publications

2026

1. arXiv

   Characterization of Dirichlet-to-Neumann maps via the Born approximation (Preprint)

   Carlos Castro, Fabricio Macià, Cristóbal Meroño, and Daniel Sánchez-Mendoza

   2026 arXiv abs bib

   The problem of identifying the set of Dirichlet-to-Neumann (DtN) maps arising from conductivities on a smooth domain, among operators acting on functions on the boundary, is a challenging issue in the mathematical analysis of the Calderón inverse problem. This question is also relevant in specific applications since, as the inverse problem is ill-posed, numerically reconstructing a conductivity from the knowledge of its DtN map is particularly delicate. In this article, we address this issue by proving that any DtN map arising from a radial conductivity in the unit ball of \mathbbR^d admits an exact representation as a linearized DtN map for a uniquely determined integrable function, that we call the Born approximation. This gives a strong necessary condition for an operator to be a DtN map arising from a radial conductivity. In particular, our results are a starting point towards developing a rigorous foundation for numerous linearization-based methods that are commonly used in the numerical solution of the Calderón inverse problem. We also characterize the Born approximation as a solution to a generalized moment problem that is formally well-defined even for non-radial conductivities. We investigate the uniqueness and structure of general non-radial solutions to this moment problem on the unit disk and provide an algorithm to numerically reconstruct the Born approximation in this setting. We provide numerical experiments to test the resolution and robustness of the Born approximation in different situations.

   @misc{Characterization_Born,
     author = {Castro, Carlos and Macià, Fabricio and Meroño, Cristóbal and Sánchez-Mendoza, Daniel},
     title = {Characterization of Dirichlet-to-Neumann maps via the Born approximation (Preprint)},
     year = {2026},
     eprint = {arXiv:2601.11975},
   }

2. Stable factorization of the Calderón problem via the Born approximation

   Thierry Daudé, Fabricio Macià, Cristóbal Meroño, and François Nicoleau

   J. Lond. Math. Soc. (2), 2026 doi arXiv abs

   The Born approximation of a potential in the context of the Calderón inverse problem is an object that can be formally defined in terms of spectral data of the Dirichlet-to-Neumann map of the corresponding Schrödinger operator. In this article, we prove, in the case of radial potentials in the Euclidean ball and any fixed energy, that the Born approximation is well-defined as a compactly supported radial distribution, and that the Calderón problem can be reformulated as recovering a potential from its Born approximation. In addition, we show that the Born approximation depends locally on the potential and captures exactly its singularities, and that the functional that maps the Born approximation to the potential is Hölder continuous. We also prove that the Born approximation converges to the potential in the high-energy limit. Moreover, we give an explicit formula for the Fourier transform of the Born approximation at any fixed energy, and illustrate how it can be used as the basis of an accurate procedure to approximate a potential from its Dirichlet-to-Neumann map.

2025

1. arXiv

   The spectrum of Dirichlet-to-Neumann maps for radial conductivities (Preprint)

   Thierry Daudé, Fabricio Macià, Cristóbal J. Meroño, and François Nicoleau

   2025 arXiv abs bib

   The problem of characterizing sequences of real numbers arising as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has received considerable attention. In this work, the authors study this question for DtN maps associated with rotation-invariant elliptic operators on the Euclidean ball. They show that the spectrum can be decomposed into a universal term determined by the boundary values of the conductivity and a sequence of Hausdorff moments of an integrable function, called the Born approximation of the conductivity. They further prove that the Born approximation is locally determined near the boundary by the conductivity, yielding a local uniqueness result for the Calderón problem. A Hölder stability result is also established for the map recovering the conductivity from its Born approximation in suitable Sobolev spaces. Finally, the regularity properties of the Born approximation and their dependence on the conductivity are investigated.

