The point scatterer approximation for wave dynamics

Given an open, bounded and connected set $\Omega\subset\mathbb{R}^{3}$ and its rescaling $\Omega_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$ (\varepsilon^{-2}\chi_{\Omega_{\varepsilon}}+\chi_{\mathbb{R}^{3}\backslash\Omega_{\varepsilon}})\partial_{tt}u=\Delta u+f $$ with initial data and source supported outside $\Omega_{\varepsilon}$; here, $\chi_{S}$ denotes the characteristic function of a set $S$. We provide the first-order $\varepsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\infty}((0,1/\varepsilon^{\tau}),L^{2}(\mathbb{R}^{3})) $-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(\Omega)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.


Introduction
Let Ω ⊂ R 3 be open, bounded and connected, y 0 ∈ Ω and let ε ∈ (0, 1) be a small-scale parameter.Denoting with Ω ε the rescaled domain (1.1) Ω ε := {y = y 0 + ε(x − y 0 ) , x ∈ Ω} , we introduce the contrast function modeling a sharp discontinuity of a medium across the interface ∂Ω ε .The perturbed wave equation describes the interaction between a scalar wave and a small inhomogeneity having high contrast with respect to an homogeneous background.Here, the relative speed of propagation inside Ω ε is given by the number ε; hence, small values of ε correspond to a local regime of small relative speed of propagation.We are interested in the asymptotic behavior of the solutions of the Cauchy problem for (1.2) as ε ց 0. Under the time-harmonic assumption u(t, x) = e itω u ω (x), the corresponding stationary equation writes as (1.3) ∆ + (ε −2 χ Ωε + χ R 3 \Ωε ) ω 2 u ω = 0 .
It is known that, depending on the incident frequency ω, the scattering solutions u sc ω of (1.3) may concentrate around Ω ε as ε ց 0. Namely, specific values of ω, referred to as resonant 1 frequencies, are associated to a scattering enhancement localized at y 0 .At these specific frequencies, the dominant part of the scattered field assumes the form (1.4) u sc ω (x) ∼ u in ω (y 0 ) Λ Ω (ω) where Λ Ω (ω) is a scattering coefficient depending on the physical parameters.Such asymptotic, provided in [3] under far-field approximations, describes a diffusion from a point scatterer placed at y 0 .The point scatterer model, introduced by Foldy in [10] and further developed by Lax, [19], consists in using a (1.4)-like formula as an heuristic approximation for the stationary scattering from a small obstacles placed at y 0 .Such an approximation is commonly adopted to describe scattering from small Dirichlet obstacles.An interpretation, discussed in [14], shows that the Foldy-Lax description of a point scatterer corresponds to modeling the interaction between the wave and the scatterer in terms of a point perturbation of the Laplacian, a class of singular perturbations studied in [1], with a frequency-dependent scattering length.It is worth to remark that such model has no direct counterpart in the time domain setting.However, since the point scatterer approximation provides an effective limit for the asymptotic regime at specific frequencies, the asymptotic analysis of the wave dynamics generated by solutions of (1.2) at small scale may suggests a physically coherent definition of a point scatterer in the time domain.
The analysis of the stationary problem (1.3) in the small scale limit ε ց 0 enlightens the emergency of a discrete set of scattering resonances at resonant frequencies ω k related to the inverse spectral points of the Newton potential operator , where λ k ∈ σ d (N 0 ), see [3], [8].This suggests that the generator of the dynamics may have spectral resonances at the energies λ −1 k .Such a spectral problem is considered in [20], where this picture is validated and precise estimates for the imaginary parts of resonances are provided.In particular, under suitable assumptions on Ω, it is shown that at each λ k ∈ σ d (N 0 ) there corresponds a unique resonance of −(ε 2 χ Ωε + χ R 3 \Ωε )∆ converging to λ −1 k with imaginary part of order ε.According to the theory of second-order Cauchy problems with self-adjoint generators, the solutions of (1.2) express in terms of time propagators which are related to the inverse Laplace transform of the resolvent operator (−(ε 2 χ Ωε + χ R 3 \Ωε )∆ + z 2 ) −1 , see, e.g., [6,Sec I.3.14].Hence, the relevant properties of the dynamical system are encoded in the spectral profile of (ε 2 χ Ωε +χ R 3 \Ωε )∆, including eigenvalues and resonances.In view of the results from [20], we expect that in the small-scale regime, the wave dynamics (1.2) may be governed by a (possibly finite) number of resonant states whose survival time, defined by the imaginary part of spectral resonances, is of order 1/ε.In this connection, the role of resonances on the asymptotic dynamics may be relevant on a large time-scale.
In this work we focus on a direct asymptotic analysis of the dynamics generated by solutions of the Cauchy problem for the wave equation (1.2 ).The main results concern both the homogeneous case, considered in Theorem 5.1, and the non-homogeneous case, in Theorem 5.2.In Theorem 5.3 a model for the effective dynamics in the small-scale limit is provided in both cases.We next present an overview of the outcomes for the homogeneous Cauchy problem; more detailed results on such a case as well on the one in presence of a source term are provided in the aforementioned theorems.
Let u ε and u free denote the classical solutions of the Cauchy problems (here and in the following Assume that φ, ψ ∈ H 6 (R 3 ) are supported outside Ω ε and define h : [0, +∞) → R depending on these Cauchy data by Notice that, by Kirchhoff's formula, h(t) = ∆u free (t, y 0 ).Then, for any τ ∈ (0, 1  11 ) there exists ε 0 > 0 small enough such that the expansion holds for all ε ∈ (0, ε 0 ) and the remainder allows the estimate Here, H is the Heaviside function and q(t) := +∞ k=1 q k (t), with each q k (t) solving the Cauchy problem where e k ∈ L 2 (Ω) is an eigenfunction of N 0 corresponding to the eigenvalue λ k .
