arXiv subjects

Fabricio Macià

Publications and source records attributed to Fabricio Macià.

At least 19 recordsLinked to original sources

arXiv2026

Stable factorization of the Calderón problem via the Born approximation

In this article, we prove the existence of the Born approximation in the context of the radial Calderón problem for Schrödinger operators. The Born approximation naturally appears as the linear component of a factorization of the Calderón problem; we show that the non-linear part, obtaining the potential from the Born approximation, enjoys several interesting properties. First, this map is local, in the sense that knowledge of the Born approximation in a neighborhood of the boundary is equivalent to knowledge of the potential in the same neighborhood, and, second, it is Hölder stable. This proves that the ill-posedness of the Calderón problem arises from the linear step, which consists in computing the Born approximation from the DtN map by solving a Hausdorff moment problem. Moreover, we present an effective algorithm to compute the potential from the Born approximation. Finally, we use the Born approximation to obtain a partial characterization of the set of DtN maps for radial potentials. The proofs of these results do not make use of Complex Geometrical Optics solutions or their analogs; they are based on results from inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of $A$-amplitude introduced by Barry Simon.

math.AP↗

arXiv2026

Mathematical analysis of one-layer neural network with fixed biases, a new activation function and other observations

We analyze a simple one-hidden-layer neural network with ReLU activation functions and fixed biases, with one-dimensional input and output. We study both continuous and discrete versions of the model, and we rigorously prove the convergence of the learning process with the $L^2$ squared loss function and the gradient descent procedure. We also prove the spectral bias property for this learning process. Several conclusions of this analysis are discussed; in particular, regarding the structure and properties that activation functions should possess, as well as the relationships between the spectrum of certain operators and the learning process. Based on this, we also propose an alternative activation function, the full-wave rectified exponential function (FReX), and we discuss the convergence of the gradient descent with this alternative activation function.

cs.LG↗

arXiv2026

Characterization of Dirichlet-to-Neumann maps via the Born approximation

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 $\mathbb{R}^d$ 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.

math.NA↗

arXiv2026

The spectrum of Dirichlet-to-Neumann maps for radial conductivities

The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this question in the simple setting of DtN maps associated with a rotation-invariant elliptic operator $\nabla \cdot (γ\nabla \centerdot )$ in the ball in Euclidean space. We show that the spectrum of such a DtN operator can be expressed as a universal term, determined solely by the boundary values of the conductivity $γ$, plus a sequence of Hausdorff moments of an integrable function, which we call the Born approximation of $γ$. We also show that this object is locally determined from the boundary by the corresponding values of the conductivity, a property that implies a local uniqueness result for the Calderón Problem in this setting. We also give a stability result: the functional mapping the Born approximation to its conductivity is Hölder stable in suitable Sobolev spaces. Finally, in order to refine the characterization of the Born approximation, we analyze its regularity properties and their dependence on the conductivity.

math.AP↗

arXiv2026

The Born approximation for the fixed energy Calderón problem

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.

math.AP↗

arXiv2022

Concentration of Quasimodes for Perturbed Harmonic Oscillators

In this work, concentration properties of quasimodes for perturbed semiclassical harmonic oscillators are studied. The starting point of this research comes from the fact that, in the presence of resonances between frequencies of the harmonic oscillator and for a generic bounded perturbation, the set of semiclassical measures for quasimodes of sufficiently small width is strictly smaller than the corresponding set of semiclassical measures for the unperturbed system. In this work we precise the description of this set taking into account the Diophantine properties of the vector of frequencies, the separation between clusters of eigenvalues in the spectrum produced by the presence of a perturbation, and the dynamical properties of the classical Hamiltonian flow generated by the average of the symbol of the perturbation by the harmonic oscillator flow. In particular, for the perturbed two-dimensional periodic harmonic oscillator, we characterize the set of semiclassical measures of quasimodes with a width that is critical for this problem. Two of the main ingredients in the proof of these results, which are of independent interest, are: (i) a novel construction of quasimodes based on propagation of coherent states by several commuting flows and (ii) a general quantum Birkhoff normal form for harmonic oscillators.

math.AP↗

arXiv2022

The Born approximation in the three-dimensional Calderón problem

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 $d\ge 3$. 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 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.

math.AP↗

arXiv2022

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

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 \subset \mathbb{R}^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 $L^{\infty}$ conductivities, and we 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 $\partial B_R$. We then investigate how increasing the radius $R$ affects the quality of the Born approximation, and the existence of a scattering limit as $R\to \infty$. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $-Δ+q$, and strong recovery of singularity results are observed in this case.

math.AP↗

arXiv2022

Effective mass theorems with Bloch modes crossings

We study a Schr{ö}dinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wavelength comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

math.AP↗

arXiv2020

Localization and delocalization of eigenmodes of Harmonic Oscillators

We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.

