arXiv subjects

Pedro Caro

Publications and source records attributed to Pedro Caro.

At least 19 recordsLinked to original sources

arXiv2026

The initial-to-final inverse problem for the heat operator

We study an inverse problem for the heat equation in a medium that generates internal heat at a rate proportional to the heat density. The goal consists of determining the heat-generation coefficient $V$ from the knowledge of the initial-to-final map, which assigns to every initial heat density its final heat density. We state and prove a uniqueness result for the heat operator in a region modelled by $\mathbb{R}^n$ with $n \geq 2$. We assume that $V$ is bounded and decays super-exponentially. Our approach relies on constructing exponentially-growing solutions for the corresponding heat operator. A key contribution of our approach is to provide a weighted $L^2$-estimate, which follows from the moment generating function of a Gaussian random variable. This extends the initial-to-final-state inverse problem, previously studied for the Schr\"odinger equation, to the parabolic setting.

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arXiv2026

The initial-to-final-state inverse problem with critically-singular potentials

The Schr\"odinger equation in high dimensions describes the evolution of a quantum system. Assume that we are given the evolution map sending each initial state $f\in L^2(\mathbb{R}^n)$ of the system to the corresponding final state at a fixed time $T$. The main question we address in this paper is whether this initial-to-final-state map uniquely determines the Hamiltonian $-\Delta+V$ that generates the evolution. We restrict attention to time-independent potentials $V$ and show that uniqueness holds provided $V \in L^1(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)$, with $q>1$ if $n=2$ or $q\geq n/2$ if $n\geq 3$. This should be compared with the results of Caro and Ruiz, who proved that in the time-dependent case, uniqueness holds under the stronger assumption that the potential exhibits super-exponential decay at infinity, for both bounded and unbounded potentials. This paper extends earlier work of the same authors, where uniqueness was obtained for bounded time-independent potentials with polynomial decay at infinity. Here we only require $L^1$-type decay at infinity and allow for $L^q$-type singularities. We reach this improvement by providing a refinement of the Kenig-Ruiz-Sogge resolvent estimate, which replaces the classical Agmon-H\"ormander estimates used previously. Crucially, the time-independent setting allows us to avoid the use of complex geometrical optics solutions and thereby dispense with strong decay assumptions at infinity.

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arXiv2025

The initial-to-final-state inverse problem with unbounded potentials and Strichartz estimates

The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. We formulated this problem to establish a theoretical framework that would explain the viability of data-driven prediction in quantum mechanics. In a previous work, we analysed this inverse problem for Hamiltonians of the form $-\Delta + V$ with an electric potential $V = V({\rm t}, {\rm x})$, and we showed that uniqueness holds whenever the potentials are bounded and decay super-exponentially at infinity. In this paper, we extend this result for unbounded potentials. One of the key steps consists in proving a family of suitable Strichartz estimates -- including the corresponding endpoint of Keel and Tao. In the context of the inverse Calder\'on problem this family of inequalities corresponds to the Carleman inequality proved by Kenig, Ruiz and Sogge. Haberman showed that this inequality can be also retrieved as an embedding of a suitable Bourgain space. The corresponding Bourgain space in our context do not capture the mixed-norm Lebesgue spaces of Strichartz inequalities. In this paper, we give a counterexample that justifies this fact, and shows the limitations of Bourgain spaces to address the initial-to-final-state inverse problem.

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arXiv2025

The initial-to-final-state inverse problem with time-independent potentials

The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and motivated by the data-driven prediction problem in quantum mechanics. Caro and Ruiz analysed the question of uniqueness for Hamiltonians of the form $-\Delta + V$ with an electric potential $V = V(\mathrm{t}, \mathrm{x})$ that depends on the time and space variables. In this context, they proved that uniqueness holds in dimension $n \geq 2$ whenever the potentials are bounded and have super-exponential decay at infinity. Although their result does not seem to be optimal, one would expect at least some degree of exponential decay to be necessary for the potentials. However, in this paper, we show that by restricting the analysis to Hamiltonians with time-independent electric potentials, namely $V = V(\mathrm{x})$, uniqueness can be established for bounded integrable potentials exhibiting only super-linear decay at infinity, in any dimension $n \geq 2$. This surprising improvement is possible because, unlike Caro and Ruiz's approach, our argument avoids the use of complex geometrical optics (CGO). Instead, we rely on the construction of stationary states at different energies -- this is possible because the potential does not depend on time. These states will have an explicit leading term, given by a Herglotz wave, plus a correction term that will vanish as the energy grows. Besides the significant relaxation of decay assumptions on the potential, the avoidance of CGO solutions is important in its own right, since such solutions are not readily available in more complicated geometric settings.

