arXiv subjects

François Nicoleau

Publications and source records attributed to François Nicoleau.

At least 19 recordsLinked to original sources

arXiv2026

Inverse Semiclassical Scattering at Fixed Energy

We investigate inverse scattering at a fixed energy for semiclassical Schrödinger operators with smooth potentials. For compactly supported potentials satisfying a natural virial condition, we prove that $ \liminf_{h\to0} \|S_1(λ,h)-S_2(λ,h)\|<\sqrt{2} $ implies $V_1=V_2$, provided the fixed energy $λ$ lies above both potentials. We then consider radial short-range repulsive potentials. Under a monotonicity assumption on the radial force, we show that the semiclassical differential cross section at a single fixed energy, up to $o(1)$ as $h\to0$, determines the potential throughout the classically accessible region. Finally, we obtain an analogous rigidity result for nontrapping compactly supported perturbations of the Euclidean metric under a strict convexity assumption.

math.AP↗

arXiv2026

Spectral Asymptotics of a Singular Harmonic Oscillator with Aharonov--Bohm Flux

Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.

math.SP↗

arXiv2026

A Local-to-Global Propagation Principle for Dirichlet-to-Neumann Maps

We establish four local-to-global propagation results for Dirichlet--to--Neumann maps. Our first two results are proved in the general setting of smooth compact Riemannian manifolds with boundary. The first shows that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component $Γ$, then equality of the corresponding local Dirichlet--to--Neumann maps on a nonempty open subset of $Γ$ propagates to equality of the associated global Dirichlet--to--Neumann maps on all of $Γ$. The proof combines unique continuation and self-adjointness arguments. The second replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet--to--Neumann maps. The proof relies on the spectral unique continuation theory of Jerison--Lebeau, through the formulation of Le~Rousseau--Lebeau. Our third and fourth results establish local-to-global propagation principles under Ingham-type quasi--analytic spectral assumptions. Assuming that the boundary manifold is respectively a compact Riemannian symmetric space or a compact quasi--analytic Riemannian manifold, they rely on the propagation theorems of Ganguly--Thangavelu and of Bhowmik--Pradhan. As an application, we consider a class of conformally warped product metrics. In this setting, the local Borg--Marchenko theorem and Weyl--Titchmarsh theory relate the required Ingham-type spectral decay to a suitable quasi--analytic boundary closeness of the conformal factors, yielding new local-to-global uniqueness results for Dirichlet--to--Neumann maps.

math.AP↗

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

A Sharp Regularity Threshold for Uniqueness in Riemannian Calderón-type Problems

We prove a sharp regularity threshold for uniqueness in two anisotropic Calderón-type inverse problems in dimension $n\ge 3$. The main setting is the Riemannian Schrödinger problem with fixed scalar potential: for a prescribed nonconstant analytic function $V$, we study whether the Dirichlet-to-Neumann map of $-Δ_g+V$ on a domain $Ω\subset\mathbb{R}^n$ determines the unknown metric $g$. The natural gauge is the group of boundary-fixing diffeomorphisms preserving $V$. We show that, while analytic metrics are uniquely determined modulo this gauge by a minor adaptation of the Lassas--Uhlmann reconstruction theorem, uniqueness fails densely in every non-analytic Gevrey class $G^σ$, $σ>1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overlineΩ$. We also prove the analogous sharp threshold for the anisotropic Calderón problem at fixed nonzero frequency, thereby upgrading the previously known finite-regularity counterexamples to Gevrey and $C^\infty$ regularity. The two constructions use different scalar mechanisms: for fixed potentials, the nonconstant potential itself provides a local coordinate, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Thus analyticity is the exact threshold for uniqueness in both problems.

math.AP↗

arXiv2026

Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk

We derive a three-term asymptotic expansion for the lowest eigenvalue of the magnetic Laplace and Steklov operators in the exterior of the unit disk in the strong magnetic field limit. This improves recent results of Helffer-Nicoleau (2025) based on special function asymptotics, and extends earlier works by Fournais-Helffer (2006), Kachmar (2006), and R. Fahs, L. Treust, N. Raymond, S. Vũ Ng\d{o}c (2024). Notably, our analysis reveals how the third term encodes the dependence on the magnetic flux. Finally, we investigate the weak magnetic field limit and establish the flux dependence in the asymptotics of Kachmar-Lotoreichik-Sundqvist (2025).

math.SP↗

arXiv2026

The Spectral Shift Function for Non-Self-Adjoint Perturbations

This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class and relatively trace-class perturbations. We extend the Lifshits-Kre__n trace formula to non-self-adjoint operators under suitable assumptions on the spectrum and the behavior of the resolvent. The role of spectral singularities is carefully analyzed, and we provide a generalization of the SSF using functional calculus. Finally, we apply our results to Schr{ö}dinger operators with complex-valued short-range potentials in dimension three. Toy models illustrate properties that one might hope to extend to general cases. In particular, they suggest that the SSF carries information on the presence of complex eigenvalues.

math-ph↗

arXiv2026

Inverse Spectral Analysis of Singular Radial AKNS Operators

We study an inverse spectral problem for singular AKNS operators based on spectral data associated with two distinct values of the effective angular momentum parameter $κ\,$. Our main focus is the local inverse problem near the zero potential. For the pairs $(κ_1,κ_2)=(0,1)$, $(1,2)$ and $(0,3)\,$, we establish local uniqueness. For $(0,2)\,$, we prove that the Fréchet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.