The effective dynamics given by 0) = ψ , where δ y 0 denotes the Dirac delta distribution supported at the point y 0 .The spherical wave component of u ε,eff has the space profile of a Green function centered at y 0 modulated by a function whose evolution in time is determined by the Cauchy data through the solution of (1.7).
It is worth to remark that (1.5) holds for the specific class of H 6 (R 3 ) Cauchy data with support away from y 0 .This condition is needed to control the high-energy contribution to the inverse Laplace transform involved in the computation of the dynamics.It is unclear if, for less regular data which may concentrate their L 2 -mass close to y 0 , a different asymptotic may occur.
Since the solution t → u ε (t, •) defines a continuous flow in the Sobolev space H 2 (R 3 ) ⊂ C b (R 3 ) and since (1.8) is smooth outside y 0 but neither bounded nor continuous there, the small remainder r ε (t, •) must compensate such loss of regularity close to y 0 .It is worth to remark that this is not in contradiction with the estimates (1.6), however, such circumstance prevents to control the errors on a more regular Sobolev scale.
Although (1.5)-(1.7)does not explicitly involve resonant states, the resonant energies λ −1 k appears in the definition of the modulation functions q k (t).This expansion holds on a large time scale which is, however, shorter than the estimated resonances life-time.We expect that (1.5)-(1.7)can be recasted in terms of quasi-bound states.Let us notice in turns that, representing the quantum and the classical-waves dynamics for point perturbations of the Laplacian in terms of resonant states presents well-known technical asperities (involving a careful estimate of the contributions to the time-propagator from energies close to resonances) and very few general results have been provided in this sense, see [2].The small-scale expansions and the effective model presented in this work are new.In particular, the asymptotic formulae resulting from our analysis are given with global-in-space estimates of the errors on a large time scale.Such formulae provide, in our opinion, a useful contribution in applications involving time domain data.Let us recall, at this concern, that the interaction of classical waves with micro-resonators is attracting the increasing interest of different branches of applied physics, where the manipulation of the dynamics at micro-scales has several relevant applications.For instance, these resonant-scattering phenomena have been used for the design of acoustic and electromagnetic metamaterials, or in the realization of contrast agents for various imaging strategies.Moreover, they show to have a potential impact in classical and quantum information processing.A vast literature has been devoted to these topics, see [11], [13], [18], [21], [24] to cite a few.
While the small-ε asymptotic of the stationary scattering problem (1.3) has been well understood, only partial results have been provided so far for the corresponding dynamics.An attempt to derive a Foldy-Lax model in the time domain has been recently discussed in the two-dimensional case using Dirichlet discs of small size, see [16].As regards the model equation (1.3), a possible approach to time-domain expansions was argued in [3] relying on the previous works [4] and [5,Appendix B].This, however, would produce an expansion only valid after truncating the high-frequency components of the fields.To the best of our knowledge, none explicit formula has been produced within such an approach, neither the relation with the asymptotic expansion of the full time-domain solution has been discussed so far.In [23], the equation (1.2), excited by a source term but with trivial Cauchy data, have been considered in the perspective of using dielectric micro-resonators as contrast agents for imaging in the time domain.The authors use known estimates in the Laplace-transform domain to get the asymptotic expansion of the solution on a finite time interval with a point-wise estimate of the error holding far from the scatterer location, while, it deserves noting, none effective limit model has been discussed.In the limit of their validity, the interesting asymptotic expansions provided in [23] are coherent with the ones provided here, enlightening the role of the Newton eigenvalues in the construction of the dynamics.
Remark 1.1.Without loss of generality, thanks to the translational invariance of the equations, throughout the text we set for simplicity y 0 = 0 in the definition (1.1) of the small domain Ω ε ; the statements of the Theorems 5.1, 5.2 and 5.3 easily adapt to the case of a generic point y 0 .
• • denotes the norm in a space of square integrable functions like L 2 (Ω) and L 2 (R 3 ) with scalar product •, • ; • also denotes the operator norm for bounded linear operators acting between such spaces.Norms in different spaces are specified with the appropriate subscripts.
• B(X, Y ) denotes the Banach space of bounded operators from the Banach space X to the Banach space Y ; B(X, X) ≡ B(X).
• L denotes the Laplace transform.
• H(t) denotes the Heaviside function.
• ̺(A) and σ(A) denote the resolvent set and the spectrum of the self-adjoint operator A; σ d (A) denotes its discrete spectrum.
• H 2k (R 3 ), k ∈ N, denotes the Hilbert-Sobolev space of order 2k, i.e. the space of square integrale functions f : R 3 → C such that ∆ k f is square integrable, ∆ k denoting the k-th power of the Laplacian.

