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Applied Mathematics Research eXpress (2009) 2009:1-13, doi:10.1093/amrx/abp003 published on November 6, 2009

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Semiclassical Resolvent Estimates in Chaotic Scattering

Stéphane Nonnenmacher¹ and Maciej Zworski²

¹ Institut de Physique Théorique, CEA/DSM/PhT, Unité de recherche associée au CNRS, 91191 Gif-sur-Yvette, France
² Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, USA

Correspondence: Correspondence to be sent to: snonnenmacher{at}cea.fr

	Abstract

TOP Abstract 1. Statement of Results 2. Review of the... 3. Resolvent Estimates References

 
We prove resolvent estimates for semiclassical operators such as – h² + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h^{– M} in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrödinger propagation and to energy decay of solutions to wave equations.

Received for publication May 4, 2009. Accepted for publication September 9, 2009.

	1. Statement of Results

TOP Abstract 1. Statement of Results 2. Review of the... 3. Resolvent Estimates References

 
In this short note we prove a resolvent estimate in the pole-free strip for operators whose classical Hamiltonian flows are hyperbolic on the sets of trapped trajectories (trapped sets), and the latter are assumed to be sufficiently filamentary—see (1.4) for the precise condition. The proof is based on the arguments of [21] and we refer the reader to Section 3 of that article for the preliminary material and assumptions on the operator.

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 nontrapping 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 Schrödinger equation. These in turn are important in the investigation of nonlinear waves in nonhomogeneous 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).

An example of an operator to which our methods apply is given by the semiclassical Schrödinger operator

(1.1)

is a symmetric positive definite matrix representing a (possibly nontrivial) metric on , , and V(x) is a potential function. We assume that the geometry and the potential are "trivial" outside a bounded region:

This operator is hence associated with a short-range scattering situation. We refer to [21, Section 3.2] for the complete set of assumptions that allow long-range perturbations, at the expense of some analyticity assumptions standard for the definition of resonances; see [25] and references therein. We note that for V – 1, P(h)u = 0 is the Helmholtz equation for a Laplace–Beltrami operator, with h = 1/, playing the rôle of wavelength.

Such operators have a purely continuous spectrum near the origin, and their truncated resolvent (P(h) – z)^–1 () can be meromorphically continued from to , with poles of finite multiplicity called resonances. In the semiclassical limit h << 1, the distribution of resonances depends on the properties of the classical flow generated by the Hamiltonian

that is the flow (x, ) exp t H_p(x, ) associated with the Hamiltonian vector field

(when V – 1 the Hamiltonian flow corresponds to the geodesic flow on .) More precisely, the properties of the resolvent (P(h) – z)^–1 near z = 0 are influenced by the nature of flow on the energy shell { p(x, ) = 0}. A lot of attention has been given to nontrapping flows, that is flows for which the trapped set

(1.2)

is empty. In that case, for > 0 small enough and any C > 0, the resolvent is pole free in a strip [– , ] – i [0, Ch], and satisfies the bound [17, 18]:

On the opposite, there exist cases of "strong trapping" for which the trapped set has a positive volume; resonances can then be exponentially close to the real axis, and the norm of the resolvent be of order e^{C/ h} for z [– , ] [3, 6, 28].

In this note we are considering an intermediate situation, namely, the case where the trapped set (1.2) is a (locally maximal) hyperbolic set. This means that K is a compact, flow-invariant set with no fixed point, such that at any point K the tangent space splits into the neutral (), stable (E^–), and unstable (E⁺) directions:

This decomposition is preserved through the flow. The (un)stable directions are characterized by the following properties:

Such trapped sets are easy to construct. The simplest case consists in a single unstable periodic orbit, but we will rather consider the more general case where K is a fractal set supporting a chaotic flow; such a set contains countably many periodic orbits, which are dense on the set of nonwandering points NW(K) K [15].

