Bloch electron). Mathematical results explaining existence of resonances created by a barrier (here formed by a corral of CO molecules) can be found in [194].
Figure 2 shows an experimental setup for microwave cavities used to study scattering resonances for chaotic systems. Density of resonance was investigated in this setting in [216] and that is related to semiclassical upper bounds in Sect. 3.4. In [13] dependence of resonance free strips on dynamical quantities was confirmed experimentally and Sect. 3.2 contains related mathematical results and references. The experimental and numerical findings in [13] are presented in Fig. 10.
Figure 12 shows the profile of gravitational waves recently detected by the Laser Interferometer GravitationalWave Observatory (LIGO) and originating from a binary black hole merger. Resonances for such waves are known by the name of quasinormal modes in physics literature and are the characteristic frequencies of the waves emitted during the ringdown phase of the merger, when the resulting single black hole settles down to its stationary state – see for instance [66, 76, 157] and Sect. 3.3.
2 Potential scattering in three dimensions
Operators of the form \( P_V :=  \Delta + V \), where \( \Delta = \partial _{x_1}^2 + \partial _{x_2}^2 + \partial _{x_3}^2 \), and where V is bounded and compactly supported, provide a setting in which one can easily present basic theory. Despite the elementary set up, interesting open problems remain—see Sect. 2.7. In this section we prove meromorphic continuation of the resolvent of \( P_V\), \( R_V ( \lambda ) := ( P_V \lambda ^2 )^{1} \), give a sharp bound on the number of resonances in discs, show existence of resonance free regions and justify the expansion (1.2). We also explain the method of complex scaling in the simplest setting of one dimension. These are the themes which reappear in Sects. 3, 4 when we discuss some recent advances. However, in the setting of this section we can present them with complete proofs. In Sect. 2.4 we also review some results in obstacle scattering as they fit naturally in our narrative.
2.1 The free resolvent
We conclude this section with
Theorem 1
Proof
2.2 Meromorphic continuation and definition of resonances
Combining (2.14) and (2.18) with (2.15) we proved
Theorem 2
This gives a mathematical definition of scattering resonances:
Definition 1
Definition 2
A function \( u \in \Pi _{\lambda _0} ( L^2_{\mathrm{comp}\,} ( {\mathbb {R}}^3 ) ) \subset L^2_{\mathrm{loc}\,} ( {\mathbb {R}}^3 ) \) is called a generalized resonant state. If \( ( P_V  \lambda _0^2 ) u = 0 \) then u is called a resonant state. This is equivalent to \( u \in ( P_V \lambda _0^2 )^{J1} \Pi _{\lambda _0} ( L^2_{\mathrm{comp}\,} ( {\mathbb {R}}^3 ) ) \).
When V is real valued then \( P_V \) is a selfadjoint operator (with the domain given by \( H^2 ( {\mathbb {R}}^3 ) \)). The poles of \( R_V ( \lambda ) \) in \( {\mathrm{Im}}\,\lambda > 0 \) correspond to negative eigenvalues of \( P_V \) and in (2.23) we have \( J = 1\). This of course is consistent with (2.24) since for \( {\mathrm{Im}}\,\lambda _0 > 0 \), \( R_0 ( \lambda _0 ) f \in L^2 ( {\mathbb {R}}^3 ) \).
2.3 Upper bound on the number of resonances
We will prove the following optimal bound
Theorem 3
In the case of \(  \Delta + V \) on \( {\mathbb {R}}^n \), the discrete spectrum is replaced by the discrete set of resonances. Hence the bound (2.26) is an analogue of the Weyl law. Except in dimension one—see (2.27)—the issue of asymptotics or even optimal lower bounds remains unclear at the time of writing of this survey. We will discuss lower bounds and existence of resonances in Sect. 2.7.
Hence our goal is to find a suitable h and to prove (2.30). Before doing it we recall some basic facts about trace class operators and Fredholm determinants. We refer to [80, Appendix B] for proofs and pointers to the literature.
Remark
It is not difficult to see that we could, just as in [278], take \( \det ( I  ( V R_0 ( \lambda ) \rho )^2 ) \). But the power 4 makes some estimates more straightforward. We could also have taken a modified determinant of \( I + V R_0 ( \lambda ) \rho \) [80, Theorem 3.23, §B.7] and that would give an entire function whose zeros are exactly the resonances of V. That works in dimension three but in higher dimensions the modified determinants grow faster than (2.30)—see [278].
Proof of Theorem 3
The key to fighting exponential growth when \( {\mathrm{Im}}\,\lambda < 0 \) is analyticity near infinity which is used implicitely in (2.41)–(2.44). That is a recurrent theme in many approaches to the study of resonances—see 2.6.
Resonance counting has moved on significantly since these early results. Some of the recent advances will be reviewed in Sect. 3.4, see also [279] for an account of other early works. The most significant breakthrough was Sjöstrand’s discovery [240] of geometric upper bounds on the number of resonances in which the exponent is no longer the dimension as in (2.26) and (2.29) but depends on the dynamical properties of the classical system.
2.4 Resonance free regions
The imaginary parts of resonances are interpreted as decay rates of the corresponding resonant states—see Sect. 2.5 for a justification of that in the context of the wave equation. If there exists a resonance closest to the real axis then (assuming we can justify expansions like (1.2)—see Theorems 5,12) its imaginary part determines the principal rate of decay of waves—see the end of this section for some comments on that. If waves are localized in frequency then imaginary parts of resonances with real parts near that frequency should determine the decay of those waves.
Here we present the simplest case:
Theorem 4
Suppose that \( V \in L^\infty ( {\mathbb {R}}^3 ) \), \(\mathrm{supp}\,V \subset B ( 0 , R_0 ) \), and that \( R_V ( \lambda ) \) is the scattering resolvent of Theorem 2.
