Inverse problem for the Schrödinger operator in an unbounded strip (original) (raw)
Inverse Problems for Ultrahyperbolic Schr\"odinger Equations
Cornell University - arXiv, 2017
In this paper, we establish a global Carleman estimate for an Ultrahyperbolic Schrödinger equation. Moreover, we prove Hölder stability for the inverse problem of determining a coefficient or a source term in the Ultrahyperbolic Schrödinger equation by some lateral boundary data.
Eigenvalue Bounds for a Class of Schrödinger Operators in a Strip
Journal of Mathematics, 2018
This paper is concerned with the estimation of the number of negative eigenvalues (bound states) of Schrödinger operators in a strip subject to Neumann boundary conditions. The estimates involve weighted L1 norms and LlnL norms of the potential. Estimates involving the norms of the potential supported by a curve embedded in a strip are also presented.
Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights
Inverse Problems, 2008
, investigated some inverse problems for the evolution Schrödinger equation by means of Carleman inequalities proved under a strict pseudoconvexity condition. We show here that new Carleman inequalities for the Schrödinger equation may be derived under a relaxed pseudoconvexity condition, which allows us to use degenerate weights with a spatial dependence of the type ψ(x) = x · e, where e is some fixed direction in R N . As a result, less restrictive boundary or internal observations are allowed to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from a single boundary or internal measurement.
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.
Journal de Mathématiques Pures et Appliquées, 2009
In this paper, we investigate the inverse problem of determining the potential of the dynamical Schrödinger equation in a bounded domain from the data of the solution in a subboundary over a time interval. Assuming that in a neighborhood of the boundary of the spatial domain, the potential is known and without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove a logarithmic stability estimate for the inverse problem with a single measurement on an arbitrarily given subboundary.
Eigenvalue bounds for a class of Schroedinger operators in a strip
arXiv (Cornell University), 2018
This paper is concerned with the estimation of the number of negative eigenvalues (bound states) of Schrödinger operators in a strip subject to Neumann boundary conditions. The estimates involve weighted L 1 norms and L ln L norms of the potential. Estimates involving the norms of the potential supported by a curve embedded in a strip are also presented.