Our results will depend on the "thickness" of the trapped set, formulated in terms of a certain dynamical object, the topological pressure. We refer to [21, Section 3.3] and texts on dynamical systems [15, 29] for the general definition of the pressure, recalling only a definition valid in the present case. Let f C⁰(K). Then the pressure of f with respect to the Hamiltonian flow on K is given by

(1.3)

where the sum runs over all periodic orbits of periods T T, and is a point on the orbit . The function f we will be using is a multiple of the (infinitesimal) unstable Jacobian of the flow on K:

We can now formulate our main result:

Theorem. Suppose that P (h) satisfies (1.1) or the more general assumptions of [21, Section 3.2]. Suppose also that the Hamiltonian flow is hyperbolic on the trapped set K, and that the topological pressure

(1.4)

Then for any and > 0, there exist () > 0 and h() > 0such that the cutoff resolvent (P (h) – z)^–1 , , continues analytically to the strip

For , this resolvent is polynomially bounded in h:

(1.5)

For any s [0, 1], the pressure measures relative strengths of the complexity of the flow on K (i.e., the number of periodic orbits), and the instability of the trajectories (through the Jacobian). For s = 0, only measures the complexity, it is the topological entropy of the flow, which is generally positive. On the opposite, is negative, it represents the "classical decay rate" of the flow. The intermediate value can take either sign, depending on the "thickness" of K. In dimension n = 2 the condition (1.4) is equivalent to the statement that the Hausdorff dimension of K p^–1 (0)is less than 2. Since the energy surface p^–1 (0) has dimension 3 and the minimal dimension of a nonempty K is 1, the condition means that we are less than "half-way" and Kis filamentary. Trapped sets with dimensions greater than 2 are referred to as bulky.

The first part of the theorem is the main result of [21], see Theorem 3 there. Here we use the techniques developed in that article to prove (1.5). For the Laplacian outside several convex obstacles on (satisfying a condition guaranteeing strict hyperbolicity of the flow) with Dirichlet or Neumann boundary condition, the theorem was proved by Ikawa [14], with the pressure being only implicit in the statement that gave an explicit condition on distances and sizes of the obstacles. For more recent developments in that setting, see [2, 19, 22].

In particular, for z on the real axis, the bound (1.5) gives

(1.6)

This result was already given in [21, Theorem 5] with a less direct proof. It has been generalized to a larger class of manifolds in [9] and (1.5) provides no new insight in that setting.

One of the applications of (1.6) in the case of the Laplacian is a local smoothing with a minimal loss [7] in the Schrödinger evolution (see [4] for the original application in the setting of obstacle scattering):

One can also deduce from (1.6) a Strichartz estimate [7, 5] useful to prove the existence of solutions for some related semilinear Schrödinger equations.

In the case of the Laplacian (V – 1), the estimate in a strip (1.5) has important consequences regarding the energy decay for the wave equation—see [4, 8, 12] and references given therein. In the odd dimension n 3, it implies that the local energy of the waves decays exponentially in time. The same type of energy decay (also involving a pressure condition) has been recently obtained by Schenck in the setting of the damped wave equation on a compact manifold of negative curvature [24].

To prove (1.5) we use several methods and intermediate results from [21]. Using estimates from [21, Section 7], we show in Section 3 how to obtain a good parametrix for the complex-scaled operator, which leads to an estimate for the resolvent. As was pointed out to us by Burq, the construction of the parametrix for the outgoing resolvent was the, somewhat implicit, key step in the work of Ikawa [14] on the resonance gap for several convex obstacle. That insight led us to reexamine the consequences of [21].

We follow the notation of [21] with precise references given as we go along. For the needed aspects of semiclassical microlocal analysis [21, Section 3] and the references to [10] and [11] should be consulted.

	2. Review of the Hyperbolic Dispersion Estimate

TOP Abstract 1. Statement of Results 2. Review of the... 3. Resolvent Estimates References

 
The central "dynamical ingredient" of the proof is a certain dispersion estimate relative to a modification of P(h), which we will now describe.

