The polynomial estimate on the **resolvent** **in** the pole free strip below the real axis (1.5) provides a direct proof of the estimate on the real axis (1.6), and that estimate is only logarithmically weaker than the similar bound **in** the non-trapping case (that is, the case where all classical trajectories escape to infinity). Through an argument going back to Kato, and more recently to Burq, that estimate is crucial for obtaining local smoothing and Strichartz **estimates** for the Schr¨odinger equation. These **in** turn are important **in** the investigation of nonlinear waves **in** non-homogeneous trapping media. Also, as has been known since the work of Lax-Phillips, the estimate **in** the complex domain is useful for obtaining exponential decay of solutions to wave equations (see the paragraph following (1.6) for some references to recent literature).

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1 Introduction and main results
The long range **scattering** theory of the Laplace-Beltrami operator on asymptotically Euclidean or conical manifolds has been widely studied. It has reached a point where our global understanding of the spectrum, **in** particular the behaviour of the **resolvent** at low, medium and high frequencies, allows to extend to curved settings many results which are well known on R n . We have typically **in** mind global **in** time Strichartz **estimates** [33, 22, 26, 18, 36] or various instances of the local energy decay [2, 4, 34, 35, 6, 32, 8] which are important tools **in** nonlinear PDE arising **in** mathematical physics. We refer to the recent paper [32] which surveys **resolvent** **estimates** (or limiting absorption principle) and some of their applications **in** this geometric framework.

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1. Introduction and statement of results
The purpose of this paper is to extend the **semiclassical** **resolvent** **estimates** obtained recently **in** [8], [12], [13] and [14] for the Schr¨ odinger operator **in** the Euclidean space R n to a large class of non-compact, connected Riemannian manifolds (M, g), n = dim M ≥ 2, with a smooth, compact boundary ∂M (which may be empty) and a smooth Riemannian metric g. We will consider manifolds of the form M = X ∪ Y , where X is a compact, connected Riemannian manifold with boundary ∂X = ∂M ∪ ∂Y , while Y is of the form Y = [r 0 , ∞) × S with metric

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1. Introduction
Motivated by earlier works of Lebeau on the asymptotic properties of the damped wave equation [22], Sj¨ostrand initiated **in** [31] the spectral study of this partial differential equa- tion on compact Riemannian manifolds. He proved that eigenfrequencies verify a Weyl asymptotics **in** the high frequency limit [31, Th. 0.1] – see also [25, 26] for earlier related contributions of Markus and Matsaev. Moreover, he showed that eigenfrequencies lie **in** a strip of the complex plane which can be completely determined **in** terms of the average of the damping function along the geodesic flow [31, Th. 0.0 and 0.2] – see also [22, 28]. Following [31], showing these results turns out to be the particular case of a more system- atic study of a nonselfadjoint **semiclassical** problem which has since then been the object of several works. More precisely, it was investigated how these generalized eigenvalues are asymptotically distributed inside the strip determined by Sj¨ostrand and how the dynamics of the underlying classical Hamiltonian influences this asymptotic distribution. Mostly two questions have been considered **in** the literature. First, one can ask about the precise distribution of eigenvalues inside the strip and this question was addressed both **in** the completely integrable framework [12, 13, 14, 19, 15, 16, 17, 18] and **in** the **chaotic** one [1]. Second, it is natural to focus on how eigenfrequencies can accumulate at the boundary of the strip and also to get **resolvent** **estimates** near the boundary of the strip. Again, this question has been explored both **in** the integrable case [4, 13, 6, 2, 5] and **in** the **chaotic** one [7, 30, 27, 8, 29, 20].

