arXiv subjects
Publications and source records attributed to Roman Novikov.
In this work we study weighted Radon transforms in multidimensions. We introduce an analog of Chang approximate inversion formula for such transforms and describe all weights for which this formula is exact. In addition, we indicate possible tomographic applications of inversion methods for weighted Radon transforms in 3D.
We consider the one-dimensional Schrödinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line $x\ge 0$. For this equation at fixed positive energy we give explicit formulas for finding the full complex valued reflection coefficient to the left from appropriate phaseless scattering data measured on the left, i.e. for $x<0$. Using these formulas and known inverse scattering results we obtain global uniqueness and reconstruction results for phaseless inverse scattering in dimension $d=1$.
We consider quantum and acoustic wave propagation at fixed frequency for compactly supported scatterers in dimension $d\ge 2$. In these framework we give explicit formulas for phase recovering from appropriate phaseless scattering data. As a corollary, we give global uniqueness results for quantum and acoustic inverse scattering at fixed frequency without phase information.
We consider phaseless inverse scattering for the Schrödinger equation with compactly supported potential in dimension $d\ge 2$. We give explicit formulas for solving this problem from appropriate data at high energies. As a corollary, we give also a global uniqueness result for this problem with appropriate data on a fixed energy neighborhood.
We give effectivized Holder-logarithmic energy and regularity dependent stability estimates for the Gel'fand inverse boundary value problem in dimension $d=3$. This effectivization includes explicit dependance of the estimates on coefficient norms and related parameters. Our new estimates are given in $L^2$ and $L^\infty$ norms for the coefficient difference and related stability efficiently increases with increasing energy and/or coefficient difference regularity. Comparisons with preceeding results are given.
We give formulas and equations for finding generalized scattering data for the Schrödinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map.
We give new global stability estimates for monochromatic inverse acoustic scattering. These estimates essentially improve estimates of [P. Hahner, T. Hohage, SIAM J. Math. Anal., 33(3), 2001, 670-685] and can be considered as a solution of an open problem formulated in the aforementioned work.
We study the impedance boundary map (or Robin-to-Robin map) for the Schrodinger equation in open bounded demain at fixed energy in multidimensions. We give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map.
We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation $-Δψ+ V(x)ψ= E ψ$ at fixed positive $E$, where $V$ is a matrix-valued potential. The first is the Gel'fand inverse problem on a bounded domain $D$ at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane $\R^2$. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential $V$ is reconstructed with Lipschitz stability by these algorithms up to $O(E^{-(m-2)/2})$ in the uniform norm as $E \to +\infty$, under the assumptions that $V$ is $m$-times differentiable in $L^1$, for $m \geq 3$, and has sufficient boundary decay.
We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for point (delta-type) potentials in two dimensions. In particular, we obtain the first explicit examples of such eigenfunctions with contour singularity in spectral parameter at a fixed real energy.
We study the multi-channel Gel'fand-Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-Δψ+ v(x) ψ= 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.
In this article we show that the large time asymptotics for the Grinevich-Zakharov rational solutions of the Novikov-Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum of localized travel waves (solitons).
We show that Novikov--Veselov equation (an analog of KdV in dimension 2 + 1) does not have exponentially localized solitons at negative energy.
In this note we show that the Novikov-Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) has no exponentially localized solitons ( in the two-dimensional sense).
We describe all weighted Radon transforms on the plane for which the Chang approximate inversion formula is precise. Some subsequent results, including the Cormack type inversion for these transforms, are also given.
We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain.
In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov--Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.
We prove new global stability estimates for the Gel'fand-Calderon inverse problem in 3D. For sufficiently regular potentials this result of the present work is a principal improvement of the result of [G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153-172].