An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation (original) (raw)
Inverse Problems, 2013
Let Ω ⊂ R 2 be a bounded domain with ∂Ω ∈ C ∞ and L be a positive number. For a three dimensional cylindrical domain Q = Ω × (0, L), we obtain some uniqueness result of determining a complex-valued potential for the Schrödinger equation from partial Cauchy data when Dirichlet data vanish on a subboundary (∂Ω \ Γ) × [0, L] and the corresponding Neumann data are observed on Γ × [0, L], where Γ is an arbitrary fixed open set of ∂Ω.
Reconstructing the potential for the one-dimensional Schrodinger equation from boundary measurements
IMA Journal of Mathematical Control and Information, 2014
We consider the inverse problem of the determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the whole boundary. Provided that source is generic using the Boundary Control method we recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We adapt the algorithm to the situation when only the finite number of eigenvalues are known and provide the result on the convergence of the method.
An inverse problem for Schrödinger equations with discontinuous main coefficient
2008
This article concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of ℝ N with Dirichlet data and discontinuous principal coefficient a (x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface.
Stability estimate in an inverse problem for non-autonomous magnetic Schrödinger equations
Applicable Analysis, 2011
We consider the inverse problem of determining the time dependent magnetic field of the Schrödinger equation in a bounded open subset of R n , n ≥ 1, 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 n times changing initial value suitably.
Journal of Functional Analysis, 2010
We consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schrödinger equation in a bounded smooth domain of R n with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichletto-Neumann map associated to the solutions of the magnetic Schrödinger equation. We prove in dimension n 2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic Schrödinger equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.
2015
We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}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.
Uniqueness and stability in an inverse problem for the Schrödinger equation
Inverse Problems, 2007
We study the Schrödinger equation iy + ∆y + qy = 0 in Ω × (0, T ) with Dirichlet boundary data y| ∂Ω×(0,T ) and real valued initial condition y| Ω×{0} and we consider the inverse problem of determining the potential q(x), x ∈ Ω when ∂y ∂ν | Γ0×(0,T ) is given. Here Ω is an open bounded domain of R N , Γ 0 is an open subset of ∂Ω satisfying a suitable geometrical condition and T > 0. More precisely, from a global Carleman estimate we prove a stability inequality between p − q and ∂y(q) ∂ν − ∂y(p) ∂ν with appropriate norms.
Reconstructing the potential for the 1D Schrödinger equation from boundary measurements
2011
We consider the inverse problem of the determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the whole boundary. Provided that source is generic using the Boundary Control method we recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We adapt the algorithm to the situation when only the finite number of eigenvalues are known and provide the result on the convergence of the method.
2012
Abstract We study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this article, we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy.