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Andrea Mantile

Publications and source records attributed to Andrea Mantile.

At least 19 recordsLinked to original sources
arXiv2026

Resolvent, spectrum and resonances for the acoustic operator with piecewise constant coefficients

We study the acoustic operator $A_{v,ρ}:=v^{2}ρ\nabla\!\cdotρ^{-1}\nabla$ with transmission conditions at the boundary of $Ω=Ω_{1}\cup\dots\cupΩ_{n}$, where the $Ω_{\ell}$'s are connected disjoint open bounded Lipschitz domains, the positive functions $v$ and $ρ$ are constant on each connected component of $Ω$ and $v=ρ=1$ on ${\mathbb R}^{3}\backslash\overlineΩ$. Through a formula for the resolvents difference $(-A_{v,ρ}+z)^{-1}-(-Δ+z)^{-1}$, we provide a Limiting Absorption Principle, determine the spectrum, which turns out to be purely absolutely continuous, and, in the case the connected components of $Ω$ are of class ${\mathcal C}^{1,α}$, characterize the resonance set. The second part of the paper is devoted to the case where $Ω=Ω(\varepsilon)$ is connected with a small size $\varepsilon$ and the $\varepsilon$-analytic functions $v=v(\varepsilon)$ and/or $ρ=ρ(\varepsilon)$ converge to $0_{+}$ inside $Ω(\varepsilon)$ as $\varepsilon\downarrow 0$; there, we provide the analytic $\varepsilon$-expansions of the resonances of $A_{v,ρ}$ according to different choices of the rate of convergence towards zero of the material parameters.

math.AP↗
arXiv2024

The point scatterer approximation for wave dynamics

Given an open, bounded and connected set $Ω\subset\mathbb{R}^{3}$ and its rescaling $Ω_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$ (\varepsilon^{-2}χ_{Ω_{\varepsilon}}+χ_{\mathbb{R}^{3}\backslashΩ_{\varepsilon}})\partial_{tt}u=Δu+f $$ with initial data and source supported outside $Ω_{\varepsilon}$; here, $χ_{S}$ 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^τ),L^{2}(\mathbb{R}^{3})) $-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(Ω)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.

math-ph↗
arXiv2024

Body resonances for classical waves

We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}χ_{Ω_{\varepsilon}}+χ_{Ω^{c}_{\varepsilon}})Δ$ modeling, through the wave equation $(\partial_{tt}+H(\varepsilon))u=0$, the dynamics of acoustic waves in the presence of a small inhomogeneity of size $\varepsilon$ having high contrast $\varepsilon^{-2}$. In particular, we give precise results on the localization of the resonances of $H(\varepsilon)$ and their first-order $\varepsilon$-expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set $Ω$ whose rescaling of size $\varepsilon$ defines $Ω_{\varepsilon}$.

math.SP↗
arXiv2023

Resonances and inverse problems for energy-dependent potentials on the half-line

We consider Schrödinger equations with linearly energy-depending potentials which are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under suitable regularity assumptions. Then, we consider a specific class of energy-dependent Schrödinger equations without eigenvalues, defined with Miura potentials and boundary conditions at the origin. We solve the inverse resonance problem in this case and describe sets of iso-resonance potentials and boundary condition parameters. Our strategy consists in exploiting a correspondance between Schrödinger and Dirac equations on the half-line. As a byproduct, we describe similar sets for Dirac operators and show that the scattering problem for Schrödinger equation or Dirac operator with an arbitrary boundary condition can be reduced to the scattering problem with the Dirichlet boundary condition.

math-ph↗
arXiv2023

Scattering theory with both regular and singular perturbations

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(Δ_B,Δ)$, where $Δ$ is the free self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $Δ_B$ is a self-adjoint operator in a class of Laplacians with both a regular perturbation, given by a short-range potential, and a singular one describing boundary conditions (like Dirichlet, Neumann and semi-transparent $δ$ and $δ'$ ones) at the boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_B$ and a Krein-like formula for the resolvent difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices) and the singular perturbations.

math-ph↗
arXiv2022

Inverse wave scattering in the time domain for point scatterers

Let $Δ_{α,Y}$ be the bounded from above self-adjoint realization in $L^{2}({\mathbb R}^{3})$ of the Laplacian with $n$ point scatterers placed at $Y=\{y_{1},\dots,y_{n}\}\subset{\mathbb R}^{3}$, the parameters $(α_{1},\dotsα_{n})\equivα\in {\mathbb R}^{n}$ being related to the scattering properties of the obstacles. Let $u^{α,Y}_{f_ε}$ and $u^{\varnothing}_{f_ε}$ denote the solutions of the wave equations corresponding to $Δ_{α,Y}$ and to the free Laplacian $Δ$ respectively, with a source term given by the pulse $f_ε(x)=\sum_{k=1}^{N}f_{k}\,φ_ε(x-x_{k}) $ supported in $ε$-neighborhoods of the points in $X_{N}=\{x_{1},\dots, x_{N}\}$, $X_{N}\cap Y=\varnothing$. We show that, for any fixed $λ>\supσ(Δ_{α,Y})$, there exits $N_{\circ}\ge 1$ such that the locations of the points in $Y$ can be determined by the knowledge of the finite-dimensional scattering data operator $F^{N}_λ:{\mathbb R}^{N}\to{\mathbb R}^{N}$, $N\ge N_{\circ}$, $$ (F^{N}_λf)_{k}:=\lim_{ε\searrow 0}\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^{α,Y}_{f_ε}(t,x_{k})-u^{\varnothing}_{f_ε}(t,x_{k})\big)\,dt\,. $$ We exploit the factorized form of the resolvent difference $(-Δ_{α,Y}+λ)^{-1}-(-Δ+λ)^{-1}$ and a variation on the finite-dimensional factorization in the MUSIC algorithm; multiple scattering effects are not neglected.

