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数理解析研究所講究録
第2074巻 2018年 157-160

RESONANCES FOR OBSTACLES IN HYPERBOLIC SPACEPETER HINTZ AND MACIEJ ZWORSKI

We consider scattering by star‐shaped obstacles in hyperbolic space and show thatfor the Dirichlet problem resonances satisfy a universal bound|{\rmIm}$\lambda$|\displaystyle\geq\frac{1}{2}which is optimal in dimension 2. In odd dimensions we also show that|{\rmIm}$\lambda$|\displaystyle\geq\frac{$\mu$}{$\rho$},for a universal constant $\mu$ , where $\rho$ is the radius of a ball containing thc obstacle;
this gives an improvement for small obstacles. That gives lower bounds on the rate of
exponential dccay of waves outsidc of the obstacle.
In dimensions 3 and higher the proofs follow the classical vector field approach of
Morawetz, while in dimension 2 we obtain our bound by working with spaces comingfrom gencral relativity. The lattter approach is inspired by the works of Vasy [Va13]
and Hintz‐Vasy [\mathrm{H}\mathrm{i}\mathrm{V}\mathrm{a}\mathrm{l}5] . We also show that in odd dimcnsions resonances of small
obstacles are close, in a suitable sense, to Euclidean resonances. The full account of
the results in presented in [\mathrm{H}\mathrm{i}\mathrm{Z}\mathrm{w}\mathrm{l}7\mathrm{a}].

For $\kappa$>0 we define hyperbolic n ‐space with constant curvature -$\kappa$^{2} as(\mathbb{H}_{ $\kappa$}^{n}, g_{ $\kappa$})=(\mathbb{R}^{n}, dr^{2}+s_{ $\kappa$}^{2}h) ,(1)where (r, $\omega$) are polar coordinates on \mathbb{R}^{n}, h= h ( $\omega$ , du) is the round metric on \mathbb{S}^{n-1},
and s_{ $\kappa$}(r)=$\kappa$^{-1}\sinh( $\kappa$ r) . We include Euclidean space as the case of $\kappa$=0, s_{0}(r)=r.
Suppose that \mathcal{O} \subset \mathbb{R}^{rb} \simeq \mathb {H}_{$\kap a$}^{n} is a boundcd open set with smooth boundary, anddenote by

P_{ $\kappa$}:=-\displaystyle \triangle_{g_{ $\kappa$} -(\frac{n-1}{2})^{2}$\kappa$^{2}(2)

the self‐adjoint operator on L^{2} ( \mathb {H}_{ $\kap a$}^{n}\backslash \mathcal{O} , dvolg ) with domain$\kappa$\mathcal{D}(P_{ $\kappa$}):=H^{2}(\mathbb{H}_{ $\kappa$}^{n}\backslash \mathcal{O})\cap H_{0}^{1}(\mathbb{H}_{ $\kappa$}^{n}\backslash \mathcal{O}).

The resolvent of P_{ $\kappa$}, $\kappa$>0,

R_{ $\kappa$}( $\lambda$) :=(P_{ $\kappa$}-$\lambda$^{2})^{-1} : L^{2}(\mathb {H}_{ $\kap a$}^{n}\backslash \mathcal{O})\rightar ow L^{2}(\mathb {H}_{ $\kap a$}^{n}\backslash \mathcal{O}) ,{\rm Im} $\lambda$>0 ,continues meromorphically to a family of operators defined on \mathb {C} :R_{ $\kap a$}( $\lambda$):L_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p} ^{2}(\mathb {H}_{ $\kap a$}^{n}\backslash \mathcal{O})\rightar ow L_{1\mathrm{o}\mathrm{c} ^{2}(\mathb {H}_{ $\kap a$}^{n}\backslash \mathcal{O})

.

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PETER HINTZ AND MACIEJ ZWORSKIFIGURE 1. Left: a star‐shaped obstacle in the Poincaré disc with reso‐nances satisfying a universal bound {\rm Im} $\lambda$\leq -\displaystyle \frac{1}{2} . Right: resonances of adisk with radius R=1 in \mathbb{H}^{2} . Highlighted are resonances corresponding
to the angular momentum \ell=12.
For $\kappa$=0 , the same result is true when n is odd; in even dimensions the continuation
takes place on the logarithmic plane.