Proof
Geometry  Resonance free region 

Arbitrary obstacle  (a) [37]; optimal for obstacles with elliptic closed orbits [254, 256, 260] 
Two convex obstacles  (b) with a pseudolattice of resonances below the strip [104, 139], hence optimal 
Several convex obstacles  (b) with M determined by a topological pressure of the chaotic system [103, 140]; an improved gap [210] 
Smooth nontrapping obstacles  (c) with arbitrarily large M [179, 262], using propagation of singularities [137, Chapter 24]; 
Analytic nontrapping obstacles  (d) with \( \beta = \frac{1}{3} \) [12, 215] based on propagation of Gevrey3 singularities [164]; optimal for convex obstacles 
Smooth strictly convex obstacles  (d) with \( \beta = \frac{1}{3} \) and \( \gamma \) determined by the curvature [124, 151, 246]; Weyl asymptotics for resonances in cubic bands \( \sim r^{n1} \) [247] 
Another rich set of recent results concerns scattering by “thin” barriers modeled by delta function potentials (possibly energy dependent) supported on hypersurfaces in \( {\mathbb {R}}^n \). For example, refinements of the Melrose–Taylor parametrix techniques give, in some cases, the optimal resonance free region defined by \( F ( x ) = x^{ \beta } \), \( \beta > 0 \) and a mathematical explanation of a Sabine law for quantum corrals [17]. See the works of Galkowski [96, 97, 98] and Smith–Galkowski [100] where references to earlier literature on resonances for transmission problems can also be found, and Fig. 14 for an illustration.
2.5 Resonance expansions of scattered waves
Theorem 5
In (2.53) we did not aim at the optimality of the condition \( t > 10 R_1 \): using propagation of singularities it suffices to take \( t > 2 R_1 \).
Proof of Theorem 5
We now want to deform the contour in (2.56). For that we choose r large enough so that all the resonances with \( {\mathrm{Im}}\,\lambda >  a  \delta \log ( 1 +  \lambda  ) \) are contained in \(  \lambda  \le r \). If we choose \( \delta < \frac{1}{2 R_0 } \) that is possible thanks to Theorem 4.
The case of arbitrary \( f \in H^1_{\mathrm{comp} } \) and \(g \equiv 0\) follows by replacing \({\sin t\lambda }/{\lambda }\) by \(\cos t\lambda \) in the formula for w(t, x). \(\square \)
For Breit–Wigner type formula near a single resonance in the semiclassical limit see Gérard–Martinez [105] and Gérard–Martinez–Robert [106] and for high energy results and results for clouds of resonances, Petkov–Zworski [211, 212] and Nakamura–Stefanov–Zworski [194].
We finally comment on expansions on the case of the Schrödinger equation. In that case the dispersive nature of the equation makes it harder to justify resonance expansions. It can be done when a semiclassical parameter is present, see Burq–Zworski [40] and Nakamura–Stefanov–Zworski [194], or when one considers a localized resonance—see Merkli–Sigal [185], Soffer–Weinstein [253] and references given there.
2.6 Complex scaling in dimension one
In Sect. 2.2 we established meromorphic continuation using analytic Fredholm theory. That allowed us to obtain general bounds on the number of resonances and to justify resonance expansions. To obtain more refined results relating geometry to the distribution of resonances, as indicated in Sect. 1.3, we would like to use spectral theory of partial differential equations. And for that we would like to have a differential operator whose eigenvalues would be given by resonances. That operator should also have suitable Fredholm properties. We would call this an effective meromorphic continuation.
The method of complex scaling produces a natural family of nonselfadjoint operators whose discrete spectrum consists of resonances. It originated in the work of AguilarCombes [3], BalslevCombes [10] and was developed by Simon [239], Hunziker [138], HelfferSjöstrand [125], Hislop–Sigal [133] and other authors. For very general compactly supported “black box” perturbations and for large angles of scaling it was studied in [245] while the case of long range black box perturbations was worked out in [241]. The method has been extensively used in computational chemistry—see Reinhardt [222] for a review. As the method of perfectly matched layers it reappeared in numerical analysis—see Berenger [19].
In this section we present the simplest case of the method by working in one dimension.^{3} The idea is to consider \( D_x^2 \) as a restriction of the complex second derivative \( D_z^2 \) to the real axis thought of as a contour in \( \mathbb {C}\). This contour is then deformed away from the support of V so that \( P = D_x^2 + V ( x ) \) can be restricted to it. This provides ellipticity at infinity at the price of losing selfadjointness.
This argument can be reversed and consequently we basically proved [80, Theorem 2.20]:
Theorem 6
2.7 Other results and open problems
For \( V \in {{\mathcal {C}}^\infty _{\mathrm{c}}}( {\mathbb {R}}^n ) \), real valued, existence of infinitely many resonances was proved by Melrose [184] for \( n = 3 \) and by Sá Barreto–Zworski [230] for all odd n (including for any superexponentially decaying potentials). The surprising fact that there exist complex valued potentials in odd dimensions greater than or equal to three that have no resonances at all was discovered by Christiansen [46].
The most frustrating open problem is existence of an optimal lower bound. It is not clear if we can expect an asymptotic formula.
Conjecture 1
All of the above questions can be asked in even dimensions and for obstacle problems in which case the bound (2.26) was established by Melrose [182] for odd dimensions and by Vodev [266, 267] in even dimension. A remarkable recent advance is due to Christiansen [48] who proved that for any obstacle in even dimensions we always have \( r^n \) growth for the number of resonances—see that paper for other references concerning lower bounds in even dimension.
Existence of resonances can be considered as a primitive inverse problem: a potential with no resonances is identically zero. In one dimension or in the radial case finer inverse results have been obtained: see Korotyaev [160, 161], Brown–Knowles–Weikard [36], Bledsoe–Weikard [18], Datchev–Hezari [55] and references given there.
Another recent development in the study of resonances for compactly supported potentials concerns highly oscillatory potentials \( V ( x ) = W ( x , x/\varepsilon ) \) where \( W : {\mathbb {R}}^n \times {\mathbb {R}}^n/ \mathbb {Z}^n \rightarrow {\mathbb {R}}\) (or \(\mathbb {C}\)) is compactly supported in the first set of variables. Precise results for \( n = 1 \) were obtained by Duchêne–Vukićević–Weinstein [63]: resonances are close to resonances of \( W_0 ( x ) := \int _{{\mathbb {R}}^n/\mathbb {Z}^n} W ( x , y ) dy \), and the difference is given by \( \varepsilon ^4 \alpha + \mathcal {O} ( \varepsilon ^5 ) \) where, in the spirit of homogenization theory, \( \alpha \) can be computed using an effective potential. In a remarkable followup Drouot [61] generalized this result to all odd dimensions and obtained a full asymptotic expansion in powers of \(\varepsilon \).
3 Some recent developments
We will now describe some recent developments in meromorphic continuation, resonance free regions, resonance counting and resonance expansions for semiclassical operators \(  h^2 \Delta _g + V \) on Riemannian manifolds with Euclidean and nonEuclidean infinities. We will also formulate some conjectures: some quite realistic (even numbered) and some perhaps less so (odd numbered).^{5}
Before doing it let us mention some interesting topics which lie beyond the scope of this survey. We will discuss hyperbolic trapped sets but not homoclinic trapped sets. Although not stable under perturbations these trapped sets occur in many situations each with its own interesting structure in distribution of resonances. An impressively precise study of this has been made by Bony–Fujiie–Ramond–Zerzeri [28]. In our survey of counting results we will concentrate on fractal Weyl laws and we refer to Borthwick–Guillarmou [32] for recent results on resonance counting in geometric settings. In the semiclassical Euclidean setting Sjöstrand [244] obtained precise asymptotic counting results in the case of random potentials. The introduction of randomness should be a beginning of a new development in the subject. We also do not discuss the important case of resonances for magnetic Schrödinger operators and refer to Alexandrova–Tamura [4], Bony–Bruneau–Raikov, [26] and Tamura [258, 259] for some recent results and pointers to the literature. We do not address issues around threshold resonances (see Jensen–Nenciu [150] and references given there) and their importance in dispersive estimates (see the survey by Schlag [233]) and for nonlinear equations (for an example of a linearly counterintuitive phenomenon, see [135, Figure 6]). Finally, for the role that shape resonances have had in the study of blowup phenomena we defer to Perelman [208] and Holmer–Liu [134] and to the references given there.
The section is organized as follows: in Sect. 3.1 we review Vasy’s method for defining resonances for asymptotically hyperbolic manifolds. In Sect. 3.2 we first review general results on resonance free regions and then describe the case of hyperbolic (in the dynamical sense) trapped sets. As applications of results in Sects. 3.1 and 3.2 we discuss expansions of waves in black hole backgrounds in Sect. 3.3. The last Sect. 3.4 is devoted to mathematical study of fractal Weyl laws with some references to the growing physics literature on that subject.
3.1 Meromorphic continuation in geometric scattering
In Sect. 2.2 we showed how to continue the resolvent meromorphically across the spectrum using analytic Fredholm theory. In Sect. 2.6 we presented the method of complex scaling which provides an effective meromorphic continuation in the sense that resonances are identified with eigenvalues of nonselfadjoint Fredholm operators. As one can see already in the one dimensional presentation that method is closely tied to the structure of the operator near infinity.
With motivation coming from general relativity in physics (see Sect. 3.3) and from analysis on locally symmetric spaces it is natural to consider different structures near infinity. Here we will discuss complete asymptotically hyperbolic Riemannian manifolds modeled on the hyperbolic space near infinity. For more general hyperbolic ends see Guillarmou–Mazzeo [119] where meromorphy of the resolvent was established using analytic Fredholm theory without providing an effective meromorphic continuation as defined above. We also remark that complex scaling method is possible in the case of manifolds with cusps. For a subtle application of that see a recent paper by Datchev [52] where existence of arbitrarily wide resonance free strips for negative curvature perturbations of \( \langle z \mapsto z + 1\rangle \backslash \mathbb {H}^2 \) is established. For a recent analysis of a higher rank symmetric spaces and references to earlier works see Mazzeo–Vasy [177, 178] and Hilgert–Pasquale–Przebinda [126].
We will follow the presentation of [282] and for simplicity will not consider the semiclassical case. That of course is essential for applications and can be found in a textbook presentation of [80, Chapter 5].

in order to reduce the investigation to the study of regularity we should conjugate \(  \Delta _g \) by the weight \( y_1^{i \lambda + \frac{n}{2} } \).

the desired smoothness properties should be stronger in the sense that the functions should be smooth in \( (y_1^2 , y' ) \).
Definition 3
Since the dependence on \( \lambda \) in \( P ( \lambda ) \) occurs only in lower order terms we can replace \( P ( \lambda ) \) by P(0) in (3.16). Hence the definition of \( \mathscr {X}_s \) is independent of \( \lambda \).
We can now state the result about mapping properties of \( P ( \lambda ) \):
Theorem 7
Let \( \mathscr {X}_s , \mathscr {Y}_s \) be defined in (3.16). Then for \( {\mathrm{Im}}\,\lambda >  s  \frac{1}{2} \) the operator \( P ( \lambda ) : \mathscr {X}_s \rightarrow \mathscr {Y}_s \), has the Fredholm property, that is \(\dim \{ u \in \mathscr {X}_s : P ( \lambda ) u = 0 \} < \infty \), \( \dim \mathscr {Y}_s / P ( \lambda ) \mathscr {X}_s < \infty \), and \( P ( \lambda ) \mathscr {X}_s \) is closed.
Moreover for \( {\mathrm{Im}}\,\lambda > 0 \), \( \lambda ^2 + (\frac{n}{2} )^2 \notin \mathrm{Spec}\,(  \Delta _g ) \) and \( s > {\mathrm{Im}}\,\lambda  \frac{1}{2} \), \( P ( \lambda ) : \mathscr {X}_s \rightarrow \mathscr {Y}_s \) is invertible. Hence, for \( s \in {\mathbb {R}}\) and \( {\mathrm{Im}}\,\lambda >  s  \frac{1}{2} \), \( \lambda \mapsto P ( \lambda )^{1} : \mathscr {Y}_s \rightarrow \mathscr {X}_s , \) is a meromorphic family of operators with poles of finite rank.