The model operator and the corresponding wave equation
The wave equation (1.2) re-writes as where the linear operator in L 2 (R 3 ) is defined by It corresponds to an additive perturbation of the Laplacian: We introduce the Hilbert space the corresponding scalar product and norm are denoted by •, • ε and • ε respectively; By the inequalities one gets the equivalence, holding in the Banach space sense, where Hence, where the Neumann series on the left converges in B(L 2 (R 3 )) for any z ∈ C\(−∞, 0] such that Re(z) ≥ 0. Furthermore, for such a z, by simple algebraic manipulations, one gets For any self-adjoint operator A and for any z ∈ ̺(A) ∩ C\{0}, one has Iterating, one gets By A(ε)u = ∆u for any u supported outside Ω ε , by (3.4) and by applying (3.8) to both Given a non-positive self-adjoint operator A in a Hilbert space H, we consider the Cauchy problem for the corresponding wave equation, i.e., (3.9) where both the data φ and ψ are in H.We say that u ∈ C(R + ; H) is a mild solution of (3.9) whenever for any t ≥ 0. By [6, Proposition 3.14.4,Corollary 3.14.8and Example 3.14.16], the unique mild solution of (3.9) is given by where the sine and cosine operator functions are defined through the functional calculus for the self-adjoint A. By [12,Theorem 7.4 in Chapter 2, Section 8], if φ ∈ dom(A) and ψ ∈ dom((−A) 1/2 ), then u in (3.10) is a classical solution, i.e., The Laplace transforms of the sine and the cosine functions give the relations Thus, by functional calculus, by the inversion of the Laplace-Stieltjes transform of Lipschitz, where c > 0 and c ′ > 0 are arbitrary.By applying these results to both A = A(ε) and A = ∆ one gets the following ), m ≥ 0 and n > 0, both having support disjoint from Ω ε ; let u ε and u free be the classical solutions of the Cauchy problems Then, where c > 0 and c ′ > 0 are arbitrary.
Therefore, by [17, Theorems 2.25 and 2.29], A(ε) converges to the free Laplacian in norm resolvent sense, i.e., for any z ∈ C\(−∞, 0], there holds If the sine and the cosine operator function of A(ε) had norms in B(L 2 (R 3 )) uniformly bounded with respect to ε, then (3.12) would imply the convergence of the solutions of the Cauchy problem for the wave equation for A(ε) to the ones for ∆, and that would be true for generic initial data (see [12,Theorem 8.6 in Section 8, Chapter 2]).However, by (3.2), these operator norms behave like 1/ε, and so, the convergence of the dynamics generated by A(ε) to the free one is not guaranteed.As we will prove in the following sections, thanks to Lemma 3.5, convergence to the free case surely holds whenever the initial data are supported outside Ω ε .

