42 research outputs found
An inverse scattering problem for short-range systems in a time-periodic electric field
We consider the time-dependent Hamiltonian on , where the external electric field and the
short-range electric potential are time-periodic with the same period.
It is well-known that the short-range notion depends on the mean value
of the external field. When , we show that the high energy limit of the
scattering operators determines uniquely . In the other case, the same
result holds in dimension for generic sghort-range potentials. In
dimension 2, one has to assume a stronger decay on the electric potential
An inverse scattering problem for the Schrödinger equation in a semiclassical process.
9 pagesWe study an inverse scattering problem for a pair of Hamiltonians on L^2 (\r^n ), where and , is a short-range potential with a regular behaviour at infinity and is the semiclassical parameter. We show that, in dimension , the knowledge of the scattering operators , , up to in {\cal{B}} (L^2(\r^n )), and which are localized near a fixed energy , determine the potential at infinity
Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces
In this paper, we study an inverse scattering problem on Liouville surfaces
having two asymptotically hyperbolic ends. The main property of Liouville
surfaces consists in the complete separability of the Hamilton-Jacobi equations
for the geodesic flow. An important related consequence is the fact that the
stationary wave equation can be separated into a system of a radial and angular
ODEs. The full scattering matrix at fixed energy associated to a scalar wave
equation on asymptotically hyperbolic Liouville surfaces can be thus simplified
by considering its restrictions onto the generalized harmonics corresponding to
the angular separated ODE. The resulting partial scattering matrices consists
in a countable set of 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
for the radial ODE in which the generalized angular momentum is seen as the
spectral parameter. Using the Complex Angular Momentum method and recent
results on 1D inverse problem from generalized Weyl-Titchmarsh functions, we
show that the knowledge of the reflection operators at a fixed non zero energy
is enough to determine uniquely the metric of the asymptotically hyperbolic
Liouville surface under consideration.Comment: 39 p
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{\"o}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 and reflection
coefficients and of a Dirac wave having a fixed
energy and angular momentum . For instance, the reflection
coefficients 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 , the knowledge of the reflection coefficients (resp.
) - up to a precise error term of the form 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
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.Comment: 24 page
Inverse scattering at fixed energy in de Sitter-Reissner-Nordström black holes
40 pagesIn this paper, we consider massless Dirac fields propagating in the outer region of de Sitter-Reissner-Nordström black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix at a fixed energy . More precisely, we consider the partial wave scattering matrices (here is the fixed energy and denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass , the square of the charge and the cosmological constant of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients , or the reflexion coefficients (resp. ), for all where is a subset of that satisfies the Müntz condition . Our main tool consists in complexifying the angular momentum and in studying the analytic properties of the "unphysical" scattering matrix in the complex variable . We show in particular that the quantities , and belong to the Nevanlinna class in the region \{z \in \C, \ Re(z) >0 \} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstrution formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect
Inverse scattering in de Sitter-Reissner-Nordstr\"om black hole spacetimes
In this paper, we study the inverse scattering of massive charged Dirac
fields in the exterior region of (de Sitter)-Reissner-Nordstr\"om black holes.
First we obtain a precise high-energy asymptotic expansion of the diagonal
elements of the scattering matrix (i.e. of the transmission coefficients) and
we show that the leading terms of this expansion allows to recover uniquely the
mass, the charge and the cosmological constant of the black hole. Second, in
the case of nonzero cosmological constant, we show that the knowledge of the
reflection coefficients of the scattering matrix on any interval of energy also
permits to recover uniquely these parameters.Comment: 44 page
Inverse scattering in de Sitter-Reissner-Nordström black hole spacetimes
44 pagesIn this paper, we study the inverse scattering of massive charged Dirac fields in the exterior region of (de Sitter)-Reissner-Nordström black holes. First we obtain a precise high-energy asymptotic expansion of the diagonal elements of the scattering matrix (i.e. of the transmission coefficients) and we show that the leading terms of this expansion allows to recover uniquely the mass, the charge and the cosmological constant of the black hole. Second, in the case of nonzero cosmological constant, we show that the knowledge of the reflection coefficients of the scattering matrix on any interval of energy also permits to recover uniquely these parameters