We denote the set of poles of R_{ $\kappa$}( $\lambda$) (included according to their multiplicities)by {\rm Res}(\mathcal{O}, $\kappa$) . The elements of {\rm Res}(\mathcal{O}, $\kappa$) are called scattering resonances and they
determine decay and oscillations of reflected waves outside of \mathcal{O}- sce [Zw17] for a recent
survey and references. In the odd‐dimensional Euclidean case their study goes backto classical works of Lax‐Phillips [\mathrm{L}\mathrm{a}\mathrm{P}\mathrm{h}68] and Morawetz [\mathrm{M}\mathrm{o}66\mathrm{a}] , and the relation
between the distribution of resonances and the geometry of obstacles has been muchstudied, especially for high energies (|{\rm Re} $\lambda$|\rightarrow\infty)- see [Zw17, §2.4].
When the obstacle is star‐shaped, a universal lower bound on resonance widths,|{\rm Im} $\lambda$| , can be given in terms of the radius of the support of the obstacle. Follow‐

ing earlier contributions of Morawetz [\mathrm{M}\mathrm{o}66\mathrm{a}],[\mathrm{M}\mathrm{o}66\mathrm{b}] ,[Mo72] and using Lax‐Phillips
theory [\mathrm{L}\mathrm{a}\mathrm{P}\mathrm{h}68] , Ralston [Ra78] obtained the bound\displaystyle \mathcal{O}\subset B_{\mathb {R}^{n} (x_{0}, $\rho$) \Rightar ow \inf_{ $\lambda$\in \mathrm{R} $\kap a$(\mathcal{O},0)}|{\rm Im} $\lambda$| \geq$\rho$^{-1}(4)

for odd n\geq 3 . Remarkably this bound is optimal in dimensions three and five—seeFig. 2 and [\mathrm{H}\mathrm{i}\mathrm{Z}\mathrm{w}\mathrm{l}7\mathrm{b}] for a discussion of this result.

In this paper we investigate analogues of (4) for \mathcal{O} \subset B_{\mathbb{H}_{ $\kappa$}^{n} (x_{0}, $\rho$) . The first resultshows that the resonance widths have a universal lower bound independent of thediameter of the obstacle. Intuitively this is due to the fact that infinity is much“larger” in the hyperbolic case.
Theorem 1. Suppose that \mathcal{O}\subset \mathb {H}_{ $\kap a$}^{n} is a star‐shapel obstacle. Then\displaystyle\inf_{$\lambda$\in\mathrm{R}ae\mathrm{s}(\mathcal{O},$\kap a$)}|{\rmIm}$\lambda$|\geq$\kap a$/2 .

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RESONANCES FOR OBSTACLES IN HYPERBOLIC SPACE

Whcn n\geq 3 the proof is based on the vector field method of Morawetz; to obtain anargument valid also when n=2 (where the estimate is sharp when \mathcal{O}=\emptyset ) we use anapproach based on ideas from general relativity and estimates on resonant states. The
hyperbolic spacc version of Morawetz’s estimate for n\geq 3 and a slight refinement of theargument from [\mathrm{M}\mathrm{o}66\mathrm{a}] gives an improvement for small obstacles in odd dimensions;this is due to the sharp Huyghens principle.
Theorem 2. Suppose that \mathcal{O} \subset \mathb {H}_{ $\kap a$}^{n} is a \mathcal{S}tar‐shaped obstacle and that n \geq 3 is odd.
Then\displaystyle \mathcal{O}\subset B_{\mathrm{H}_{ $\kap a$}^{n} (x_{0}, $\rho$) \Rightar ow \inf_{ $\lambda$\in{\rm Res}(\mathcal{O}, $\kap a$)}|{\rm Im} $\lambda$| \geq $\mu \rho$^{-1}(6)

for a universal constant $\mu$.
Remark. Jcns Marklof suggested a formulation of Theorems 1 and 2 which does notdepend on $\kappa$ : there exist constants c_{n} such that for star‐shaped obstacles \mathcal{O}\subset \mathb {H}_{ $\kap a$}^{n}, nodd,\displayst le\mathcal{O}\subsetB_{\mathb {H}_{$\kap a$}^{n}(x_{0}, $\rho$)\Rightarow\inf_{$\lambda$\in{\rmRes}(\mathcal{O},$\kap a$)}|{\rmI }$\lambda$|\geqc_{n}\frac{\mathrm{v}\mathrm{o}1(\parti lB_{\mathb {H}_{$\kap a$}^{n}(0,$\rho$)}{\mathrm{v}\mathrm{o}1(B_{\mathb {H}_{$\kap a$}^{n}(0,$\rho$) }.

