arXiv subjects

Damien Gobin

Publications and source records attributed to Damien Gobin.

7 recordsLinked to original sources

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 $\kappa\,$. Our main focus is the local inverse problem near the zero potential. For the pairs $(\kappa_1,\kappa_2)=(0,1)$, $(1,2)$ and $(0,3)\,$, we establish local uniqueness. For $(0,2)\,$, we prove that the Fr\'echet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.

math.AP↗

arXiv2025

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

We consider an inverse spectral problem for radial Schr\"odinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a M\"untz-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↗

arXiv2022

A Green's function for the source-free Maxwell equations on $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$

We compute a Green's function giving rise to the solution of the Cauchy problem for the source-free Maxwell's equations on a causal domain $\mathcal{D}$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $\mathcal{D}$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $\mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on $AdS^5 \times \mathbb{S}^2$, which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on $\mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $\mathcal{D}\subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell's equations.

math.AP↗

arXiv2017

Inverse scattering at fixed energy for radial magnetic Schr{ö}dinger operators with obstacle in dimension two

We study an inverse scattering problem at fixed energy for radial magnetic Schr{ö}dinger operators on R^2 \ B(0, r\_0), where r\_0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B respectively. If (V, B) and (\tilde{V} , \tilde{B}) are two couples belonging to C, we then show that if the corresponding phase shifts $δ$\_l and \tilde{$δ$}\_l (i.e. the scattering data at fixed energy) coincide for all l $\in$ L, where L $\subset$ N^$\star$ satisfies the M{ü}ntz condition \sum\_{l$\in$L} \frac{1}{l} = +$\infty$, then V (x) = \tilde{V}(x) and B(x) = \tilde{B}(x) outside the obstacle B(0, r\_0). The proof use the Complex Angular Momentum method and is close in spirit to the celebrated B{ö}rg-Marchenko uniqueness Theorem.

math-ph↗

arXiv2016

Inverse scattering at fixed energy on three-dimensional asymptotically hyperbolic St{ä}ckel manifolds

In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic St{ä}ckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the Helmholtz equation can be separated into a system of a radial ODE and two angular ODEs. We can thus decompose the full scattering operator onto generalized harmonics and the resulting partial scattering matrices consist in a countable set of $2 \times 2$ matrices whose coefficients are the so-called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions associated with the radial ODE. Using a novel multivariable version of the Complex Angular Momentum method, we show that the knowledge of the scattering operator at a fixed non-zero energy is enough to determine uniquely the metric of the three-dimensional St{ä}ckel manifold up to natural obstructions.

math.AP↗

arXiv2015

Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

In this paper, we adapt the well-known \emph{local} uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schr{ö}dinger equation to prove \emph{local} uniqueness results in the setting of inverse \emph{metric} problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients $T(λ,n)$ and reflection coefficients $L(λ,n)$ and $R(λ,n)$ of a Dirac wave having a fixed energy $λ$ and angular momentum $n$. For instance, the reflection coefficients $L(λ,n)$ correspond to the scattering experiment in which a wave is sent from the \emph{left} end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed $λ\not=0$, the knowledge of the reflection coefficients $L(λ,n)$ (resp. $R(λ,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with $B\textgreater{}0$ - determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude $B$ of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.

math-ph↗

arXiv2014

Inverse scattering at fixed energy for massive charged Dirac fields in de Sitter-Reissner-Nordström black holes

In this paper, we consider massive charged Dirac fields propagating in the exterior region of de Sitter-Reissner-Nordström black holes.We show that the parameters of such black holes are uniquely determined by the partial knowledge of the corresponding scattering operator $S(λ)$ at a fixed energy $λ$.More precisely, we consider the partial wave scattering operators $S(λ,n)$ (here $λ\in \mathbb{R}$ is the energy and $n \in \mathbb{N}^{\star}$ denotes the angular momentum) defined as the restrictions of the full scattering operator on a well chosen basis of spin-weighted spherical harmonics.We prove that the knowledge of the scattering operators $S(λ,n)$, for all $n \in \mathcal{L}$, where $\mathcal{L}$ is a subset of $\mathbb{N}^{\star}$ that satisfies the Müntz condition $\sum\_{n \in \mathcal{L}} \frac{1}{n} = + \infty$, allows to recover the mass, the charge and the cosmological constant of a dS-RN black hole.The main tool consists in the complexification of the angular momentum $n$ and in studying the analytic properties of the "unphysical" corresponding data in the complex variable $z$.

math-ph↗