In view of (3.11) this immediately recovers the results of Mazzeo–Melrose [176] and Guillarmou [115] in the case of even metrics (Guillarmou showed that for generic noneven metrics global meromorphic continuation does not hold; he also showed that the method of [176] does provide a meromorphic continuation to \( \mathbb {C}\setminus i \mathbb {N}^* \) for all asymptotically hyperbolic metrics and to \( \mathbb {C}\) for the even ones):
Theorem 8
Suppose that (M, g) is an even asymptotically hyperbolic manifold and that \( R ( \lambda ) \) is defined by (3.7). Then \( R (\lambda ) : {\dot{\mathcal {C}}^\infty }( M ) \rightarrow {{\mathcal {C}}^\infty }( M ) \), continues meromorphically from \( {\mathrm{Im}}\,\lambda > \frac{n}{2} \) to \( \mathbb {C}\) with poles of finite rank.
The actual proof of (3.17) is based on the analysis of the Hamilton flow of the principal symbol of \( P ( \lambda ) \), \( x_1 \xi _1^2 + \xi '^2_{h(x)} \in {{\mathcal {C}}^\infty }( T^* X ) \), and of positive commutator estimates depending on lower order terms—that is where the dependence on \( \lambda \) comes from. For that we refer to [282, §4], [80, §E.5.2].
One weakness of the method lies in the fact that it provides effective meromorphic continuation only in strips, even though the resolvent is meromorphic in \( \mathbb {C}\). Hence, results which involve larger regions (such as asymptotics of resonances for convex obstacles [152, 247] where resonances lie in cubic regions, \( {\mathrm{Im}}\,\lambda \sim   \mathrm{Re}\,\lambda ^{\frac{1}{3}} \)) are still inaccessible in the setting of asymptotically hyperbolic manifolds (or even \( \mathbb {H}^n \setminus \mathcal {O} \)). Analyticity near infinity should play a role when larger regions are considered and towards that aim we formulate a conjecture which could perhaps interest specialists in analytic hypoellipticity. It is an analytic analogue of (3.17) and it also has a microlocal version:
Conjecture 2
We remark that in the analytic case the operator \( P ( \lambda ) \) belongs to the class of Fuchsian differential operators studied by Baouendi–Goulaouic [11] and that the conjecture is true for \( P ( \lambda ) = 4 ( x_1 D_{x_1}^2  ( \lambda + i ) D_{x_1 } )  \Delta _h \), where h is a metric on \( \partial M\), independent of \( x_1 \) and \( \lambda \notin  i \mathbb {N}^* \) [166].
3.2 Resonance free regions
The asymptotic parameter h is supposed to be small and we will consider resonances of P(h) near a fixed energy level. When \( V \equiv 0 \) then the limit \( h \rightarrow 0 \) corresponds to the high energy limit but even in that case the link with physical intuitions around classical/quantum correspondence is useful.
Hamiltonian flow  Resonance free region and resolvent bounds 

General case  (a) [38, 43, 51, 224, 236, 268], [80, §6.4]; optimal for shape resonances [95, 125], and for “resonances from quasimodes” [102, 254, 260], [80, §7.3]; corresponding cutoff resolvent bounds (cf.(2.46)) also optimal [54] 
Normally hyperbolic trapping  (b) with \( \gamma \) given by a “Lyapounov exponent” [69, 108, 202, 274], [80, §6.3]; optimal for one closed hyperbolic orbit [107] and for rnormally hyperbolic trapping [67] 
Hyperbolic trapped set  (b) with \( \gamma \) determined by a topological pressure of the trapped set [200]; an improved gap expected; known for hyperbolic quotients [75, 191, 257] 
Smooth nontrapping  
aGevrey nontrapping  (d) with \( \beta = 1 1 /a \) [225] 
Analytic nontrapping  (d) with \( \beta = 0 \) [125, 240]; optimal with density \(\sim h^{n} \) via Sjöstrand’s trace formula [242], [80, §7.4] 
(See Definitions 4, 5 for definitions hyperbolic and normally hyperbolic trapped sets and (3.29), (3.36) for definitions of “Lyapounov exponenents” and topogical pressure respectively; smooth, Gevrey and analytic refers to the regularity of the coefficients of the operator P.)
The first case we will consider is that of hyperbolic trapped sets:
Definition 4
For more on this in dynamical systems see [154, §17.4]. One important property is the stability of this condition. We also remark that the existence of the metric \( \Vert \bullet \Vert _\rho \) in part (iii) or (3.28) follows from the same estimates for some metric with \( C e^{ \lambda t } \) on the right hand side. Examples include \( p = \xi _g^2 \) where the curvature is negative near the trapped set or \( p = \xi ^2 + V ( x ) \) where V is a potential given by several bumps (see [202, Figure 1]). In obstacle scattering trapped sets are hyperbolic for several convex obstacles satisfying Ikawa’s noneclipse condition (see [198, (1.1)]).
The analogue of Ikawa’s result for several convex obstacles [140], postulated independently in the physics literature by Gaspard–Rice [103], was established for operators of the form (3.20) by Nonnenmacher–Zworski [200, 201]. Thanks to the results of Sect. 3.1 the result also holds for \( h^2 \Delta _g \) on even asymptotically hyperbolic manifolds. In that case it generalizes celebrated results of Patterson and Sullivan (see [31]) formulated using dimension of the limit set (3.31). That gap is shown as the top line on the right graphs in Fig. 21.
Theorem 9
It is not clear if the pressure condition (3.32) is needed to obtain a resonance strip—see Theorem 11 below. What seems to be clear is that \( \mathcal {P}_E ( \frac{1}{2} ) \) is a robust classical quantity determining the strip while the improvements require analysing quantum effects. The upper bound (3.34) is optimal for \( {\mathrm{Im}}\,z = 0 \) thanks to a general result of Bony–Burq–Ramond [27], [80, §7.1].
Applications of the estimate (3.34) include local smoothing estimates with a logarithmic loss by Datchev [50] and Strichartz estimates with no loss by Burq–Guillarmou–Hassell [39]. The method of proof was used by Schenck [232] to obtain decay estimates for damped wave equations and by Ingremeau [141] to describe distorted plane waves in quantum chaotic scattering.
