30 research outputs found
Rigidity of broken geodesic flow and inverse problems
Consider broken geodesics alpha([0, 1]) on a compact Riemannian manifold (M, g) with boundary of dimension n >= 3. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic alpha([0, 1]) starting at and ending to the boundary partial derivative M we know the starting point and direction (alpha(0), alpha'(0)), the end point and direction (alpha(1), alpha'(1)), and the length 1. We show that this data determines uniquely, up to an isometry, the manifold (M, g). This result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium, and arises in several applications including geophysics and medical imaging. As an example we consider the inverse problem for the radiative transfer equation (or the linear transport equation) with a nonconstant wave speed. Assuming that the scattering kernel is everywhere positive, we show that the boundary measurements determine the wave speed inside the domain up to an isometry
Inverse problem for wave equation with sources and observations on disjoint sets
We consider an inverse problem for a hyperbolic partial differential equation
on a compact Riemannian manifold. Assuming that and are
two disjoint open subsets of the boundary of the manifold we define the
restricted Dirichlet-to-Neumann operator . This
operator corresponds the boundary measurements when we have smooth sources
supported on and the fields produced by these sources are observed
on . We show that when and are disjoint but
their closures intersect at least at one point, then the restricted
Dirichlet-to-Neumann operator determines the
Riemannian manifold and the metric on it up to an isometry. In the Euclidian
space, the result yields that an anisotropic wave speed inside a compact body
is determined, up to a natural coordinate transformations, by measurements on
the boundary of the body even when wave sources are kept away from receivers.
Moreover, we show that if we have three arbitrary non-empty open subsets
, and of the boundary, then the restricted
Dirichlet-to-Neumann operators for determine the Riemannian manifold to an isometry. Similar result is proven
also for the finite-time boundary measurements when the hyperbolic equation
satisfies an exact controllability condition
Approximate quantum cloaking and almost trapped states
We describe families of potentials which act as approximate cloaks for matter
waves, i.e., for solutions of the time-independent Schr\"odinger equation at
energy , with applications to the design of ion traps. These are derived
from perfect cloaks for the conductivity and Helmholtz equations, by a
procedure we refer to as isotropic transformation optics. If is a potential
which is surrounded by a sequence of approximate
cloaks, then for generic , asymptotically in (i) is both
undetectable and unaltered by matter waves originating externally to the cloak;
and (ii) the combined potential does not perturb waves outside the
cloak. On the other hand, for near a discrete set of energies, cloaking
{\it per se} fails and the approximate cloaks support wave functions
concentrated, or {\it almost trapped}, inside the cloaked region and negligible
outside. Applications include ion traps, almost invisible to matter waves or
customizable to support almost trapped states of arbitrary multiplicity.
Possible uses include simulation of abstract quantum systems, magnetically
tunable quantum beam switches, and illusions of singular magnetic fields.Comment: Revised, with new figures. Single column forma
Electromagnetic wormholes via handlebody constructions
Cloaking devices are prescriptions of electrostatic, optical or
electromagnetic parameter fields (conductivity , index of refraction
, or electric permittivity and magnetic permeability
) which are piecewise smooth on and singular on a
hypersurface , and such that objects in the region enclosed by
are not detectable to external observation by waves. Here, we give related
constructions of invisible tunnels, which allow electromagnetic waves to pass
between possibly distant points, but with only the ends of the tunnels visible
to electromagnetic imaging. Effectively, these change the topology of space
with respect to solutions of Maxwell's equations, corresponding to attaching a
handlebody to . The resulting devices thus function as
electromagnetic wormholes.Comment: 25 pages, 6 figures (some color
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 . We give two basic constructions for cloaking a region
contained in a domain from measurements of Cauchy data of waves at \p
\Omega; we pay particular attention to cloaking not just a passive object, but
an active device within , interpreted as a collection of sources and sinks
or an internal current.Comment: Final revision; to appear in Commun. in Math. Physic
Uniform stability estimates for the discrete Calderon problems
In this article, we focus on the analysis of discrete versions of the
Calderon problem in dimension d \geq 3. In particular, our goal is to obtain
stability estimates for the discrete Calderon problems that hold uniformly with
respect to the discretization parameter. Our approach mimics the one in the
continuous setting. Namely, we shall prove discrete Carleman estimates for the
discrete Laplace operator. A main difference with the continuous ones is that
there, the Carleman parameters cannot be taken arbitrarily large, but should be
smaller than some frequency scale depending on the mesh size. Following the
by-now classical Complex Geometric Optics (CGO) approach, we can thus derive
discrete CGO solutions, but with limited range of parameters. As in the
continuous case, we then use these solutions to obtain uniform stability
estimates for the discrete Calderon problems.Comment: 38 pages, 2 figure
Schrodinger's Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics
The advent of transformation optics and metamaterials has made possible
devices producing extreme effects on wave propagation. Here we give theoretical
designs for devices, Schr\"odinger hats, acting as invisible concentrators of
waves. These exist for any wave phenomenon modeled by either the Helmholtz or
Schr\"odinger equations, e.g., polarized waves in EM, pressure waves in
acoustics and matter waves in QM, and occupy one part of a parameter space
continuum of wave-manipulating structures which also contains standard
transformation optics based cloaks, resonant cloaks and cloaked sensors. For EM
and acoustic Schr\"odinger hats, the resulting centralized wave is a localized
excitation. In QM, the result is a new charged quasiparticle, a \emph{quasmon},
which causes conditional probabilistic illusions. We discuss possible solid
state implementations.Comment: 36 pages, 3 figure
Inverse problems with partial data for a magnetic Schr\"odinger operator in an infinite slab and on a bounded domain
In this paper we study inverse boundary value problems with partial data for
the magnetic Schr\"odinger operator. In the case of an infinite slab in ,
, we establish that the magnetic field and the electric potential can
be determined uniquely, when the Dirichlet and Neumann data are given either on
the different boundary hyperplanes of the slab or on the same hyperplane. This
is a generalization of the results of [41], obtained for the Schr\"odinger
operator without magnetic potentials. In the case of a bounded domain in ,
, extending the results of [2], we show the unique determination of the
magnetic field and electric potential from the Dirichlet and Neumann data,
given on two arbitrary open subsets of the boundary, provided that the magnetic
and electric potentials are known in a neighborhood of the boundary.
Generalizing the results of [31], we also obtain uniqueness results for the
magnetic Schr\"odinger operator, when the Dirichlet and Neumann data are known
on the same part of the boundary, assuming that the inaccessible part of the
boundary is a part of a hyperplane
Limiting Carleman weights and anisotropic inverse problems
In this article we consider the anisotropic Calderon problem and related
inverse problems. The approach is based on limiting Carleman weights,
introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean
case. We characterize those Riemannian manifolds which admit limiting Carleman
weights, and give a complex geometrical optics construction for a class of such
manifolds. This is used to prove uniqueness results for anisotropic inverse
problems, via the attenuated geodesic X-ray transform. Earlier results in
dimension were restricted to real-analytic metrics.Comment: 58 page
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Inverse boundary value problems for the perturbed polyharmonic operator
We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (-Î)m, m â„ 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in ân, n â„ 3. Notice that the corresponding result does not hold in general when m = 1. © 2013 American Mathematical Society