The first modification of P (h) comes from the method of complex scaling reviewed in [21, Section 3.4]. For any fixed, sufficiently large R₀ > 0, it results in the operator P(h), with the following properties. To formulate them, put

(2.1)

Then

(2.2)

(2.3)

Here and below we set the following notation for the resolvents:

and (2.3) shows the meromorphic continuation of R (z, h) to , guaranteed by the Fredholm property of P(h) – z.

The operator P(h) is further modified by an exponential weight, G^w = G^w (x, h D),

where > 0 is a fixed small number. The modified operator is obtained by conjugation:

(2.4)

This operator has the same spectrum as P(h)and has the following properties:

(2.5)

(2.6)

The main reason for introducing the weight G is to ensure the bound (2.6). The specific choice of G is explained in [21, Section 6.1]. In particular, Gvanishes in some neighborhood of the trapped set K, and the operator exp ( G^w (x, h D)) is an h-pseudodifferential operator B^w (x, h D), with symbol satisfying

As a result, if the spatial cutoff is supported away from , the calculus of the semiclassical pseudodifferential operators ensures that

(2.7)

From now on, our objective will be to estimate the norm .

We consider a final modification of P_,(h) near the zero energy surface. Let be supported in p^–1 ((– 3 /2, 3 /2) and equal to 1 in p^–1 (– , ). Define

(2.8)

and the associated propagator

(2.9)

The crucial ingredients in proving (1.5) are good upper bounds for the norms

where M > 0 is fixed but large, and

(2.10)

From the bound (2.6) on the imaginary part of , we obviously get an exponential control on the propagator:

(2.11)

The reason to conjugate P with the weight G^w was indeed to ensure this exponential bound. Together with the hyperbolic dispersion bound (2.13), this exponential bound would suffice to get a polynomial bound in (1.5), for some (unknown) L > 0. To obtain the explicit value,

for the exponent, we need to improve (2.11) into the following uniform bound.Lemma 2.1. Let satisfy the conditions (2.10). Then, there exist h₀, C₀ > 0 such that,

(2.12)

Before proving this lemma, we state the major consequence of our dynamical assumptions for the classical flow on K, namely, its hyperbolicity and the "filamentary" nature of K (expressed through (1.4)). It is a hyperbolic dispersion estimate, which was explicitly written only in a model case [21, Proposition 9.1], but can be easily drawn from [21, Proposition 6.3], in the spirit of [21, Section 6.4] As above, we take as in (2.10). For any > 0 we set . For any 0 < h < h(), we then have

(2.13)

The constant M is arbitrarily large, and M₃ can be taken as large as we wish, provided we choose M₁ in (2.1) large enough depending on M. If the pressure is negative, one can take small enough to ensure > /2 > 0. The above estimate is then sharper than (2.12) for times beyond the Ehrenfest time

(2.14)

The large constant M will always be chosen (much) larger than c_E.

Proof of Lemma 2.12. To motivate the proof, we start with a heuristic argument for the bound (2.12). As mentioned above, the exponential bound (2.11) is due to the fact that the imaginary part of can take positive values of order (2.6). However, the construction of the weight Gshows that outside a bounded region of phase space of the form

the imaginary part of is negative up to errors.

The radius R₁ above is large enough, so that V_pos lies at finite distance from the trapped set. As a result, any trajectory crossing the region V_pos will only spend a bounded time in that region. For this reason, the propagator U(t) on a large time t >> 1 will "accumulate" exponential growth only during a uniformly bounded time.

We now provide a rigorous proof, using ideas and results from [21, Section 6.3]. The phase space is split using a smooth partition of unity:

These four functions have specific localization properties:

• for b = 0, 1, 2;
  
• is localized outside p^–1 ((–3/4, 3/4));
  
• ₁ is supported near K, in particular, its support does not intersect V_pos;
  
• ₂is supported away from K but inside {| x| < R₂ + 1};
  
• ₀ is supported near spatial infinity, that is on {| x| > R₂ – 1} where the operator is absorbing(the imaginary part of its symbol is negative).
  