Operator estimates
In the following, we use the identification provided by the unitary map u → 1 R 3 \Ωε u ⊕ 1 Ωε .Here and below, given the measurable domain D ⊂ R 3 we denote by 1 D : L 2 (R 3 ) → L 2 (D) the bounded linear operator given by the restriction to D; then, the adjoint 1 In such a framework, the free resolvent rewrites as the operator block matrix By the identity making use of (4.1), we get Hence, for any z ∈ C such that z 2 ∈ C\(−∞, 0], introducing the notations This gives Introducing the unitary dilation operator Ωε and then, defining Proof. 4.1.The Newton potential operator of Ω.Now, we introduce the not negative, symmetric operator N 0 in L 2 (Ω) defined by By Sobolev's inequality (see, e.g., [22, Appendix 2 to I.1]) and so N 0 is Hilbert-Schmidt and hence compact (see, e.g., [22,Theorem A.28]).Since −∆N 0 u = u, one gets ker(N 0 ) = {0}.By the spectral theory of compact symmetric operators (see, e.g., [15,Section 6]), σ d (N 0 ) = σ(N 0 )\{0} and the orthonormal sequence {e k } +∞ 1 of eigenvectors corresponding to the discrete spectrum is an orthonormal base of ker(N 0 ) ⊥ = L 2 (Ω).We denote by {λ k } +∞ 1 , the set of eigenvalues (counting multiplicities) indexed in decreasing order.One has λ 1 = N 0 and λ k ց 0. Furthermore, for any u ∈ L 2 (Ω) and for any bounded measurable function f Then, 1 + z 2 N 0 has a bounded inverse and has a bounded inverse and, by the functional calculus for N 0 , .
Proof. 1) 2) By the continuous embedding of H 2 (Ω) into the space of Hölder-continuous functions of order 1  2 , one gets 3) By the inequalities, and by interpolation, one obtains Then, by the Sobolev embedding and by Hölder's inequality with 1 p = s 3 , so that 1 p + 1 q = 1 2 , one gets

Asymptotic dynamics
By (3.4), (4.7) and by the results in the previous section, one has Then, combining the estimates provided in Lemmata 4.2-4.7 with Lemma 3.5, one gets Theorem 5.1.Let φ and ψ be in H 6 (R 3 ) and supported outside Ω ε ; let u ε and u free be the classical solutions of the Cauchy problems Then, for any τ ∈ (0, 1  11 ), τ ′ ∈ (0, 1  9 ) and for any ε > 0 such that where and the e n 's and the λ n 's are the eigenvectors and the eigenvalues of N 0 .
Proof.By Lemma 3.5, there holds, for any φ ∈ H 2(m ′ +1) (R 3 ), m ′ ≥ 0, and where c > 0 and c ′ > 0 are arbitrary.Using Corollary 3.4, one has, for any 1 and, for any 0 ≤ m < m ′ , where one gets φ , Now, we re-write K z 2 (ε) by using (5.1) and provide estimates on the various terms.We begin by considering the evolution of the initial datum ψ.
As regards the inhomogeneous Cauchy problem, one has the following )) be such that f (t) is supported outside Ω ε for any t ∈ R + ; let u ε and u free be the mild (classical if furthermore f ∈ C(R + , H 2 (R 3 )) ) solutions of the inhomogeneous Cauchy problems then, for any τ ∈ 0, and for any ε > 0 such that where and the e n 's and the λ n 's are the eigenvectors and the eigenvalues of N 0 .
If f ∈ L 1 (R + , H 6 (R 3 )), then, for any τ ∈ (0, 1  11 ) and for any ε > 0 as above, sup Proof.By [9, Theorem 4.1 and Lemma 4.2 in Section II.4] one gets the existence of the solutions with the stated regularity and Therefore, by the same reasonings as in the proof of Theorem 5.1,

This gives sup
and so in this case Similarly to the proof of Theorem 5.1, and so, by Fubini's theorem, The proof is then concluded by Lemma 6.3.
where q(t) := ∞ k=1 q k (t) and, for any k ≥ 1, q k (t) solves the Cauchy problem and the e n 's and the λ n 's are the eigenvectors and the eigenvalues of N 0 .Then, there exist τ ∈ (0, 1  11 ) and τ ′′ ∈ (0, 1  2 ) such that Proof.By Theorems 5.1 and 5.2, it suffices to show that, for any k ≥ 1, q k (t) := q 1,k (t) + q k,2 (t) + q k,3 (t) , solves (5.5),where where u ∆ free denotes the solution of the Cauchy problem for the free wave equation with initial data ∆φ, ∆ψ and source ∆f (t); such a solution coincides with ∆(u free + u free ).

5. 1 .Theorem 5 . 3 .
Effective dynamics.In this subsection, φ, ψ and f (t) are as in Theorems 5.1 and 5.2, δ 0 denotes the Dirac delta distribution supported at the origin and Mu(t) denotes the spherical mean of the continuous function u over the sphere {x ∈ R 3 : |x| = t}, i.e. (here σ t denotes the surface measure)Mu(t) := 1 4πt 2 |x|=t u(x) dσ t (x) ;u free and u free denote the solutions of the homogeneous and inhomogeneous Cauchy problem as in Theorems 5.1 and 5.2 respectively.Let u ε and u ε,eff be the solutions of the inhomogeneous Cauchy problems