The trapped sets to which Theorem 9 applies are typically fractal. In fact, in dimension 2 the condition \( \mathcal {P}_E ( \frac{1}{2}) < 0 \) is equivalent to the trapped set being filamentary in the sense that \( \dim K_E < 2 \), that is the trapped set is below the mean of the maximal dimension 3 and the minimal dimension 1 (direction of the flow). We will now consider another case in which the trapped set is smooth but with hyperbolicity in the transversal direction:
Definition 5
This dynamical configuration is stable under perturbations under a stronger addition condition of rnormal hyperbolicity. Roughly speaking that means that the flow on \( K_J \) has weaker expansion and contraction rates than the flow in the transversal directions—see [67, 132] for precise a definition and (3.48) below for an example. Normally hyperbolic trapping occurs in many situations: for instance, for null flows for black hole metrics, see [76], Sect. 3.3, and in molecular dynamics, see [202, Remark 1.1], [111, 234]. Another important example comes from contact Anosov flows lifted to the cotangent bundle, see [261], [202, Theorem 4], Sect. 4.4. The following general result was proved by Nonnenmacher–Zworski [202]:
Theorem 10
For more general trapped sets but for analytic coefficients of P and without the estimate (3.34), the resonance free region (3.37) was obtained early on by Gérard–Sjöstrand [108]. Sometimes, for instance in the case of black holes, \( d^\perp = 1 \) (see 3.35), \( \Gamma _J^\pm \) (see 3.24) are smooth, and \( T_\rho K_J \oplus E_\rho ^+ = T_\rho \Gamma ^{\pm }_J \). In that case a resonance free region \( {\mathrm{Im}}\,z >  h /C \) and the bound (3.34) was obtained by Wunsch–Zworski [274]. An elegant proof with the width (3.36) and a sharp resolvent estimate was given by Dyatlov [69][80, §6.3]. The width given by \( \nu _{\min } \) is essentially optimal as one can already see from the work of Gérard–Sjöstrand [107] on the pseudolattice of resonances generated by a system with one hyperbolic closed orbit. A general result was provided by Dyatlov [67]: if \( K_E \) is rnormally hyperbolic for any r and if the maximal expansion rate \( \nu _{\max }\) satisfies \( \nu _{\max } < 2 \nu _{\min } \), then the resonance free region is essentially optimal in the sense that in the strip below \( {\mathrm{Im}}\,z = h \nu _{\min }/2 \) there exist infinitely many resonances—see Fig. 22.
We now discuss improvements over the pressure gap. The first such improvement for scattering resonances was achieved by Naud [191] who extended Dolgopyat’s method [59] to the case of convex cocompact quotients and showed that there exists \( \gamma > 0 \) such that when \( \delta < \frac{1}{2} \) (see 3.31) then there are no resonances other than \( i ( \delta  \frac{1}{2} ) \) with \( {\mathrm{Im}}\,\lambda > \delta  \frac{1}{2}  \gamma \). The Dolgopyat method was further developed by Stoyanov [257] for higher dimensional quotients (as an application of general results) and by Oh–Winter [203] for uniform gaps for arithmetic quotients. Petkov–Stoyanov [210] also adapted Dolgopyat’s method to the case of several convex obstacles. All of these results assume that \( \mathcal {P}_E ( \frac{1}{2}) \le 0 \).
Open quantum maps have been studied in physics and mathematics. They are quantizations of symplectic relations on \( {\mathbb {T}}^2 = {\mathbb {R}}^2 / \mathbb {Z}^2 \) where the symplectic relations have features of Poincaré maps in scattering theory: see Nonnenmacher [196, §5] for a general introduction, [74, §1.4] for references to the physics literature, Nonnenmacher–Sjöstrand–Zworski [197] for a reduction of chaotic scattering problems to quantizations of Poincaré maps and Nonnenmacher–Zworski [199] for quantization of piecewise smooth relations.
The case of \( \gamma = \frac{1}{2}  \delta \) is easy to establish but a finer analysis of FUP gives [74]:
Theorem 11
A numerical illustration of Theorem 11 is shown in Fig. 23. In the example presented there the numerically computed best exponent in FUP (3.43) is sharp. Some other examples in [74] show however that it is not always the case. We will return to open quantum maps in Sect. 3.4.
Theorem 11 is the strongest gap result for hyperbolic trapped sets and it suggests that a resonance free strip of size h exists for all operators P(h) with such trapped sets:
Conjecture 3
3.3 Resonance expansions in general relativity
In this section we will describe some recent results concerning expansions of solutions to wave equations for black hole metrics in terms of quasinormal modes (QNM). That is the name by which scattering resonances go in general relativity [157]. We want to emphasize the connection to the results of Sects. 3.1, 3.2 and in particular to the r–normally hyperbolic dynamics [274].
We will consider the case of Kerr–de Sitter, or asymptotically Kerr–de Sitter metrics. These model rotating black holes in the case of positive cosmological constant \( \Lambda > 0 \). From the mathematical point of view that makes infinity “larger” and provides exponential decay of waves which makes a rigorous formulation of expansions easier. When one adds frequency localization weaker expansions are still possible in the Kerr case—see [68, Theorem 2], [76, (13)]. References to the extensive mathematics literature in the case of \( \Lambda = 0 \) can be found in [68].
For Schwarzschild–de Sitter black holes, QNM were described by Sá Barreto–Zworski [231]: at high frequencies they lie on a pseudolattice as in the case of one hyperbolic closed orbit [107] but with multiplicities coming from the spherical degeneracy. Dyatlov [66, 67] went much further by describing QNM for Kerr–de Sitter black holes: for small values of rotations he showed a Zeemanlike splitting of multiplicities predicted in the physics literature, and for perturbations of Kerr–de Sitter black holes, he obtained a counting law (see Figs. 22, 24). In particular, for small values of a the results of [66] show that there are no modes with \( {\mathrm{Im}}\,\lambda \ge 0 \). For Kerr black holes ((3.46) with \( \Lambda = 0 \)) ShlapentokhRothman [238] showed that this is the case in the full range \( a < M \). This suggests the following
Conjecture 4
For a, \( \Lambda \) and M satisfying \( ( 1  \Lambda a^2/3 )^3 > 9 \Lambda M^2 \) and in the shaded region of Fig. 24 (or in some other “large” range of values), any \( \lambda \ne 0 \) for which (3.49) holds satisfies \( {\mathrm{Im}}\,\lambda < 0 \) and \( \lambda = 0 \) is a simple resonance.