math-ph↗
arXiv2022

On the Origin of Minnaert Resonances

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size $\varepsilon$), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency $ω_{M}$. Here we explain the origin of such a phenomenon: at first we show that the scattering of an incident wave of frequency $ω$ is described by a self-adjoint $ω$-dependent Schrödinger operator with a singular $δ$-like potential supported at the inhomogeneity interface. Then we show that, in the low energy regime (corresponding in our setting to $\varepsilon\ll1$) such an operator has a non-trivial limit (i.e., it asymptotically differs from the Laplacian) if and only if $ω=ω_{M}$. The limit operator describing the non-trivial scattering process is explicitly determined and belongs to the class of point perturbations of the Laplacian. When the frequency of the incident wave approaches $ω_{M}$, the scattering process undergoes a transition between an asymptotically trivial behaviour and a non-trivial one.

math.AP↗
arXiv2020

Inverse wave scattering in the Laplace domain: a factorization method approach

Let $Δ_Λ\le λ_Λ$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle $Ω$. Let $u^Λ_{f}$ and $u^{0}_{f}$ denote the solutions of the wave equations corresponding to $Δ_Λ$ and to the free Laplacian $Δ$ respectively, with a source term $f$ concentrated at time $t=0$ (a pulse). We show that for any fixed $λ>λ_Λ\ge 0$ and any fixed $B\subset\subset{\mathbb R}^{n}\backslash\barΩ$, the obstacle $Ω$ can be reconstructed by the data $$ F^Λ_λf(x):=\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^Λ_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,. $$ A similar result holds in the case of screens reconstruction, when the boundary conditions are assigned only on a part of the boundary. Our method exploits the factorized form of the resolvent difference $(-Δ_Λ+λ)^{-1}-(-Δ+λ)^{-1}$.

math.AP↗
arXiv2019

Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

This paper concerns the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field. In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are present to confirm the inversion schemes.

math-ph↗
arXiv2019

Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces

We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\widetildeΔ,Δ)$, where $Δ$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetildeΔ$ is one of its singular perturbations, i.e., such that the set $\{u\in H^{2}({\mathbb R}^{3})\cap \text{dom}(\widetildeΔ)\, :\, Δu=\widetildeΔu\}$ is dense. Typically $\widetildeΔ$ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at $Γ=\partialΩ$, where $Ω\subset{\mathbb R}^{3}$ is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on $Σ\subsetΓ$, a relatively open subset with a Lipschitz boundary. We show that either $Γ$ or $Σ$ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.

math.AP↗
arXiv2019

Asymptotic Completeness and S-Matrix for Singular Perturbations

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

math-ph↗
arXiv2018

Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $Σ\subseteqΓ=\partialΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $δ$-type potential supported on the surface $Γ$ and of strength $α\in L^{p}(Γ)$, $p>2$.

math.AP↗
arXiv2017

Adiabatic evolution and shape resonances

Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter $\varepsilon $ with $\ln\varepsilon \asymp -1/h$, where $h$ denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length $\varepsilon ^{-N}$ with an error ${\cal O}(\varepsilon ^N)$. Here $N>0$ is arbitrary.

math-ph↗
arXiv2017

Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

We deal with the time-harmonic acoustic waves scattered by a large number of small holes, of maximal radius $a, a<<1$, arbitrary (i.e. not necessarily periodically) distributed in a bounded part of a homogeneous background. We show that as their number $M$ grows following the law $M:=M(a):=O(a^{-s}), \; a<<1$, the collection of these holes has one of the following behaviors: 1. if $s<1$, then the scattered fields tend to vanish as $a$ tends to zero, i.e. the cluster is a soft one. 2. if $s=1$, then the cluster behaves as an equivalent medium modeled by a refraction index, supported in a given bounded domain $Ω$, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. 3. if $s>1$, then the cluster behaves as a totally reflecting extended body, modeled by a bounded and smooth domain $Ω$, i.e. the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. These approximations are provided with explicit error estimates in terms of $a,\; a<<1$.

math-ph↗
arXiv2017

Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space ${\mathbb{R}}^{n}$. Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, $δ$ and $δ'$-type, either assigned on $Γ$ or on $Σ\subsetΓ$.

math-ph↗
arXiv2016

Time dependent delta-prime interactions in dimension one

We solve the Cauchy problem for the Schrödinger equation corresponding to the family of Hamiltonians $H_{γ(t)}$ in $L^{2}(\mathbb{R})$ which describes a $δ'$-interaction with time-dependent strength $1/γ(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapstoγ(t)$ belongs to the fractional Sobolev space $H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.

math-ph↗
arXiv2015

A linearized model of quantum transport with interface conditions in the adiabatic regime

We introduce a modified model where h-dependent artificial interface conditions, occurring at the boundary of an interaction region, allow to obtain adiabatic approximations for the relevant resonant states connected to the quantum transport problem. Under positivity assumptions on the potential, we show that this modification produces a small perturbation of the dynamics on any time scale. The result extends to the corresponding non-autonomous case, allowing us to consider the adiabatic evolution problem in the modified setting. In this framework, we give an expansion formula for the small-h asymptotics of the adiabatic variable, where the error introduced on the adiabatic dynamics by the interface conditions is polynomially small w.r.t. h, independently from the adiabatic time scale. This result provides with a rigorous mathematical framework for the interface conditions approach to the analysis of the adiabatic transport problem in the quantum wells regime.

math-ph↗