math.AP↗

arXiv2020

Observability results related to fractional Schrödinger operators

We establish observability inequalities for various problems involving fractional Schrödinger operators $(-Δ)^{α/2}+V$, $α>0$, on a compact Riemannian manifold. Observability from an open set for the corresponding fractional Schrödinger evolution equation with $α>1$ is proved to hold as soon as the observation set satisfies the Geometric Control Condition; it is also shown that this condition is necessary when the manifold is the $d$-dimensional sphere equipped with the standard metric. This is in stark contrast with the case of eigenfunctions. We construct potentials on the two-sphere with the property that there exist two points on the sphere such that eigenfunctions of $-Δ+V$ are uniformly observable from an arbitrarily small neighborhood of those two points. This condition is much weaker than the Geometric Control Condition, which is necessary for uniform observability of eigenfunctions for the free Laplacian on the sphere. The same result also holds for eigenfunctions of $(-Δ)^{α/2}+V$, for any $α>0$.

math.AP↗

arXiv2019

Wigner measures and effective mass theorems

We study a semi-classical Schr{ö}dinger equation which describes the dynamics of an electron in a crystal in the presence of impurities. It is well-known that under suitable assumptions on the initial data, the wave function can be approximated in the semi-classical limit by the solution of a simpler equation, the effective mass equation. Using Floquet-Bloch decomposition and with a non-degeneracy condition on the critical points of the Bloch bands, as it is classical in this subject, we establish effective mass equations for more general initial data. Then, when the critical points are degenerated (which may occur in dimension strictly larger than one), we prove that a similar analysis can be performed, leading to a new type of effective mass equations which are operator-valued and of Heisenberg form. Our analysis relies on Wigner measure theory and, more precisely, to its applications to the analysis of dispersion effects.

math.AP↗

arXiv2018

Semiclassical analysis of dispersion phenomena

Our aim in this work is to give some quantitative insight on the dispersive effects exhibited by solutions of a semiclassical Schr{ö}dinger-type equation in R d. We describe quantitatively the localisation of the energy in a long-time semiclassical limit within this non compact geometry and exhibit conditions under which the energy remains localized on compact sets. We also explain how our results can be applied in a straightforward way to describe obstructions to the validity of smoothing type estimates.

math.AP↗

arXiv2017

Two-microlocal regularity of quasimodes on the torus

We study the regularity of stationary and time-dependent solutions to strong perturbations of the free Schrödinger equation on two-dimensional flat tori. This is achieved by performing a second microlocalization related to the size of the perturbation and by analysing concentration and nonconcentration properties at this new scale. In particular, we show that sufficiently accurate quasimodes can only concentrate on the set of critical points of the average of the potential along geodesics.

math.AP↗

arXiv2017

Observability and quantum limits for the Schrödinger equation on the sphere

In this note, we describe our recent results on semiclassical measures for the Schr{ö}dinger evolution on Zoll manifolds. We focus on the particular case of eigenmodes of the Schr{ö}dinger operator on the sphere endowed with its canonical metric. We also recall the relation of this problem with the observability question from control theory. In particular, we exhibit examples of open sets and potentials on the 2-sphere for which observability fails for the evolution problem while it holds for the stationary one. Finally, we give some new results in the case where the Radon transform of the potential identically vanishes.

math.AP↗

arXiv2015

Concentration and non-concentration for the Schrödinger evolution on Zoll manifolds

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential to the Laplacian on the sphere results on the existence of geodesics $γ$ such that $δ_γ$ cannot be obtained as a semiclassical measure for some sequence of eigenfunctions. We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.

math.AP↗

arXiv2015

Wigner measures and observability for the Schrödinger equation on the disk

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a "structure theorem", expressing the restriction of the Wigner measures on each invariant torus in terms of {\em second-microlocal measures}. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the $L^2$-norm of a solution on any open subset intersecting the boundary (resp. the $L^2$-norm of the Neumann trace on any nonempty open set of the boundary) controls its full $L^2$-norm (resp. $H^1$-norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.

math.AP↗

arXiv2015

Delocalization of quasimodes on the disk

This note deals with semiclassical measures associated to {(sufficiently accurate)} quasimodes $(u_h)$ for the Laplace-Dirichlet operator on the disk. In this time-independent set-up, we simplify the statements of our preprint arXiv:1406.0681 and their proofs. We describe the restriction of semiclassical measures to every invariant torus in terms of two-microlocal measures. As corollaries, we show regularity and delocalization properties for limit measures of $|u_h|^2 dx$: these are absolutely continuous in the interior of the disk and charge every open set intersecting the boundary.

math.AP↗