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A. G. Ramm

Publications and source records attributed to A. G. Ramm.

198 records · Page 11Linked to original sources
arXiv2000

Uniqueness theorems for inverse obstacle scattering in Lipschitz domains

An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness of the spectrum of the corresponding Laplacian in a bounded domain. Proof of the uniqueness results is based on the fact that the Hilbert space of square integrable functions is separable.

math-ph↗
arXiv1999

An approximate method for solving inverse scattering problem with fixed-energy phase shifts

An approximate method is proposed for the recovery of a compactly supported spherically-symmetric potential from the set of fixed-energy phase-shifts known for all angular momenta. The method reduces the inverse scattering problem to a moment problem which is solved numerically. The above reduction is done approximately and it is more accurate for larger angular momenta or for smaller potentials.

math-ph↗
arXiv1999

Existence and uniqueness of the scattering solutions in the exterior of rough domains

Existence and uniqueness of the scattering solutions is proved for a class of bounded rough obstacles which is much larger than the class of Lipschitz obstacles. Integral equations method is not used. The approach is based on the representation theorem for selfadjoint operators via closed symmetric quadratic forms and on the limiting absorption principles. For the Dirichlet boundary condition no assumptions on the smoothness of the boundary are needed, for the Neumann boundary condition and the Robin boundary condition some assumptions on the smoothness of the boundary are needed.

math-ph↗
arXiv1999

Inequalities for the minimal eigenvalue of the Laplacian in an annulus

It is proved that the minimal Dirichlet eigenvalue of the Laplacian in an annulus is a monotonically decreasing function of the displacement of the center of the smaller disc. The maximal value of the minimal eigenvalue is attained when the annulus is formed by two concentric discs.

math-ph↗
arXiv1999

Continuous regularization of nonlinear ill-posed problems

A general method for solving nonlinear ill-posed problems is developed. The method consists of solving a Cauchy problem with a regularized operator and proving that the solution of this problem tends, as time grows, to a solution of the original nonlinear stationary problem. Examples of applications of the general method are given. Convergence theorems are proved.

math-ph↗
arXiv1999

Remark on the theory of reproducing kernels

Foundations of the theory of Hilbert spaces with reproducing kernels are discussed. It is demonstrated that the claims in the papers of S.Saitoh and in his book "Theory of reproducing kernels and applications, Pitman research notes, 189, Longman, New York, 1988, that a characterization of the range of arbitrary linear transform is obtained in his works, is not correct.

math-ph↗
arXiv1999

Inverse problem for an inhomogeneous Schrödinger equation

An inverse problem is considered for an inhomogeneous Schrödinger equation. Assuming that the potential vanishes outside a finite interval and satisfies some other technical assumptions, one proves the uniqueness of the recovery of this potential from the knowledge of the wave function at the ends of the above interval for all energies. An algorithm is given for the recovery of the potential from the above data.

math-ph↗
arXiv1999

Continuous analog of Gauss-Newton method

A continuous analog of Gauss-Newton method for solving nonlinear ill-posed problems is proposed. Its converegence is proved. A numerical example is presented to demonstrate efficiency of the propsed method.

math-ph↗
arXiv1999

Stability estimates in inverse scattering

Mathematically rigorous inversion method is developed to recover compactly supported potentials from the fixed-energy scattering data in three dimensions. Error estimates are given for the solution. An algorithm for inversion of noisy discrete fixed-energy #D scattering data is developed and its error estimates are obtained

math-ph↗
arXiv1999

Example of two different potentials which have practically the same fixed-energy phase shifts

It is shown that the Newton-Sabatier procedure for inverting the fixed-energy phase shifts for a potential is not an inversion method but a parameter-fitting procedure. Theoretically there is no guarantee that this procedure is applicable to the given set of the phase shifts, if it is applicable, there is no guaran- tee that the potential it produces generates the phase shifts from which it was reconstructed. Moreover, no generic potential, specifically, no potential which is not analytic in a neighborhood of the positive real semiaxis can be reconstructed by the Newton-Sabatier procedure. A numerical method is given for finding spherically symmetric compactly supported potentials which produce practically the same set of fixed-energy phase shifts for all values of angular momentum. Concrete example of such potentials is given.

math-ph↗
arXiv1999

Property C for ODE and applications to inverse problems

Property C stands for completeness of the set of products of solutions to homogeneous linear differential equations. property C is proved in various formulations for Schrödinger operators. Many applications of this property to inverse problems and inverse scattering problems are given. It is shown what part of the fixed-energy phase shifts determines a compactly supported potential uniquely. It is shown that the Newton-Sabatier method is not really an inversion method but a parameter-fitting procedure and it is proved that this procedure cannot recover a generic potential, in particular, any potential which is not analytic in a neighborhood of the positive real axis.

math-ph↗
arXiv1999

Inequalities for the derivatives

It is proved that one cannot approximate stably the first derivative of a smooth function given noisy values of this function and a bound on this function and its first derivative. Such an approximation is shown to be possible if an a priori bound is known for a fractional derivative of order greater than one. An algorithm is proposed for such a stable approximation and error estimates for the proposed algorithm are given. Under certain assumptions it is proved that this algorithm is best possible among all linear and nonlinear algorithms.

math-ph↗
arXiv1999

Finding small inhomogeneities from surface scattering data

A method is given for finding small inhomogeneities from surface scattering data. The method consists in localization of the positions and finding the intensities of these small inhomogeneities. The case of acoustic wave scattering is studied, but the method is applicable to electromagnetic waves as well.

math-ph↗
arXiv1999

An inverse problem for point inhomogeneities

We study quantum scattering theory off $n$ point inhomogeneities ($n\in\bbN$) in three dimensions. The inhomogeneities (or generalized point interactions) positioned at $\{ξ_1,...,ξ_n\}\subset\bbR^3$ are modeled in terms of the $n^2$ (real) parameter family of self-adjoint extensions of $-Δ\big|_{C^\infty_0(\bbR^3\backslash\{ξ_1,...,ξ_n\})}$ in $L^2(\bbR^3)$. The Green's function, the scattering solutions and the scattering amplitude for this model are explicitly computed in terms of elementary functions. Moreover, using the connection between fixed energy quantum scattering and acoustical scattering, the following inverse spectral result in acoustics is proved: The knowledge of the scattered field on a plane outside these point-like inhomogeneities, with all inhomogeneities located on one side of the plane, uniquely determines the positions and boundary conditions associated with them.

math.SP↗