   @misc{Spectrum_DtN,
     author = {Daudé, Thierry and Macià, Fabricio and Meroño, Cristóbal J. and Nicoleau, François},
     title = {The spectrum of Dirichlet-to-Neumann maps for radial conductivities (Preprint)},
     year = {2025},
     eprint = {arXiv:2510.22585},
   }

2. The Born approximation for the fixed energy Calderón problem

   Fabricio Macià, Cristóbal Meroño, and Daniel Sánchez-Mendoza

   Proc. Roy. Soc. Edinburgh Sect. A., 2025 doi arXiv abs

   The Born approximation of a potential in the context of the Calderón inverse problem is an object that can be formally defined in terms of spectral data of the Dirichlet-to-Neumann map of the corresponding Schrödinger operator. In this article, we prove, in the case of radial potentials in the Euclidean ball and any fixed energy, that the Born approximation is well-defined as a compactly supported radial distribution, and that the Calderón problem can be reformulated as recovering a potential from its Born approximation. In addition, we show that the Born approximation depends locally on the potential and captures exactly its singularities, and that the functional that maps the Born approximation to the potential is Hölder continuous. We also prove that the Born approximation converges to the potential in the high-energy limit. Moreover, we give an explicit formula for the Fourier transform of the Born approximation at any fixed energy, and illustrate how it can be used as the basis of an accurate procedure to approximate a potential from its Dirichlet-to-Neumann map.

2024

1. Solving ill-posed inverse problems via the born approximation

   Fabricio Maciá, and Cristóbal J. Meroño

   In Advances in partial differential equations and control, 2024 doi arXiv abs

2. IPI

   The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

   Juan A. Barceló, Carlos Castro, Fabricio Macià, and Cristóbal J. Meroño

   Inverse Probl. Imaging, 2024 doi arXiv abs bib

   In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball B_R ⊂\mathbbR^3 of radius R , in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for L^∞ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary ∂B_R . We then investigate how increasing the radius R affects the quality of the Born approximation, and the existence of a scattering limit as R\to ∞. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator -∆+q , and strong recovery of singularity results are observed in this case.

   @article{MR4685806,
     author = {Barcel\'o, Juan A. and Castro, Carlos and Maci\`a, Fabricio and Mero\~no, Crist\'obal J.},
     title = {The {B}orn approximation in the three-dimensional {C}alder\'on
                   problem {II}: {N}umerical reconstruction in the radial case},
     journal = {Inverse Probl. Imaging},
     fjournal = {Inverse Problems and Imaging},
     volume = {18},
     year = {2024},
     number = {1},
     pages = {183--207},
     issn = {1930-8337,1930-8345},
     mrclass = {35R30 (35P25 65N21)},
     mrnumber = {4685806},
     doi = {10.3934/ipi.2023029},
     url = {https://doi.org/10.3934/ipi.2023029},
     keywords = {Calderón problem, Born approximation, inverse problems, numerical reconstruction, Dirichlet to Neumann map}
   }

2022

1. JFA

   The Born approximation in the three-dimensional Calderón problem

   Juan A. Barceló, Carlos Castro, Fabricio Macià, and Cristóbal J. Meroño

   J. Funct. Anal., 2022 doi arXiv abs bib

   Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators −Δ+q. We study the Born approximation of q in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension d≥3. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map associated to −Δ+q, which is closely related to a particular moment problem. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis of spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.

   @article{MR4477941,
     author = {Barcel\'o, Juan A. and Castro, Carlos and Maci\`a, Fabricio and Mero\~no, Crist\'obal J.},
     title = {The {B}orn approximation in the three-dimensional {C}alder\'on
                   problem},
     journal = {J. Funct. Anal.},
     fjournal = {Journal of Functional Analysis},
     volume = {283},
     year = {2022},
     number = {12},
     pages = {Paper No. 109681, 43},
     issn = {0022-1236,1096-0783},
     mrclass = {35Q60 (42B10 43A90)},
     mrnumber = {4477941},
     doi = {10.1016/j.jfa.2022.109681},
     url = {https://doi.org/10.1016/j.jfa.2022.109681},
     keywords = {Inverse problems, Calderón problem, Eigenvalues, Born approximation}
   }

2. JDE

   Rotational smoothing

   Pedro Caro, Cristóbal J. Meroño, and Ioannis Parissis

   J. Differential Equations, 2022 doi arXiv abs bib

   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 Rn 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 Rn, or a solution operator for the transport equation in Rn. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.

