Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ 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 $\Omega$. Let $u^{\Lambda}_{f}$ and $u^{0}_{f}$ denote the solutions of the wave equations corresponding to $\Delta_{\Lambda}$ and to the free Laplacian $\Delta$ respectively, with a source term $f$ concentrated at time $t=0$ (a pulse). We show that for any fixed $\lambda>\lambda_{\Lambda}\ge 0$ and any fixed $B\subset\subset{\mathbb R}^{n}\backslash\overline\Omega$, the obstacle $\Omega$ can be reconstructed by the scattering data operator $$ F^{\Lambda}_{\lambda}f(x):=\int_{0}^{\infty}e^{-\sqrt\lambda\,t}\big(u^{\Lambda}_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,. $$ A similar result holds for point scatterers; in this case, the locations of the of scatterers are determined by an analog of $F^{\Lambda}_{\lambda}$ acting in a finite dimensional space.

Consider an inverse problem of reconstructing a source term from boundary measurements for the wave equation. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann-Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have non-zero averaged eigenfunctions. By introducing a Riesz basis, we reconstruct the wave field, generated before injecting the particles, on the center of the particles. Finally, from these last fields, we reconstruct the source term. A significant advantage of our approach is that we only need the measurements for a single point away from the support of the source.

We propose an approach for the simultaneous reconstruction of the electromagnetic and acoustic material parameters, in the given medium $\Omega$ where to image, using the photoacoustic pressure, measured on a single point of the boundary of $\Omega$, generated by plasmonic nano-particles. We prove that the generated pressure, that we denote by $p^{\star}(x, s, \omega)$, depending on only one fixed point $x \in \partial \Omega$, the time variable $s$, in a large enough interval, and the incidence frequency $\omega$, in a large enough band, is enough to reconstruct both the sound speed, the mass density and the permittivity inside $\Omega$. Indeed, from the behaviour of the measured pressure in terms of time, we can estimate the travel time of the pressure, for arriving points inside $\Omega$, then using the eikonal equation we reconstruct the acoustic speed of propagation, inside $\Omega$. In addition, we reconstruct the internal values of the acoustic Green’s function. From the singularity analysis of this Green’s function, we extract the integrals along the geodesics, for internal arriving points, of the logarithmic-gradient of the mass density. Solving this integral geometric problem provides us with the values of the mass density function inside $\Omega$. Finally, from the behaviour of $p^{\star}(x, s, \omega)$ with respect to the frequency $\omega$, we detect the generated plasmonic resonances from which we reconstruct the permittivity inside $\Omega$.

We consider the wave propagation in the time domain in the presence of small inhomogeneities having high contrast with respect to a homogeneous background. This can be interpreted as a reduced scalar-model for the interaction of an electromagnetic wave with dielectric nanoparticles with high refractive indices. Such composite systems are known to exhibit a transition towards a resonant regime where an enhancement of the scattered wave can be observed at specific incoming frequencies, commonly referred to as body resonances. The asymptotic analysis of the stationary scattering problem, in the high-index nanoparticles regime, recently provided accurate estimates of the resonant frequencies and useful point-scatterer expansions for the solution in the far-field approximation. A key role in this analysis is played by the Newton operator related to the inhomogeneity support, whose eigenvalues identify the inverse resonant energies. We point out that a characterization of such singular frequencies in a proper sense spectral requires the spectral analysis of the Hamiltonian associated to the time-dependent problem. Here we focus on this problem by introducing the scale-dependent Hamiltonian of the time-evolution equation. In this framework, we consider the spectral profile with a particular focus on the generalized spectrum close to the branch-cut. We show that, in this region, the resonances are located in small neighbourhoods of the eigenvalues of the inverse Newton operator and provide accurate estimates for their imaginary parts. In particular, this allows a complete computation of the time-propagator in the asymptotic regime, providing in this way the full asymptotic expansion of the time-domain solution.