The first expansion involving infinitely many QNM lying on horizontal strings of the pseudolattice of [231] was obtained by Bony–Häfner [29]. Using his precise results on the distribution of QNM for Kerr–de Sitter metrics with small values of a, Dyatlov [66] obtained similar expansions for rotating black holes. In [67] he formulated an expansion of waves for perturbations of Kerr–de Sitter metrics in terms of a microlocal projector, see also [68].
Here we will only state a simpler result which is an almost immediate consequence of Theorems 8, 10 and of gluing results of Datchev–Vasy [57]. For tensorvalued wave equations on perturbations of Schwarzschild–de Sitter spaces (including Kerr–de Sitter spaces with small values of a) and in any spacetime dimension \( n \ge 4 \) this result was obtained by Hintz [127]. A more precise formulation valid across the horizons can be found there.
Theorem 12
3.4 Upper bounds on the number of resonances: fractal Weyl laws
The standard Weyl law for the density of quantum states was already recalled in (2.29) as motivation for the upper bound on the number of resonances in discs (2.26). In this section we will describe finer upper bounds which take into account the geometry of the trapped set. They were first proved by Sjöstrand [240] for operators with analytic coefficients but in greater geometric generality than presented here.
Theorem 13
For outlines and ideas of the proofs we refer the reader to general reviews [196, §§4,7], [193] and to [248, §2], [53, §1], [198, §1].
Existence of fractal lower bounds or asymptotics has been studied numerically, first for a three bump potential by Lin [167] and then using semiclassical zeta function calculations [103] for three disc systems (see Figs. 2 and 10) by Lu–Sridhar–Zworski [170]. The results have been encouraging and led to experimental investigation by Potzuweit et al [216]: the experimentally observed density is higher than linear but does not fit the upper bound (3.56). There are many possible reasons for this, including the limited range of frequencies. (Related experiments by Barkhofen et al [13] confirmed the pressure gaps of Theorem 9, see Figs. 10 and 26). Fractal Weyl laws have been considered (and numerically checked) for various open chaotic quantum maps—see [74, 199] and references given there. The have also been proposed in other types of chaotic systems, ranging from dielectric cavities to communication and social networks—see recent review articles by Cao–Wiersig [42] and Ermann–Frahm–Shepelyansky [83] respectively. Fractal Weyl laws for mixed systems and localization of resonant states have been investigated by Körber et al [158, 159].
So far, the only rigorous lower bound which agrees with the fractal upper bound was obtained by Nonnenmacher–Zworski in a special toy model quantum map [200, Theorem 1]. Nevertheless the evidence seems encouraging enough to reiterate the following conjecture [170], [123, (4.7)]:
Conjecture 5
Numerical investigation of fractal Weyl laws lead to the observation that imaginary parts of resonances (that is resonance widths) concentrate at \( {\mathrm{Im}}\,z \approx \frac{1}{2} \mathcal {P}_E ( 1 ) h \) where \( \mathcal {P}_E ( s ) \) is defined by (3.29). The value \(  \mathcal {P}_E ( 1 ) \) gives the classical escape rate on the energy surface \( p^{1} ( E) \)—see [196, (17)]. The point made by Gaspard–Rice [103] was that the gap is given by \( \mathcal {P}_E ( \frac{1}{2} ) h < \frac{1}{2} \mathcal {P}_E ( 1 )h\) (with the latter being a more obvious guess). However most of resonances want to live near \( {\mathrm{Im}}\,z = \frac{1}{2} \mathcal {P}_E ( 1 ) h \)—see also Fig. 21.
The first rigorous result indicating lower density away from \( {\mathrm{Im}}\,\lambda = \frac{1}{2} \mathcal {P}_E ( 1 ) \) was obtained by Naud [192] who improved on the fractal bound in Theorem 13 for \( \gamma < \mathcal {P}_E ( 1 ) /2 \) in the case of convex cocompact surfaces. Dyatlov [70] provided an improved bound for convex cocompact hyperbolic quotients in all dimensions. We state his result in the case of dimension two and in the nonsemiclassical setting (resonances as poles of \( (  \Delta _g^2  \frac{1}{4}  \lambda ^2 )^{1} \)—see (3.7), (3.27) and (3.31)):
Theorem 14
The bound \( r^\delta \) comes from [122] and the improvement is in strips \( {\mathrm{Im}}\,\lambda>  \gamma >  \frac{1}{2} ( 1  \delta ) \). A comparison between (3.59) and numerically fitted exponents was shown in Fig. 21 (for the more robust counting fuction \( N ( R , \beta ) = \sum _{k=1}^{[R]} \mathcal {N} ( R  k, \beta ) \)). These numerical experiments of Borthwick–Dyatlov–Weich [70, Appendix] again suggest that Conjecture 5 (perhaps in a weaker form) holds, while the optimality of (3.59) is unclear. Some lower bounds in strips have been obtained by Jakobson and Naud [146] but they are far from the upper bounds.
Dyatlov–Jin [74] proved an analogue of Theorem 14 for open quantum maps of the form shown in Fig. 23. That paper can be consulted for more numerical experiments and interesting conjectures.
Here we make a weak (and hopefully accessible) conjecture which in the case of quotients follows from Theorem 14:
Conjecture 6
Jakobson and Naud [147] made a bolder conjecture that for convex cocompact hyperbolic surfaces there are only finitely many resonances with \( {\mathrm{Im}}\,\lambda >  \frac{1}{2} ( 1 \delta ) \)—see Fig. 21. We make an equally bold but perhaps more realistic
Conjecture 7
4 Pollicott–Ruelle resonances from a scattering theory viewpoint
As signatures of a chaotic systems the dynamical resonances \( \lambda _j \)’s were introduced by Pollicott [213] and Ruelle [227]—see Fig. 13 for an example in modeling of physical phenomena. They can also be studied in the simpler setting of maps, that is for systems with discrete time—see Baladi–Eckmann–Ruelle [8] for an early study and Baladi [5, 6] for later developments.