   @article{MR4332040,
     author = {Caro, Pedro and Mero\~no, Crist\'obal J. and Parissis, Ioannis},
     title = {Rotational smoothing},
     journal = {J. Differential Equations},
     fjournal = {Journal of Differential Equations},
     volume = {306},
     year = {2022},
     pages = {101--151},
     issn = {0022-0396,1090-2732},
     mrclass = {35Q49},
     mrnumber = {4332040},
     doi = {10.1016/j.jde.2021.10.018},
     url = {https://doi.org/10.1016/j.jde.2021.10.018},
     keywords = {Rotational smoothing, Multipliers with singular symbols, Riesz potentials, Cauchy transform, Special orthogonal group}
   }

2021

1. AHP

   The fixed angle scattering problem with a first-order perturbation

   Cristóbal J. Meroño, Leyter Potenciano-Machado, and Mikko Salo

   Ann. Henri Poincaré, 2021 doi arXiv abs bib

   We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and Salo (SIAM J Math Anal 52(6):5467–5499, 2020) and (Inverse Probl 36(3):035005, 2020) to Hamiltonians with first-order perturbations, and it is based on wave equation methods and Carleman estimates.

   @article{MR4325875,
     author = {Mero\~no, Crist\'obal J. and Potenciano-Machado, Leyter and Salo, Mikko},
     title = {The fixed angle scattering problem with a first-order
                   perturbation},
     journal = {Ann. Henri Poincar\'e},
     fjournal = {Annales Henri Poincar\'e. A Journal of Theoretical and
                   Mathematical Physics},
     volume = {22},
     year = {2021},
     number = {11},
     pages = {3699--3746},
     issn = {1424-0637,1424-0661},
     mrclass = {35R30 (81U40)},
     mrnumber = {4325875},
     doi = {10.1007/s00023-021-01081-w},
     url = {https://doi.org/10.1007/s00023-021-01081-w},
     keywords = {Inverse Scattering, Inverse Problems}
   }

2. RMI

   The double dispersion operator in backscattering: Hölder estimates and optimal Sobolev estimates for radial potentials

   Cristóbal J. Meroño

   Rev. Mat. Iberoam., 2021 doi bib

   @article{MR4236805,
     author = {Mero\~no, Crist\'obal J.},
     title = {The double dispersion operator in backscattering: {H}\"older
                   estimates and optimal {S}obolev estimates for radial
                   potentials},
     journal = {Rev. Mat. Iberoam.},
     fjournal = {Revista Matem\'atica Iberoamericana},
     volume = {37},
     year = {2021},
     number = {3},
     pages = {1175--1205},
     issn = {0213-2230,2235-0616},
     mrclass = {35R30 (42B20 81U40)},
     mrnumber = {4236805},
     doi = {10.4171/rmi/1223},
     url = {https://doi.org/10.4171/rmi/1223},
     keywords = {Schrödinger equation, Inverse Scattering, Inverse Problems}
   }

2020

1. SIAM JMA

   The observational limit of wave packets with noisy measurements

   Pedro Caro, and Cristóbal J. Meroño

   SIAM J. Math. Anal., 2020 doi arXiv abs bib

   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.

   @article{MR4165940,
     author = {Caro, Pedro and Mero\~no, Crist\'obal J.},
     title = {The observational limit of wave packets with noisy
                   measurements},
     journal = {SIAM J. Math. Anal.},
     fjournal = {SIAM Journal on Mathematical Analysis},
     volume = {52},
     year = {2020},
     number = {5},
     pages = {5196--5212},
     issn = {0036-1410,1095-7154},
     mrclass = {35S05 (35R30 60H40)},
     mrnumber = {4165940},
     mrreviewer = {Akhilesh\ Prasad},
     doi = {10.1137/20M1324946},
     url = {https://doi.org/10.1137/20M1324946},
     keywords = {Wave Packets, Inverse Problems}
   }

2. Uniqueness for the inverse fixed angle scattering problem

   J.A. Barceló, C. Castro, T. Luque, C. J. Meroño, A. Ruiz, and M. C. Vilela

   J. Inverse Ill-Posed Probl., 2020 doi arXiv bib

   @article{MR4129366,
     author = {Barcel\'o, J.A. and Castro, C. and Luque, T. and Mero\~no, C. J. and Ruiz, A. and Vilela, M. C.},
     title = {Uniqueness for the inverse fixed angle scattering problem},
     journal = {J. Inverse Ill-Posed Probl.},
     fjournal = {Journal of Inverse and Ill-Posed Problems},
     volume = {28},
     year = {2020},
     number = {4},
     pages = {465--470},
     issn = {0928-0219,1569-3945},
     mrclass = {35P25 (35J05 35R30 81U40)},
     mrnumber = {4129366},
     doi = {10.1515/jiip-2019-0019},
     url = {https://doi.org/10.1515/jiip-2019-0019},
     keywords = {Inverse Scattering, Inverse Problems}
   }