math.AP↗

arXiv2026

On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta

We consider an inverse spectral problem for radial Schrödinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a Müntz-type condition uniquely determines the potential. Next, in a neighborhood of the zero potential, we prove local uniqueness from two Dirichlet spectra associated with distinct angular momenta in the cases $(\ell_1,\ell_2) = (0,1)\,, \ (1,2)$ and $(0,3)$\,. Our approach relies on an explicit analysis of the associated singular differential equation, combined with the classical Kneser--Sommerfeld formula. These results sharpen a theorem of Carlson-Shubin~(1994) and confirm, in the linearized setting and for these configurations, a conjecture originally formulated by Rundell and Sacks~(2001).

math.AP↗

arXiv2026

Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds

We study the question of stability of the global and partial anisotropic Calderón inverse problems for the class of Painlevé-Liouville Riemannian manifolds, that is compact $n$-dimensional manifolds with boundary $(M,g)$, where $M=[0,1]\times K\,$, $K$ is any smooth closed connected orientable manifold of dimension $n-1$ endowed with a Riemannian metric $g_K$, and $g=α^4 g_{0}$ is any conformal deformation of the product metric $g_{0}=dx^2+g_{K}$ on $M$ which is compatible with the Painlevé block-separability of the Laplace-Beltrami operator $Δ_{g_0}$. Given a pair of Painlevé-Liouville Riemannian manifolds $(M,g)$ and $(M,\tilde{g})$ satisfying some technical hypothesis, denoting the corresponding Dirichlet-to-Neumann maps by $Λ_{g}$ and $Λ_{\tilde{g}}$, and assuming that $\lVert Λ_{g}-Λ_{\tilde{g}}\rVert_{\mathcal{B}(H^{1/2}(\partial M), H^{-1/2}(\partial M))}\ = ε$, we show a logarithmic stability result for the global anisotropic Calderón problem which says that there exists constants $C$ and $0<θ<1$ such that $\| α- \tildeα \|_{C^{0,r}(M)} \leq C \left( \ln \frac{1}ε \right)^{-θ}$ for some $0<r<1$. Similar results are obtained for the partial anisotropic Calderón problem, corresponding to the case where the data are measured on only one connected component of the boundary.

math.AP↗

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↗

arXiv2024

Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities

Let $Ω\subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $γ$ be a smooth conductivity in $\overlineΩ$. Consider a non-zero frequency $λ_0$ which does not belong to the Dirichlet spectrum of $L_γ= -{\rm div} (γ\nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(γ_1, γ_2)$ on $\overlineΩ$, which are close to $γ$ such that the associated DN maps at frequency $λ_0$ satisfy \begin{equation*} Λ_{γ_1,λ_0} = Λ_{γ_2,λ_0}. \end{equation*}

math.AP↗

arXiv2024

Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions --

Inspired by a recent paper of G. Liu and X. Tan (2023), we would like to measure how the magnetic effect appears in the heat trace formula associated with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We propose to the reader an overview of magnetic heat trace formulas through explicit examples. On the way we obtain new formulas and in particular we calculate explicitely some non local terms and logarithmic terms appearing in the Steklov heat trace asymptotics.

math.AP↗

arXiv2023

Local Hölder Stability in the Inverse Steklov and Calderón Problems for Radial Schrödinger operators and Quantified Resonances

We obtain Hölder stability estimates for the inverse Steklov and Calderón problems for Schrödinger operators corresponding to a special class of $L^2$ radial potentials on the unit ball. These results provide an improvement on earlier logarithmic stability estimates obtained in \cite{DKN5} in the case of the the Schrödinger operators related to deformations of the closed Euclidean unit ball. The main tools involve: i) A formula relating the difference of the Steklov spectra of the Schrödinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon \cite{Si1} in his representation formula for the Weyl-Titchmarsh function, and ii) A key moment stability estimate due to Still \cite{St}. It is noteworthy that with respect to the original Schrödinger operator, the type of perturbation being considered for the amplitude function amounts to the introduction of a finite number of negative eigenvalues and of a countable set of negative resonances which are quantified explicitly in terms of the eigenvalues of the Laplace-Beltrami operator on the boundary sphere.

math.AP↗

arXiv2021

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds $(M,g)$ whose boundary $\partial M$ consists in two distinct connected components $Γ_0$ and $Γ_1$. First, we show that the Steklov eigenvalues can be divided into two families $(λ_m^\pm)_{m \geq 0}$ which satisfy accurate asymptotics as $m \to \infty$. Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Steklov eigenfunctions are exponentially localized on the whole boundary $\partial M$ as $m \to \infty$. Whenever we add an asymmetric perturbation to a symmetric warped product, we observe a flea on the elephant effect. Roughly speaking, we prove that "half" the Steklov eigenfunctions are exponentially localized on one connected component of the boundary, say $Γ_0$, and the other half on the other connected component $Γ_1$ as $m \to \infty$.

math.AP↗

arXiv2019

The anisotropic Calder{ó}n problem on 3-dimensional conformally St{ä}ckel manifolds

Conformally St{ä}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$Δ$ G $ψ$ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n--1 commuting conformal symmetry operators for the Laplace-Beltrami operator $Δ$ G. In this paper, we solve the anisotropic Calder{ó}n problem on compact 3-dimensional Riemannian manifolds with boundary which are conformally St{ä}ckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to dieomorphims that preserve the boundary.

math.AP↗

arXiv2019

On non-uniqueness for the anisotropic Calder{ó}n problem with partial data

We show that there is non-uniqueness for the Calder{ó}n problem with partial data for Riemannian metrics with H{ö}lder continuous coefficients in dimension greater or equal than three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only H{ö}lder continuous of order $$ρ$\<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M, g)$ which is not a warped product manifold, and we show that there exist in the conformal class of g an infinite number of Riemannian $\tilde{g} = c^4 g such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric g and do not satisfy the unique continuation principle.

math.AP↗

arXiv2018

Stability in the inverse Steklov problem on warped product Riemannian manifolds

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in R^d. We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.

math.AP↗