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arXiv2024

Quantitative unique continuation for non-regular perturbations of the Laplacian

In this work, we investigate the quantitative estimates of the unique continuation property for solutions of an elliptic equation $\Delta u = V u + W_1 \cdot \nabla u + \hbox{div} (W_2 u)$ in an open, connected subset of $\mathbb{R}^d$, where $d \geq 3$. Here, $V \in L^{q_0}$, $W_1 \in L^{q_1}$, and $W_2 \in L^{q_2}$ with $q_0 > d/2$, $q_1 > d$, and $q_2 > d$. Our aim is to provide an explicit quantification of the unique continuation property with respect to the norms of the potentials. To achieve this, we revisit the Carleman estimates established in [Dehman-Ervedoza-Thabouti-2023] and prove a refined version of them, and we combine them with an argument due to T. Wolff introduced in [Wolff-1992] for the proof of unique continuation for solutions of equations of the form $\Delta u = V u + W_1 \cdot \nabla u$.

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arXiv2024

Reconstruction for the Calder\'on problem with Lipschitz conductivities

We determine the conductivity of the interior of a body using electrical measurements on its surface. We assume only that the conductivity is bounded below by a positive constant and that the conductivity and surface are Lipschitz continuous. To determine the conductivity we first solve an associated integral equation locally, finding solutions in $H^1(B)$, where $B$ is a ball that properly contains the body. A key ingredient is to equip this Sobolev space with an equivalent norm which depends on two auxiliary parameters that can be chosen to yield a contraction.

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arXiv2023

An inverse problem for data-driven prediction in quantum mechanics

Data-driven prediction in quantum mechanics consists in providing an approximative description of the motion of any particles at any given time, from data that have been previously collected for a certain number of particles under the influence of the same Hamiltonian. The difficulty of this problem comes from the ignorance of the exact Hamiltonian ruling the dynamic. In order to address this problem, we formulate an inverse problem consisting in determining the Hamiltonian of a quantum system from the knowledge of the state at some fixed finite time for each initial state. We focus on the simplest case where the Hamiltonian is given by $-Δ+ V$, where the potential $V = V(\mathrm{t}, \mathrm{x})$ is non-compactly supported. Our main result is a uniqueness theorem, which establishes that the Hamiltonian ruling the dynamic of all quantum particles is determined by the prescription of the initial and final states of each particle. As a consequence, one expects to be able to know the state of any particle at any given time, without an a priori knowledge of the Hamiltonian just from the data consisting of the initial and final state of each particle.

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arXiv2022

Magnetic Reconnection in Magnetohydrodynamics

We provide examples of periodic solutions (in both 2 and 3 dimension) of the Magnetohydrodynamics equations such that the topology of the magnetic lines changes during the evolution. This phenomenon, known as magnetic reconnection, is relevant for physicists, in particular in the study of highly conducting plasmas. Although numerical and experimental evidences exist, analytical examples of magnetic reconnection were not known.

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arXiv2021

Rotational smoothing

Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes $k$-vector-space Riesz potentials in $\mathbb{R}^n$ with $k < n$, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on $k$-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in $\mathbb{R}^n$, or a solution operator for the transport equation in $\mathbb{R}^n$. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.

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arXiv2019

The observational limit of wave packets with noisy measurements

The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. The authors show how wave packets can be used to partially recover the observable from the measurements almost surely. Furthermore, they point out the limitation of wave packets to recover the remaining part of the observable, and show how the errors hide the signal coming from the observable. The recovery results are based on an ergodicity property of the errors produced by wave packets.