Employing a positive (Wick) quantization scheme (see, for instance, [16], and for the semiclassical setting, [23, Section 3.3]), _b = Op⁺_h(_b), we produce a quantum partition of unity:

The evolution U(t) is then split between time intervals of length t₀, where t₀ > 0 is large but independent of h. Using the partition of unity, we decompose the propagator at time t = Nt₀ into

Expanding the power, we obtain a sum of terms ; to understand each of such term semiclassically, we investigate whether there exist true classical trajectories following that "symbolic history," namely, sitting in at time 0, in at time t₀, etc. up to time Nt₀.

Since the energy cutoffs and have disjoint support, no classical trajectory can spend time in both supports. As a result, any sequence containing at least one index b_i = is irrelevant (meaning that the corresponding term is ) [21, Lemma 6.5].

Since any classical trajectory can travel in at most for a finite time N₀ t₀ before escaping, Lemma 6.6 of [21] shows that the relevant sequences b₁ b_N are of the form

They correspond to trajectories spending most of the time near K. One then has

uniformly for any 2N₀ N < M log (1/ h), where M₅ > 0 is large if the previous M, M_i are.

Finally, using the fact that the weight G vanishes on , [21, Lemma 6.3] shows that

where U₀(t₀) = exp (– it₀ P(h)/ h) is unitary. Hence, , while || U(N₀t₀)|| is estimated using (2.11).

	3. Resolvent Estimates

TOP Abstract 1. Statement of Results 2. Review of the... 3. Resolvent Estimates References

 
We can now prove the resolvent estimate (1.5) by constructing a parametrix for P_,(h) – z, z (h) defined in the statement of the theorem. We will use the notation

to shorten some of the formulas. We want to find an approximate solution to

First, the ellipticity away from the energy surface p^–1 (0) shows that, for as in (2.10), there exists an operator, , such that

To treat the vicinity of p^–1 (0), we put

which satisfies

(3.1)

The estimate (2.13) shows that, if , and for arbitrary M₄ > 0, one can choose M and M₃large enough such that . We can estimate the norm of T₁ by the triangle inequality,

(3.2)

and then use the bounds (2.12) for times 0 t t_Eand (2.13) for times t_E < t t_M.

When , the above integral can be estimated by the integral over the interval t [0, t_E]:

In the case , the dominant part of the integral comes from t = t_E:

We rewrite (3.1) as

From , one can show (as in [21, Lemma 6.5]) that

and also that

Putting T = T₀ + T₁ and R = R₀ + R₁ + R₂, we obtain

This means that (P_,(h) – z)can be inverted, with

The above estimates on the norms of T₀ and T₁ can be summarized by

(3.3)

Using (2.7), this proves the bound (1.5).

Remark. By using a sharper energy cutoff _h belonging to an exotic symbol class (see [27, Section 4]) and supported in the energy layer p^–1 ((– h^1–, h^1–))(as in [1]), the bound (2.13) is likely to be improved to

(3.4)

This bound becomes sharper than (2.12) around the time t'_E = c'_E log (1/ h), where

As a result, the bounds on the norm of the corresponding operator T'₁are modified accordingly. At the same time, as shown in [1, Proposition 5.4], the ellipticity away from the energy surface provides an operator T'₀ satisfying

and of norm . The norm of T' = T'₀ + T'₁is still dominated by that of T'₁, so that we eventually get

Since it is not clear that even this bound is optimal, and that proving (3.4) would require some effort, we have limited ourselves to using the established bound (2.13).

One advantage of the approach presented in this note (compared with the method of [21, Section 9]) is that, to obtain the bound (1.6), we did not have to use the complex interpolation arguments of [4] and [28].

	Acknowledgments

 
In addition to Nicolas Burq, we would like to thank Nalini Anantharaman and Jared Wunsch for helpful discussions related to [21]. This work was supported by the Agence Nationale de la Recherche [ANR-05-JCJC-0107-01, S. N]; and the National Science Foundation [DMS 0654436, M. Z.].

	References

TOP Abstract 1. Statement of Results 2. Review of the... 3. Resolvent Estimates References

 

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