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h → 0. Indeed, the fractal character of the trapped set has a strong influence on the **semiclassical** density of eigenvalues: any point situated at distance ≫ h 1/2
from Γ will be pushed out of B(0, R 1 ) through the classical dynamics (either **in** the
past or **in** the future), within a time |n| ≤ C log(1/h), where **semiclassical** methods still apply. As a result, the eigenstates of M (T, h) associated with nonnegligible

0
V 2 (x(t, w)) dt > 0. (1.5)
This condition is **in** particular satisfied when V 2 > 0 and ( 1.4 ) holds. From this point of view,
the results we are going to prove here are stronger than those given **in** the dissipative setting. **In** this setting we cannot use the dissipative version of Mourre’s commutators method. We use the same approach as **in** [AK07] instead. The idea is due to G. Lebeau [Leb96] and N. Burq [Bur02]. It is a contradiction argument. We consider a family of functions which denies the result, a **semiclassical** mesure associated to this family and finally we prove that this measure is both zero and non-zero. This idea was used **in** [Bur02] for a general self-adjoint and com- pactly supported perturbation of the laplacian. **In** [Jec04], Th. Jecko used the argument to give a new proof of ( 1.2 ) with a real-valued potential. The motivation was to give a proof which could be applied to matrix-valued operators. To allow long range potentials, the author intro- duced a bounded “escape function” which we use here. The method was then used **in** [CJK08] for a potential with Coulomb singularities and **in** [Jec05, FR08, DFJ09] for a matrix-valued operator. Let us now state the main results about the **resolvent**. An important difference with the dissipative case is that we do not know if the **resolvent** is well-defined, even on the upper half- plane C + . So **in** the results we state now, we first claim that H h has no eigenvalue **in** the

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hal-00090428, version 3 - 23 Apr 2007
ST´ EPHANE NONNENMACHER AND MATHIEU RUBIN
Abstract. We study the spectrum of quantized open maps, as a model for the reso- nance spectrum of quantum **scattering** systems. We are particularly interested **in** open maps admitting a fractal repeller. Using the “open baker’s map” as an example, we nu- merically investigate the exponent appearing **in** the Fractal Weyl law for the density of resonances; we show that this exponent is not related with the “information dimension”, but rather the Hausdorﬀ dimension of the repeller. We then consider the **semiclassical** measures associated with the eigenstates: we prove that these measures are conditionally invariant with respect to the classical dynamics. We then address the problem of clas- sifying **semiclassical** measures among conditionally invariant ones. For a solvable model, the “Walsh-quantized” open baker’s map, we manage to exhibit a family of **semiclassical** measures with simple self-similar properties.

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ccsd-00005445, version 2 - 20 Jun 2005
**SCATTERING**
ST´ EPHANE NONNENMACHER AND MACIEJ ZWORSKI
Abstract. We analyze simple models of quantum **chaotic** **scattering**, namely quantized open baker’s maps. We numerically compute the density of quantum resonances **in** the **semiclassical** r´egime. This density satisﬁes a fractal Weyl law, where the exponent is gov- erned by the (fractal) dimension of the set of trapped trajectories. This type of behaviour is also expected **in** the (physically more relevant) case of Hamiltonian **chaotic** **scattering**. Within a simpliﬁed model, we are able to rigorously prove this Weyl law, and compute quantities related to the “coherent transport” through the system, namely the conductance and “shot noise”. The latter is close to the prediction of random matrix theory.

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the values of V (x) **in** {x ∈ IR n : V (x) ≤ λ}. It follows that if we modify the potential
V outside the classical allowed region, the **scattering** amplitude remains unchanged up to
O(h ∞ ).
It seems to be difficult to prove our conjecture for a fixed energy without assuming the non-trapping condition. The difficulty comes from the estimate of the **resolvent** R(λ + i0, h) due to resonances converging exponentially to the real axis. To avoid the problem due to these resonances, one can average with respect to the energy λ. **In** [16], Yajima obtains,

[7] F. Klopp and M. Vogel, On **resolvent** **estimates** and resonance free regions for **semiclassical** Schr¨ odinger operators with bounded potentials, Pure and Applied Analysis, to appear.
[8] J. Shapiro, **Semiclassical** **resolvent** bounds **in** dimension two, Proc. Amer. Math. Soc., to appear. [9] J. Shapiro, **Semiclassical** **resolvent** bound for compactly supported L ∞