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arXiv2019

Scattering with critically-singular and $δ$-shell potentials

The authors consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. They study direct and inverse point-source scattering under the assumptions that the potentials are real-valued and compactly supported. To solve the direct scattering problem, the authors introduce two functional spaces ---sort of Bourgain type spaces--- that allow to refine the classical resolvent estimates of Agmon and Hörmander, and Kenig, Ruiz and Sogge. These spaces seem to be very useful to deal with the critically-singular and $δ$-shell components of the potentials at the same time. Furthermore, these spaces and their corresponding resolvent estimates turn out to have a strong connection with the estimates for the conjugated Laplacian used in the context of the inverse Calderón problem. In fact, the authors derive the classical estimates by Sylvester and Uhlmann, and the more recent ones by Haberman and Tataru after some embedding properties of these new spaces. Regarding the inverse scattering problem,the authors prove uniqueness for the potentials from point-source scattering data at fix energy. To address the question of uniqueness the authors combine some of the most advanced techniques in the construction of complex geometrical optics solutions.

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arXiv2019

Determination of convection terms and quasi-linearities appearing in diffusion equations

We consider the highly nonlinear and ill-posed inverse problem of determining some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion equation $\partial_tu-Δ_x u+F(x,t,u,\nabla_xu)=0$ on $Ω\times(0,T)$, with $T>0$ and $Ω$ a bounded open subset of $\mathbb R^n$, $n\geq2$, from measurements of solutions on the lateral boundary $\partialΩ\times(0,T)$. We consider both linear and nonlinear expression of $F(x,t,u,\nabla_xu)$. In the linear case, the equation can be seen as a convection-diffusion equation and our inverse problem corresponds to the unique recovery, in some suitable sense, of a time evolving velocity field associated with the moving quantity as well as the density of the medium in some rough setting described by non-smooth coefficients on a Lipschitz domain. In the nonlinear case, we prove the recovery of more general quasi-linear expression appearing in a nonlinear parabolic equation associated with more complex model. Here the goal is to determine the underlying physical low of the system associated with our equation. In this paper, we consider for what seems to be the first time the unique recovery of a general vector valued first order coefficient, depending on both time and space variable. Moreover, we provide results of full recovery of some general class of quasi-linear terms admitting evolution inside the system independently of the solution from measurements at the boundary. These last results improve earlier works of Isakov in terms of generality and precision. In addition, our results give a partial positive answer, in terms of measurements restricted to the lateral boundary, to an open problem posed by Isakov in his classic book (\textit{Inverse Problems for Partial Differential Equations}) extended to the recovery of quasi-linear terms.

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arXiv2019

Boundary determination of electromagnetic and Lamé parameters with corrupted data

We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

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arXiv2019

Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane

For potentials $V\in L^\infty(\mathbb{R}^2,\mathbb{R})$ and $A\in W^{1,\infty}(\mathbb{R}^2,\mathbb{R}^2)$ with compact support, we consider the Schrödinger equation $-(\nabla +iA)^2 u+Vu=k^2u$ with fixed positive energy $k^2$. Under a mild additional regularity hypothesis, and with fixed magnetic potential $A$, we show that the scattering solutions uniquely determine the electric potential $V$. For this we develop the method of Bukhgeim for the purely electric Schrödinger equation.

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arXiv2018

Correlation imaging in inverse scattering is tomography on probability distributions

Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same data, they have identical laws as function-valued random variables.

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arXiv2017

The Calder\'on problem with corrupted data

We consider the inverse Calder\'on problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such map. This situation corresponds to having access to infinite-precision measurements, which is totally unrealistic. In this paper, we study the Calder\'on problem assuming the data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface. Additionally, we state the rate convergence of the method. Our approach is theoretical and has a stochastic flavour.

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arXiv2016

Inverse scattering for a random potential

In this paper we consider an inverse problem for the $n$-dimensional random Schrödinger equation $(Δ-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential $q$, we uniquely determine the principal symbol of the covariance operator of $q$. Especially, for $n=3$ this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance.

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arXiv2016

Global uniqueness for the Calderón problem with Lipschitz conductivities

We prove uniqueness for Calderón's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.

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