Gunther Uhlmann, Andras Vasy
We survey finite energy inverse results in N-body scattering, and we also sketch the proof of the extension of our recent two-cluster to two-cluster three-body result to the many-body case: this requires only minor modifications. We also indicate how to extend a free-to-free inverse result in three-body scattering to the many-body case.
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We consider low energy inverse problems in three-body scattering and show that if all unknown interactions are small in an appropriate sense then the 2-cluster to 2-cluster S-matrices given at low energies determine the Fourier transform of the effective interaction in a ball.
A detailed discussion of the Krein's results (applicable for solving the inverse scattering problem) is given with complete proofs.
A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse potential scattering problem is discussed in detail, inversion formulas are derived and error estimates are obtained. Inverse obstacle scattering problem with data at a fixed frequency is studied. Uniqueness theorems and stability estimates are obtained. Inverse geophysical scattering problem is discussed. An algorithm for computing the Dirichlet-to-Neumann map f…
We analyze the inverse scattering series for diffuse waves in random media. In previous work the inverse series was used to develop fast, direct image reconstruction algorithms in optical tomography. Here we characterize the convergence, stability and approximation error of the series
In a recent paper, we developed an inverse scattering approach to the Boussinesq equation in the case when no solitons are present. In this paper, we extend this approach to include solutions with solitons.
We study explicit formulas for phaseless inverse scattering in the Born approximation at high energies for the Schrödinger equation with compactly supported potential in dimension d $\ge$ 2. We obtain error estimates for these formulas in the configuration space.
Let $q(x)$ be real-valued compactly supported sufficiently smooth function. It is proved that the scattering data $A(β,α_0,k)$ $\forall β\in S^2$, $\forall k>0,$ determine $q$ uniquely. Here $α_0\in S^2$ is a fixed direction of the incident plane wave.
We prove a fixed frequency inverse scattering result for the magnetic Schrödinger operator (or connection Laplacian) on surfaces with Euclidean ends. We show that, under suitable decaying conditions, the scattering matrix for the operator determines both the gauge class of the connection and the zeroth order potential.
We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QW.
In this paper we consider the inverse scattering problem at a fixed energy for the Schrödinger equation with a long-range potential in $\ere^d, d\geq 3$. We prove that the long-range part can be uniquely reconstructed from the leading forward singularity of the scattering amplitude at some positive energy.
We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $Σ\subseteqΓ=\partialΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $δ$-type potential supported on the surface $Γ$ and of strength $α\in L^{p}(Γ)$, $p>2$.
We consider the fixed angle inverse scattering problem and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that almost symmetric or horizontally controlled potentials are uniquely determined by their fixed angle scattering data. This is done by establishing an equivalence between the frequency domain and the time domain formulations of the problem, and by solving the time domain problem by extending the methods of [RS19] which adapts the ideas introduced in [BK81] and [IY01] on the use of Carleman estimate…
We study the inverse scattering problem for electric potentials and magnetic fields in $\ere^d, d\geq 3$, that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at some positive energy.
We consider the Aharonov-Bohm effect for the Schroedinger operator and the related inverse problem in an exterior domain the 2-dim. Euclidean space with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain and the Hamiltonian up to Gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle is convex.
In this paper we study the linearized inverse problem associated with imaging of reflection seismic data. We introduce an inverse scattering transform derived from reverse-time migration (RTM). In the process, the explicit evaluation of the so-called normal operator is avoided, while other differential and pseudodifferential operator factors are introduced. We prove that, under certain conditions, the transform yields a partial inverse, and support this with numerical simulations. In addition, we explain the recently discussed 'low-frequency artifacts' in RTM, which are naturally removed by th…
It is proved that if the scattering amplitudes at a fixed wavenumber for two obstacles from a certain large class of obstacles differ a little, than the obstacles differ a little. Error estimate is given. It is proved that there is an inversion formula for finding the obstacle from a fixed-energy scattering data.
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.
In this work we present an analogue of the inverse scattering for Canonical systems using theory of vessels and associated to them completely integrable systems. Analytic coefficients fits into this setting, significantly expanding the class of functions for which the inverse scattering exist. We also derive an evolutionary equation, arising from canonical systems, which describes the evolution of the logarithmic derivative of the tau function, associated to these systems
Stability estimates in inverse scattering
针对三维固定能量散射数据，提出了一种数学严格的逆散射方法以恢复紧支撑势，并给出了解的误差估计。
We give a brief survey for the recent development of inverse scattering theory on non-compact Riemannian manifolds. The main theme is the reconstruction of the manifold and the metric from the scattering matrix.
This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relat…
Inverse Nonlinear Scattering by a Metric
本文研究从半线性波动方程的散射算子反推含时全局双曲洛伦兹度规的反问题。
On a nonlocal non-linear inverse scattering problem
本文研究三维非线性分数阶亥姆霍兹方程的反散射问题，通过散射振幅恢复 compactly 支持的势。
Inverse scattering for the nonlinear magnetic Schrodinger equation
本文研究带磁势的非线性Schrödinger方程的逆散射问题，证明散射算子获得的散射数据能唯一确定磁势，建立了磁势的唯一性结果。
We study an inverse source scattering problem for the Schrödinger equation with a quadratic nonlinearity. In general, uniqueness of inverse source problems can not be guaranteed at a fixed energy. Therefore, additional information is required for the source in order to obtain a unique solution. By adding reference point sources, we show that a general source function could be uniquely determined from boundary measurements at a fixed wavenumber. This method does not apply to inverse source problems of linear equations since it uses the nonlinearity as a tool. The proof utilizes the method of li…
A minimization problem with free boundary and its application to inverse scattering problems
本文研究了一个具有自由边界的极小化问题，推导出Helmholtz方程的混合求积区域，并探讨了其在逆散射问题中的应用。
Split Corners in Scattering and Inverse Scattering
研究具有角奇点的源和可穿透介质的散射与逆散射，引入分裂角概念，建立相关结果并推广经典理论，得到源和介质逆散射的唯一性结果，如从单个远场测量恢复多边形凸包。
We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimension $d\geq 3$. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension $d=2$ is also given.
