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Gunther Uhlmann

Publications and source records attributed to Gunther Uhlmann.

194 records · Page 11Linked to original sources
arXiv2007

Full-wave invisibility of active devices at all frequencies

There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or "cloaking") from observation by electromagnetic (EM) waves. Here, we prove invisibility, with respect to solutions of the Helmholtz and Maxwell's equations, for several constructions of cloaking devices. Previous results have either been on the level of ray tracing [Le,PSS] or at zero frequency [GLU2,GLU3], but recent numerical [CPSSP] and experimental [SMJCPSS] work has provided evidence for invisibility at frequency $k\ne 0$. We give two basic constructions for cloaking a region $D$ contained in a domain $Ω$ from measurements of Cauchy data of waves at $\p Ω$; we pay particular attention to cloaking not just a passive object, but an active device within $D$, interpreted as a collection of sources and sinks or an internal current.

math.AP↗
arXiv2007

The X-ray transform for a generic family of curves

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves $Γ$. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of $Γ$ covers $T^*M$.

math.DG↗
arXiv2007

Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds

Let $σ$ be the scattering relation on a compact Riemannian manifold $M$ with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let $\ell$ be the length of that geodesic ray. We study the question of whether the metric $g$ is uniquely determined, up to an isometry, by knowledge of $σ$ and $\ell$ restricted on some subset $D$. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from $D$ covers $T^*M$; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that $σ$ and $\ell$ restricted to $D$ uniquely recover an isometric copy of $g$ locally near generic metrics, and in particular, near real analytic ones.

math.DG↗
arXiv2006

Integral geometry of tensor fields on a class of non-simple Riemannian manifolds

We study the geodesic X-ray transform $I_Γ$ of tensor fields on a compact Riemannian manifold $M$ with non-necessarily convex boundary and with possible conjugate points. We assume that $I_Γ$ is known for geodesics belonging to an open set $Γ$ with endpoint on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any $(x,ξ)\in T^*M$, there is a geodesic without conjugate points in $Γ$ through $x$ normal to $ξ$.

math.DG↗
arXiv2004

Boundary rigidity and stability for generic simple metrics

We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $ρ_g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set $\mathcal{G}$ of simple Riemannian metrics and an open dense set $\mathcal{U}\subset \mathcal{G}\times\mathcal{G}$, such that any two Riemannian metrics in $\mathcal{U}$ having the same distance function, must be isometric. We also prove Hölder type stability estimates for this problem for metrics which are close to a given one in $\mathcal{G}$.

math.DG↗
arXiv2003

Fixed energy inverse problem for exponentially decreasing potentials

In this paper we show that in two-body scattering the scattering matrix at a fixed energy determines real-valued exponentially decreasing potentials. This result has been proved by Novikov previously, see also the work of Novikov and Khenkin using a d-bar-equation. We present a different method, which combines a density argument and real analyticity in part of the complex momentum. The latter has been noted by Novikov and Khenkin; here we give a short proof using contour deformations. In the addendum to the paper we also supply a reference to the work of Eskin and Ralston that did not appear in the published paper since we were unaware of the relevant aspects of their work.

math.AP↗
arXiv2003

On nonuniqueness for Calderon's inverse problem

We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.

math.AP↗
arXiv2003

Inverse problems in N-body scattering

We survey finite energy inverse results in N-body scattering, and we also sketch the proof of the extension of our recent two-cluster to two-cluster three-body result to the many-body case: this requires only minor modifications. We also indicate how to extend a free-to-free inverse result in three-body scattering to the many-body case.

math.AP↗
arXiv2002

The Calderon problem for conormal potentials, I: Global uniqueness and reconstruction

In dimensions greater than or equal to three, we establish global uniqueness and obtain reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials. The potentials considered are conormal of order less than 1-k with respect to submanifolds (of arbitrary codimension k). This gives positive results for (conormal) conductivities which are Holder of any order > 1. A related problem for highly singular potentials is shown to exhibit nonuniqueness.

math.AP↗
arXiv2001

Low energy inverse problems in three-body scattering

We consider low energy inverse problems in three-body scattering and show that if all unknown interactions are small in an appropriate sense then the 2-cluster to 2-cluster S-matrices given at low energies determine the Fourier transform of the effective interaction in a ball.

math.AP↗
arXiv2000

Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform

We consider the Dirichlet-to-Neumann map associated to the Schrödinger equation with a potential in a bounded Lipschitz domain in three or more dimensions. We show that the integral of the potential over a two-plane is determined by the Cauchy data of certain exponentially growing solutions on any neighborhood of the intersection of the two-plane with the boundary.

math.AP↗