Reconstructing the potential for the 1D Schrödinger equation from boundary measurements (original) (raw)
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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.
Reconstructing the potential for the 1D Schr\
2011
We consider the inverse problem of the determining the potential in the dynamical Schr\"odinger equation on the interval by the measurement on the whole boundary. Provided that source is \emph{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
Journal of Inverse and Ill-posed Problems, 2005
We consider the problem of identifying the potential in the onedimensional Schrödinger equation with input Dirichlet data, from measured Neumann data. Knowledge of the Dirichlet to Neumann map together with spectral controllability results for the Schrödinger equation obtained using properties of exponential Riesz bases allow recovery of the spectral data. Once the the spectral data is recovered, we use the Boundary Control method to solve the identification problem.
Stability for the inverse potential problem by finite measurements on the boundary
Inverse Problems, 2001
In this paper, we investigate the inverse problem of determining the potential of the Schrödinger equation from finite measurements on the boundary. It is well known that this is an ill posed problem in the sense of Hadamard. The stability estimate is proved under the assumption that the potentials have some a priori constraints. Based on this conditional stability result, we propose one kind of Tikhonov regularization and prove the convergence rate for the regularized solution.
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.
Dynamical inverse problem for the Schrödinger equation (BC-method)
American Mathematical Society Translations: Series 2, 2005
Let IR n be a bounded domain; @ 2 C 2 ; ; q 2 C 2 () real functions, > 0. We show that for any xed T > 0 the response operator R T : f ! uj @ 0;T] of the Schr odinger system i u t + u ? qu = 0 in (0; T); uj t=0 = 0; @u @ j @ 0;T] = f determines the coe cients = (x); q = q(x); x 2 uniquely. The problem is reduced to one of recovering ; q through the boundary spectral data. The spectral data are extracted from R T by the use of a variational principle. A peculiarity of the approach (the Boundary Control method) is that it allows to solve the problem using the data on a nite time interval, avoiding a continuation beyond 0,T].
The reconstruction of a source and a potential from boundary measurements
Journal of Mathematical Analysis and Applications, 2016
We are concerned with the reconstruction of both the heat coe¢ cient and the source appearing in a parabolic equation from observations of the solution on the boundary. We propose two methods. The …rst uses a single point on the boundary and a …nal time overdetermination, while the second uses only the observations over a small region on the boundary only. These methods are based on properties of the principal Dirichlet eigenfunction and Holmgren's theorem. 1991 Mathematics Subject Classi…cation. 35R30; 35K65. Key words and phrases. inverse source problem; inverse parabolic problem, boundary inversion. 1
2000
We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrödinger operator H = − d 2 dx 2 +q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.
Journal of the Franklin Institute, 2015
We consider parameter estimation and stabilization for a one-dimensional Schrödinger equation with an unknown constant disturbance suffered from the boundary observation at one end and the non-collocated control at other end. An adaptive observer is designed in terms of measured position with unknown constant by the Lyapunov functional approach. By a backstepping transformation for infinite-dimensional systems, it is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. The numerical experiments are carried out to illustrate the proposed approach.