202 research outputs found
Stability estimate in an inverse problem for non-autonomous Schr\"odinger equations
We consider the inverse problem of determining the time dependent magnetic
field of the Schr\"odinger equation in a bounded open subset of , with , from a finite number of Neumann data, when the boundary measurement is
taken on an appropriate open subset of the boundary. We prove the Lispchitz
stability of the magnetic potential in the Coulomb gauge class by times
changing initial value suitably
Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries
Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to bounded or
unbounded subsets of the plane by confining potential barriers. The edges of
the confining potential barriers create edge currents. This is the second of
two papers in which we review recent progress and prove explicit lower bounds
on the edge currents associated with one- and two-edge geometries. In this
paper, we study various unbounded and bounded, two-edge geometries with soft
and hard confining potentials. These two-edge geometries describe the electron
confined to unbounded regions in the plane, such as a strip, or to bounded
regions, such as a finite length cylinder. We prove that the edge currents are
stable under various perturbations, provided they are suitably small relative
to the magnetic field strength, including perturbations by random potentials.
The existence of, and the estimates on, the edge currents are independent of
the spectral type of the operator.Comment: 57 page
Uniqueness and stability results for an inverse spectral problem in a periodic waveguide
Let where be a
bounded domain, and a bounded potential which is
-periodic in the variable . We study the inverse
problem consisting in the determination of , through the boundary spectral
data of the operator , acting on
, with quasi-periodic and Dirichlet boundary
conditions. More precisely we show that if for two potentials and
we denote by and the
eigenvalues associated to the operators and (that is the
operator with or ), then if as we have that ,
provided one knows also that , where . We establish also an optimal Lipschitz
stability estimate. The arguments developed here may be applied to other
spectral inverse problems, and similar results can be obtained
Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map
We examine the stability issue in the inverse problem of determining a scalar
potential appearing in the stationary Schr{\"o}dinger equation in a bounded
domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet
data is imposed on the shadowed face of the boundary of the domain and the
Neumann data is measured on its illuminated face. We establish a log log
stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp.
L2) potentials whose difference is lying in any Sobolev space of order positive
order
Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary
We consider the multidimensional inverse problem of determining the
conductivity coefficient of a hyperbolic equation in an infinite cylindrical
domain, from a single boundary observation of the solution. We prove H{\"o}lder
stability with the aid of a Carleman estimate specifically designed for
hyperbolic waveguides
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