THE POTENTIAL
IVANA ALEXANDROVA, JEAN-FRANC ¸ OIS BONY, AND THIERRY RAMOND
Abstract. We study the **scattering** amplitude for Schr¨ odinger operators at a critical energy level, which is a unique non-degenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results of [5] to analyze the contributions of the trapped trajectories. We prove a **semiclassical** expansion of the **scattering** amplitude and compute its leading term. We show that it has different orders of magnitude **in** specific regions of phase space. We also prove upper and lower bounds for the **resolvent** **in** this setting.

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are **semiclassical**-conormal to the diagonal at h = 0. As P (h, σ) is not semiclassically elliptic (but is elliptic **in** the usual sense, so it behaves as a **semiclassical** pseudodifferential operator of order −∞ for our purposes), **in** order to construct a parametrix for P (h, σ), we need to follow the flow out of **semiclassical** singularities from the conormal bundle of the diagonal. For P (h, σ) as above, the resulting Lagrangian manifold is induced by the geodesic flow, and is **in** particular, up to a constant factor, the graph of the differential of the distance function on the product space. Thus, it is necessary to analyze the geodesic flow and the distance function; here the presence of boundaries is the main issue. As we show **in** Section 2, the geodesic flow is well-behaved on B n+1 ×0 B n+1 as a Lagrangian manifold of the appropriate cotangent bundle. Further, for δ > 0 small (this is where the smallness of the metric perturbation enters), its projection to the base is a diffeomorphism, which implies that the distance function is also well-behaved. This last step is based upon the precise description of the geodesic flow and the distance function on hyperbolic space, see Section 2.

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[11].
The assumption that n = 1 outside some compact is only necessary to study the low-frequency behavior of the cut-off **resolvent** of G. Indeed, under this assumption one can easily see that this behavior is exactly the same as **in** the case when n ≡ 1, which **in** turn is well-known (e.g. see Appendix B.2 of [1]). Therefore, **in** this case the low-frequency analysis can be carried out **in** precisely the same way as **in** [9]. Most probably, the condition (1.12) with δ > 2 would be enough. The high-frequency analysis **in** our case is also very similar to that one **in** [9] with some slight modifications allowing to deduce from (1.13) the sharp decay rate ω(t) (instead of (log t) −3/4+ǫ ).

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sure dν, and to get precise informations on the speed of convergence. Taking dν(y) = δ x (y), it
turns out that this is equivalent to study the convergence of t n
h (x, dy) towards dν h,∞ . Observe
that **in** the present setting, proving pointwise convergence (h being fixed) of t n h (x, dy) towards the invariant measure is an easy consequence of some general theorem (see [8], Theorem 2, p272). The interest of our approach is to get convergence **in** a stronger topology and to obtain precise information on the behavior with respect to the **semiclassical** parameter h.

We have also extended our solution to include the limit at which the high-frequency field completely destroys the tra- jectories of the field-free system. This allows us to show that for Rydberg atoms two different mechanisms of atomic stabilization—the so-called ‘‘interference’’ @4# and ‘‘adia- batic’’ @5# stabilization models—are two limits of the same expression. As the field increases, the approximately con- stant ionization rate g ;1/n 3 , found **in** the region of interfer- ence stabilization, gives way to a decreasing g as the ampli- tude of electron oscillations **in** the external field approaches the characteristic size n 2 of the Kepler orbit.