The explicit analogy with scattering theory was emphasized by Faure–Sjöstrand [84] following earlier works by Faure–Roy–Sjöstrand [86] and Faure–Roy [85] in the case of maps. Semiclassical methods for the study of decay of correlations were also introduced by Tsujii [261] who applied FBI transform techniques to obtain precise bounds on the asymptotic gaps for certain flows (see Sect. 4.4 below). This led to rapid progress some of which is described below—to see the extent of this progress one can compare the current state of affairs to that in the early review by Eckmann [81]. For some other recent developments see also Faure–Tsujii [89].
In this section we first give a precise definition of chaotic dynamical systems and then define Pollicott–Ruelle resonances. We concentrate to the case of M compact but give indications what happens in the noncompact case. We then explain the connection to dynamical zeta functions and survey results on resonance free strips and on ounting of resonances.
4.1 Anosov dynamical systems
Let M be a compact manifold and \( \varphi _t = \exp t X : M \rightarrow M \) a \( C^\infty \) flow generated by \( X \in C^\infty ( M; T M) \).
4.2 Definition of Pollicott–Ruelle resonances
The following theorem was first proved by Faure–Sjöstrand [84] for more specific weights G and by Dyatlov–Zworski [78]. The characterization of resonant states using a wave front set condition is implicit in [84] and is stated in Dyatlov–Faure–Guillarmou [71, Lemma 5.1] and [79, Lemma 2.2]. For noncompact M with hyperbolic trapped sets for the flow \( \varphi _t \) a much more complicated analogue was proved by Dyatlov–Guillarmou [73, Theorem 2].
Theorem 15
This result remains valid for operators \( \mathbf {P} : {{\mathcal {C}}^\infty }( M , \mathcal {E}) \rightarrow {{\mathcal {C}}^\infty }( M , \mathcal {E} ) \) where \( \mathcal {E}\) is any smooth complex vector bundle and \( \mathbf {P} \) is a first order differential system with the principal part given by P. That will be important in Sect. 4.3.
Theorem 15 provides a simple definition of Pollicott–Ruelle resonances formulated in terms of the wave front set condition and action on distributions. We state it only in the scalar case:
Definition 6
Suppose M is a compact manifold, \( X \in {{\mathcal {C}}^\infty }( M ; TM ) \) generates an Anosov flow in the sense of (4.4) and \( E_u^* \) is defined in (4.7).
Our insistence of \( \frac{1}{i} \) in P comes from quantum mechanician’s attachment to selfadjoint operators: P is selfadjoint on \( L^2 ( M , dm ) \) if the flow admits a smooth invariant measure dm. In some conventions, e.g. in [71, 118], resonances are given by \( s =  i \lambda \).
Theorem 16
The last statement means that an asymptotic resonance free region for contact flows (see Sect. 4.4) is stable under random perturbations. It also comes with polynomial resolvent bounds \( ( P_\varepsilon  \lambda )^{1} = ( \lambda ^{N_0} )_{ H^{s_0} \rightarrow H^{s_0} } \). Stability of the gap for certain maps was established by Nakano–Wittsten [195].
We formulate the \( \widetilde{P}_\varepsilon \) analogue of the second part of Theorem 16 as
Conjecture 8
Suppose \( M = S^* \Sigma \), where \( ( \Sigma , g ) \) is a compact Riemannian manifold whose geodesic flow is an Anosov flow. Then the second part of Theorem 16 holds with \( \widetilde{P}_\varepsilon \) in place of \( P_\varepsilon \).
4.3 Connections to dynamical zeta functions
The first zero of \( \zeta _R \) is related to topological entropy (the value of the pressure at 0 in the notation of (3.29)) and the continuation to a small strip past that first zero was achieved by Parry–Pollicott [204]. To obtain larger strips turns out to be as difficult as obtaining global meromorphic continuation (which proceeds strip by strip; in the case of \( C^k \) flows the meromorphy only holds in a strip of size depending on k—see [109]; our microlocal arguments give that as well but with less precision). When the manifold M and the flow are real analytic that continuation was obtained by Rugh [228] and Fried [94]. For Anosov flows on smooth compact manifolds it was first established by Giulietti–Liverani–Pollicott [109] and then by Dyatlov–Zworski [78]. Dyatlov–Guillarmou [73] considered the more complicated noncompact case and essentially settled the original conjecture of Smale.
The condition (4.34) is satisfied for K given by the Schwartz kernel of \( e^{itP} = \varphi _{t}^* \), \( t > 0 \), because for Anosov flows powers of the Poincaré map (4.30), \( \mathcal {P}_\gamma ^k \), cannot have 1 as an eigenvalue. Hence (4.32) makes sense and (4.31) holds.
Condition (4.37) is an immediate consequence of [78, Proposition 3.3] (though it takes a moment to verify it: the pullback by \( ( \varphi _{t_0} \otimes {\mathrm{id}} )\) shifts things away from the diagonal):
Theorem 17
Away from the conic sets \( E_u^* \times E_s^* \) this follows from a modification of results of Duistermaat–Hörmander on propagation of singulaties [78, Proposition 2.5]. Near \( E_s^* \) we use a modification of Melrose’s propagation result [183] for radial sources [78, Proposition 2.6], and near \( E_u^* \), his propagation result for radial sinks [78, Proposition 2.7]. The property (4.18) is crucial here and implies the relation between r and \( {\mathrm{Im}}\,\lambda \) in Theorem 15. Except for the fact that \( E_u^* \) and \( E^*_s \) are typically very irregular as sets, these are the same estimates that Vasy [263] used to prove Theorem 8 presented Sect. 3.1.
Retracing our steps through (4.37), (4.36), (4.35) and (4.29) we see that we proved the theorem of Giulietti–Liverani–Pollicott [109] settling Smale’s conjecture [249, §II.4] in the case of compact manifolds. For the full conjecture in the noncompact case proved by a highly nontrivial elaboration of the above strategy, see Dyatlov–Guillarmou [73]:
Theorem 18
The Ruelle zeta function, \( \zeta _R ( \lambda ) \), defined for \( {\mathrm{Im}}\,\lambda \gg 1 \) by (4.27), continues meromorphically to \( \mathbb {C}\).