3. RMC

   Resolvent estimates for the magnetic Schrödinger operator in dimensions ≥2

   Cristóbal J. Meroño, Leyter Potenciano-Machado, and Mikko Salo

   Rev. Mat. Complut., 2020 doi arXiv abs bib

   It is well known that the resolvent of the free Schrödinger operator on weighted $L^2 spaces has norm decaying like λ^-\frac12 at energy λ. There are several works proving analogous high frequency estimates for magnetic Schrödinger operators, with large long or short range potentials, in dimensions n \ge 3. We prove that the same estimates remain valid in all dimensions n \ge 2$.

   @article{MR4082592,
     author = {Mero\~no, Crist\'obal J. and Potenciano-Machado, Leyter and Salo, Mikko},
     title = {Resolvent estimates for the magnetic {S}chr\"odinger operator
                   in dimensions {$\geq 2$}},
     journal = {Rev. Mat. Complut.},
     fjournal = {Revista Matem\'atica Complutense},
     volume = {33},
     year = {2020},
     number = {2},
     pages = {619--641},
     issn = {1139-1138,1988-2807},
     mrclass = {35J10 (35P20 47A10 58J50)},
     mrnumber = {4082592},
     mrreviewer = {L.\ V.\ Kritskov},
     doi = {10.1007/s13163-019-00316-z},
     url = {https://doi.org/10.1007/s13163-019-00316-z},
     keywords = {Inverse Scattering, Resolvent estimates}
   }

2019

1. JDE

   Recovery of the singularities of a potential from backscattering data in general dimension

   Cristóbal J. Meroño

   J. Differential Equations, 2019 doi arXiv abs bib

   We prove that in dimension n≥2 the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation qB constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform side. We also show that q−qB can be up to one derivative more regular than q in the Sobolev scale. On the other hand, we construct counterexamples showing that in general it is not possible to have more than one derivative gain, sometimes even strictly less, depending on the a priori regularity of q.

   @article{MR3926070,
     author = {Mero\~no, Crist\'obal J.},
     title = {Recovery of the singularities of a potential from
                   backscattering data in general dimension},
     journal = {J. Differential Equations},
     fjournal = {Journal of Differential Equations},
     volume = {266},
     year = {2019},
     number = {10},
     pages = {6307--6345},
     issn = {0022-0396,1090-2732},
     mrclass = {35J10 (35P25 35R30 81U40)},
     mrnumber = {3926070},
     mrreviewer = {Tomio\ Umeda},
     doi = {10.1016/j.jde.2018.11.003},
     url = {https://doi.org/10.1016/j.jde.2018.11.003},
     keywords = {Schrödinger equation, Inverse scattering, Inverse Problems}
   }

2018

1. SIAM JMA

   Fixed angle scattering: recovery of singularities and its limitations

   Cristóbal J. Meroño

   SIAM J. Math. Anal., 2018 doi arXiv abs bib

   We prove that in dimension \n \ge 2 the main singularities of a complex potential \q having a certain a priori regularity are contained in the Born approximation \q_θconstructed from fixed angle scattering data. Moreover, {q-q_θ can be up to one derivative more regular than \q in the Sobolev scale. In fact, this result is optimal. We construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for \n>3 the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.

   @article{MR3867616,
     author = {Mero\~no, Crist\'obal J.},
     title = {Fixed angle scattering: recovery of singularities and its
                   limitations},
     journal = {SIAM J. Math. Anal.},
     fjournal = {SIAM Journal on Mathematical Analysis},
     volume = {50},
     year = {2018},
     number = {5},
     pages = {5616--5636},
     issn = {0036-1410,1095-7154},
     mrclass = {35R30 (35J05 35P25 42B20 81U40)},
     mrnumber = {3867616},
     mrreviewer = {Drossos\ Gintides},
     doi = {10.1137/18M1164871},
     url = {https://doi.org/10.1137/18M1164871},
     keywords = {Schrödinger equation, Inverse scattering, Inverse Problems}
   }