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The localization of light along the 2-bounce PO clearly demonstrates that the scar modes have been excited and enhanced thanks to the interaction with the doped region.
4 Conclusion
The optical fiber with a truncated cross section appears to be a good candidate to image experi- mentally wave chaos: complementary informations about the field that propagates **in** the **chaotic** billiard are deduced from the NF and FF intensity. The ergodic behavior of generic modes has been proved experimentally and scar modes have first been observed **in** an optical fiber. Due to the inescapable diffraction at the fiber input, the properties of individual scar modes cannot be studied experimentally. Nevertheless, their signature on the probability distribution of the intensity appears to be pregnant: our preliminary investigations have shown that deviations to the ergodic theory are to be expected as soon as a scar is excited on a superposition of modes. To perform a selective excitation of scar modes **in** our multimode structure, we have investi- gated the influence of a localized gain area on the propagation of a plane-wave illumination. Our numerical results clearly show that scar modes are selectively amplified. The experimental validation is **in** progress and will be the subject of a next publication.

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V. CONCLUSION
We have developed a **semiclassical** description that accu- rately captures the properties of quantum spin ice at large length scales and allows for a systematic, perturbative expan- sion, which **in** principle can be truncated at arbitrary order. **In** particular, we have computed the speed of light to first order **in** the expansion parameter (the inverse of the spin size), and the ground-state energy to second order. Our results are **in** good quantitative agreement with recent numerical calculations **in** Ref. [ 13 ]. We find that Hartree-Fock corrections, due to photon-photon interactions, that go beyond the quadratic U (1) lattice gauge theory, significantly renormalize the speed of light and give rise to a small splitting **in** the energy of the two photon modes at intermediate wave vectors.

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which, together with several density arguments (to be invoked at different stages of the formal expansion), is enough to justify the analysis given below.
Theorem 1.7 provides an approximate description of the solution to (1.1) up to Ehrenfest time and can be seen as the analog of the results given **in** [16, 27, 8, 9, 18, 34, 35, 38] where the case of slowly varying potentials V (x) is considered. The proof does not rely on the use of pseudo-differential calculus or space space- adiabatic perturbation theory and can thus be considered to be considerably simpler from a mathematical point of view. **In** fact, our approach is similar to the one given **in** [18], which derives an analogous result for the so-called Born-Oppenheimer approximation of molecular dynamics. Note however, that we allow for more general initial amplitudes, not necessarily Gaussian. Indeed, **in** the special case where the initial envelope u 0 is a Gaussian, then its evolution u remains Gaussian, and can

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DOI: 10.1103/PhysRevB.91.064302 PACS number(s): 42.65.Ky, 42.50.Hz, 78.47.−p, 72.20.Ht
I. INTRODUCTION
The interaction of intense laser fields with solid matter has been studied since the advent of nonlinear optics. Most of it happens **in** the perturbative realm, and the major outcome of such interactions is the parametric conversion of a few pump laser photons to a single photon of higher energy, i.e., a radiation field with frequency being a harmonic of the laser frequency is created [ 1 ]. The process of harmonic generation has been revolutionized by the availability of more intense lasers capable of converting tens to hundreds of photons to produce coherent soft x-ray radiation. This phenomenon, dubbed high-order harmonic generation (HHG) [ 2 , 3 ], operates **in** a highly nonperturbative regime that cannot be understood with conventional nonlinear optics techniques. So far, research has focused on HHG **in** atomic and molecular gases. Only recently, high harmonics have been measured from the bulk of semiconductor crystals thereby opening the way to the study of nonperturbative optical phenomena **in** the condensed matter phase [ 4 – 6 ]. Experiments have been performed **in** two different wavelength domains, for mid-infrared (IR) [ 4 ] and for far-IR [terahertz (THz)] driving wavelengths [ 5 , 6 ]. Further, HHG has been studied via single- and two-color experiments. HHG oc- curs when electron-hole pairs are generated and subsequently accelerated by the same intense pump field [ 4 , 6 ]. **In** two-color experiments [ 5 ], conducted on semiconductor heterostruc- tures, bound excitons are created by resonant excitation with a weak near-infrared field, followed by high harmonic sideband generation driven by THz fields. Our analysis focuses on single-color HHG **in** semiconductor bulk materials.

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