What the method does not recover is the order of \( \zeta _R ( \lambda ) \) in the case when M and X are real analytic [94, 228]. That may be related to issues around Conjecture 2.
Conjecture 9
The analytic torsion \( T_\alpha ( \Sigma ) \) was defined by Ray and Singer using eigenvalues of an \( \alpha \)twisted Hodge Laplacian. Their conjecture that \( T_\alpha ( \Sigma ) \) is equal to the Reidemeister torsion, a topological invariant, was proved independently by Cheeger and Müller. Hence (4.41) would link dynamical, spectral and topological quantities. In the case of locally symmetric manifolds a more precise version of the conjecture was recently proved by Shen [237, Theorem 4.1] following earlier contributions by Bismut [22] and Moskovici–Stanton [190].
4.4 Distribution of Pollicott–Ruelle resonances
Exponential decay of correlations (4.3) was established by Dolgopyat [59] for systems which include geodesic flows on negatively curved compact surfaces and by Liverani [168] for all contact Anosov flows, hence in particular for geodesics flows on negatively curved compact manifolds—see (4.8), (4.9). These two papers can also be consulted for the history of the subject. See also Baladi–Demers–Liverani [7] for some very recent progress in the case of hyperbolic billiards.
We have the following result which implies exponential decay of correlations for contact Anosov flows:
Theorem 19
One could estimate s more precisely depending on the width of the strip which would give better decay of correlations results.
The lower bound in (4.44) was proved by Tsujii [261] who improved the result of Liverani at high energies. It also follows from the very general results of [202] which apply also in quantum scattering. The method of [202] is crucial in obtaining the second part of Theorem 16—see also Conjecture 8.
The finiteness of \( \nu _0 \) follows from a stronger statement in Jin–Zworski [153]: for any Anosov flow there exist strips with infinitely many PollicottRuelle resonances.
This state of affairs shows that many problems remain and the noncompact case seems to be completely open.
Footnotes
 1.
That is the flow defined by propagation along straight lines with reflection at the boundary; trapping refers to existence of trajectories which never escape to infinity.
 2.
We do not prove it here but it is a consequence of Rellich’s uniqueness theorem that for V real valued \( R_V ( \lambda ) \) has no poles in \( {\mathbb {R}}\setminus \{ 0 \} \)—see [80, §3.6].
 3.
Our presentation is based on [245, §2] and on an upublished note by Kiril Datchev http://www.math.purdue.edu/~kdatchev/res.ps.
 4.
This is consistent with the expansion (1.2) for the free wave equation: the single pole is “responsible” for the failure of the sharp Huyghens principle in that case.
 5.Small prizes are offered by the author for the first proofs of the conjectures within five years of the publication of this survey: a dinner in a restaurant for an even numbered conjecture and in a restaurant, for an odd numbered one.
 6.
Here we follow the notation of [137, Appendix B] where \( {\bar{\mathcal {C}}^\infty }( M ; V)\) denotes functions \( M \rightarrow V \), which are smoothly extendable across \( \partial M \) and \( {\dot{\mathcal {C}}^\infty }( \overline{M} ; V) \) functions which are extendable to smooth functions supported in \( \overline{M} \).
 7.
A related approach to meromorphic continuation, also motivated by the study of Antide Sitter black holes, was independently developed by Warnick [269]. It is based on physical space techniques for hyperbolic equations and it also provides meromorphic continuation of resolvents for even asymptotically hyperbolic metrics [269, §7.5].
 8.
After this survey first appeared Conjecture 2 was proved by Claude Zuily [275] who used results of Bolley–Camus [24] and Bolley–Camus–Hanouzet [25]. These methods also showed analyticity of radiation patterns of resonant states, F in (3.10), also when the metric is not even. The award of the prize (see footnote 6) took place at http://www.latabledulancaster.fr on November 30, 2016.
 9.
This method is related to a method used in chemistry to compute resonances—see [223] and [235] for the original approach and [145] for some recent developments and references. A simple mathematical result justifying this computational method is given in [283]: if \( V \in L^\infty _{\mathrm{comp}\,} ( {\mathbb {R}}^n ) \) then the eigenvalues of \(  \Delta + V  i \varepsilon x^2 \) converge to resonances of \(  \Delta + V\) uniformly on compact subsets of the region \( \arg z >  \pi /4 \).
 10.
After this survey first appeared Conjecture 3 was proved in the case of finitely generated hyperbolic surfaces by Bourgain–Dyatlov [34]. That means that for convex cocompact surfaces, \( \Gamma \setminus \mathbb {H}^2 \)—see Fig. 19 and [31]—there is a high energy gap for all values, \(0 \le \delta < 1 \), of the dimension of the limit set of \( \Gamma \) – see (3.31). This is the first result about gaps for quantum Hamiltonians for any value of the pressure \( \mathcal {P}_E ( \frac{1}{2}) \) defined in (3.29). The proof is based on the fractal uncertainty principle of [75] and fine harmonic analysis estimates related to the Beurling–Malliavin multiplier theorem, see [175] and references given there.
 11.
I cannot resist recalling that Smale referred to it as a “wild idea” and wrote “I must admit a positive answer would be a little shocking!”
 12.
See [109, Appendix B] for the modifications needed in the nonorientable case.
 13.
There are no nonzero real resonances for Anosov flows with smooth invariant measures and in particular for contact Anosov flows—for a direct microlocal argument for that see [79, Lemma 2.3].
 14.
\( \Omega ( f ) = g \) if it is not true that \( f = o(g)\).
Notes
Acknowledgements
I would like thank my collaborator on [80], Semyon Dyatlov, for allowing me to use some of the expository material from our book, including figures, in the preparation of this survey and for his constructive criticism of earlier versions. Many thanks go also to David Borthwick, Alexis Drouot, Jeff Galkowski, Peter Hintz, Jian Wang and Steve Zelditch for their comments and corrections. I am also grateful to Hari Manoharan, Ulrich Kuhl, Plamen Stefanov, David Bindel, Mickael Chekroun, Eric Heller and David Borthwick for allowing me to use Figs. 1, 2, 8, 11, 13, 14, 19, 21 and 26, respectively. The work on this survey was supported in part by the National Science Foundation grant DMS1500852.
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