FIGURE 2. Left: resonances for the ball of radius one in \mathbb{R}^{3} . For eachspherical momentum \ell they are given by solutions ofwhere

H_{\ell+1/2}^{(2)}( $\lambda$)=0

H_{ $\nu$}^{(2)} is the Hankel function of the second kind and order v . Each zeroappears as a resonance of multiplicity 2\ell+1 ; highlighted are resonances
12 . Right: resonances of thc ball with radiuscorresponding to \ell=

R=0.25 in \mathbb{H}^{3} (red) and in \mathbb{R}^{3} (blue); this illustrates Theorem 3.

We expect that $\mu$=1 in (6). (An adaptation of Ralston’s argument [Ra78] should
work but would require some buildup of scattering theory; for a proof of his crucial
estimate without using Lax‐Phillips theory, see [ \mathrm{D}_{\mathrm{c}\mathrm{v}\mathrm{Z}\mathrm{w} , Exercise 3.5].) That the esti‐

matc (6) is indcpcndent of $\kappa$ is related to rescaling: identifying an obstacle with a subset
of \mathbb{R}^{n} and denoting by x\mapsto $\varepsilon$ x the Euclidean dilation, we see that if $\sigma$ \in {\rm Res}(6\mathcal{O}, 1)
then $\varepsilon \sigma$ \in {\rm Res}(\mathcal{O}, \in) , and $\Xi \sigma$ should be close to a resonance in {\rm Res}(\mathcal{O}, 0) . So even

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though the bound (5) gets worse for small $\kappa$ , the bound in odd dimensions is close to(4) and improves for small diameters. This is illustrated by Fig. 2 and confirmed by
the following theorem:
Theorem 3. Suppose that \mathcal{O}\subset \mathbb{H}_{ $\kappa$}^{n}\simeq \mathbb{R}^{n} is an arbitrary bounded obstacle with smoothboundary and that n\geq 3 is odd. Then{\rm Res}(\mathcal{O}, $\kappa$)\rightarrow{\rm Res}(\mathcal{O}, 0) , $\kappa$\rightarrow 0,
locally uniformly and with multiplicities.
REFERENCES[DyZw] S. Dyatlov and M. Zworski, Mathematical theory of scattering resonances, book in prepara‐tion; http: // math. mit. \mathrm{e}\mathrm{d}\mathrm{u}/\sim \mathrm{d}\mathrm{y}\mathrm{a}\mathrm{t}\mathrm{l}\mathrm{o}\mathrm{v}/\mathrm{r}\mathrm{e}\mathrm{s}/[HiVa15] P. Hintz and A. Vasy. Asymptotics for the wave equation on differential forms on Kerr‐deSitter space. arXiv:1502.03179.

[HiZw17a] P. Hintz and M. Zworski, Resonances for obstacles in hyperbolic space, arXiv:l703.01384.
[HiZw17b] P. Hintz and M. Zworski, Wave decay for star‐shaped obstacles in \mathbb{R}^{3} : papers of Morawetz
and Ralston revisited, arXiv:1703.01389, to appear in Proc. Roy. Irish Acad. Sect. A.

[LaPh68] P. D. Lax and R. S. Phillips, Scattering theory, Academic Press 1968.
[Mo66a] C. Morawetz, Exponential decay of solutions of the wave equation, Comm. Pure Appl. Math.
19(1966), 439‐444.
[Mo66b] C. Morawetz, Energy Identities for the Wave Equation, New York Univ., Courant Inst. Math.
Sci., Res. Rep. No. IMM346,1966, https: //archive. org/details/energyidentities00mora[Mo72] C. Morawetz, On the modes of decay for the wave equation in the exterior of a reflecting body,
Proc. Roy. Irish Acad. Sect. A 72(1972), 113‐120.
[Ra78] J. Ralston, Addendum to: “The first variation of the scattering matrix” (J. Differential Equa‐tions 21(1976), no. 2, 378−394) by J. W. Helton and J. Ralston. J. Differential Equations 28(1978),155−162.

[Va13] A. Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr‐de Sitter spaces, with anappendix by Semyon Dyatlov, Invent. Math. 194(2013), 381‐513.
[Zw17] M. Zworski, Mathematical study of scattering resonances, Bull. Math. Sci. 7(2017), 1‐85.
E‐mail address:

phintz@berkeley.edu

E‐mail address:

zworski@math.berkeley.edu

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALiFoRNiA, BERKELEY, CA 94720, USA