In this paper we consider the inverse scattering problem for the Schr{ö}dinger operator with short-range electric potential. We prove in dimension n $\geq$ 2 that the knowledge of the scattering operator determines the electric potential and we establish H{ö}lder-type stability in determining the short range electric potential.
We consider the inverse scattering problem on the energy interval in three dimensions. We are focused on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows optimality of the logarithmic stability result of [Stefanov, 1990] (up to the value of the exponent).
We study the long-time behavior of solutions to nonlinear Schroedinger equations with some critical rough potential of inverse square type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with the Lapalacian with inverse square potential. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schroedinger equation with inverse square potential in energy space.
We prove that the scattering matrix at all energies restricted to an open subset of the boundary determines an asymptotically hyperbolic manifold modulo isometries that are equal to the identity on the open subset where the scattering matrix is known.
Consider the scattering of the two- or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source scattering problem which is to reconstruct the source function. Our results show that increasing stability can be obtained for the inverse problem by using only the Dirichlet boundary data with multi-frequencies.
We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define the radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.
We show that the scattering operator for defocusing energy critical semilinear wave equations \square u+f(u)=0, with f C-infinity and f ~ u^5, in three space dimensions, determines f.
We study the $L^2$-Sobolev space bijectivity of the direct and inverse scattering of the $3\times 3$ AKNS system associated to the Manakov system and Sasa-Satsuma equation. We establish the bijectivity on the weighted Sobolev space $H^{i,1}(\mathbb{R})$ for $i=1,2$.
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 inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.
This paper concerns the stability on the inverse source scattering problem for the one-dimensional Helmholtz equation in a two-layered medium. We show that the increasing stability can be achieved by using multi-frequency wave field at the two end points of the interval which contains the compact support of the source function.
Inverse scattering problems have many important applications. In this paper, given limited aperture data, we propose a Bayesian method for the inverse acoustic scattering to reconstruct the shape of an obstacle. The inverse problem is formulated as a statistical model using the Baye's formula. The well-posedness is proved in the sense of the Hellinger metric. The extended sampling method is modified to provide the initial guess of the target location, which is critical to the fast convergence of the MCMC algorithm. An extensive numerical study is presented to illustrate the performance of the …
In this paper, we study the inverse medium scattering problem to reconstruct unknown inhomogeneous medium from far field patterns of scattered waves. In the first part of our work, the linear inverse scattering problem was discussed, while in the second part, we deal with the nonlinear problem. The main idea is to apply the linear Kalman filter to the linearized problem. There are several ways to linearize, which introduce two reconstruction algorithms. Finally, we give numerical examples to demonstrate our proposed method.
Quantum inverse scattering for time-dependent repulsive Hamiltonians of quadratic type
针对二次型含时排斥哈密顿量下的多维逆散射问题，采用Enss-Weder含时方法，从散射算子速度极限唯一确定带类库仑奇点的短程势函数。
These notes are a considerably revised and expanded version of expository lectures given at the Fields Institute Workshop on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" in August 2017. We give a complete and self-contained treatment of inverse scattering for the defocussing cubic NLS in one-dimension, following the 2003 paper of Deift and Zhou, and the defocussing Davey-Stewartson equation in two space dimensions, following the work of Perry and more recent work of Nachman, Regev, and Tataru.
We consider the inverse problem of recovering a potential by measuring the response at a point to a source located at the same point and then varying the point on the surface of a sphere. This is a similar to the inverse back-scattering problem. We show that if the angular derivatives of the difference of two potentials having the same data is controlled by the L^2 norm of the difference of the potentials they must be equal. In particular this shows injectivity of the inverse problem for radial potentials.
We consider inverse scattering problems for the three-dimensional Hartree equation. We prove that if the unknown interaction potential $V(x)$ of the equation satisfies some rapid decay condition, then we can uniquely determine the exact value of $\partial_ξ^α\hat{V}(0)$ for any multi-index $α$ by the knowledge of the scattering operator for the equation. Furthermore, we show some stability estimate for identifying $\partial_ξ^α\hat{V}(0)$.
We study an inverse scattering problem of a perturbed biharmonic operator. we show that the high-frequency asymptotic of scattering amplitude of the biharmonic operator uniquely determine the curl A and {V -1/2 nablaA}. We also study the near-field scattering problem and show that the high-frequency asymptotic expansion up to a certain error in terms of frequency lambda recovers the same two above quantities with no additional information about A and V. We also prove stability estimates for curl A and {V -1/2 nablaA}.
We discuss the inverse uniqueness problem in phaseless scattering by counting the zeros of its modulus of the scattering amplitude. The phase linearization of scattered wave field disturbs the originally uniform distribution of the zero set. There is a connection between the perturbation of the index of refraction and the zero distribution of the modulus. We conclude the inverse uniqueness of the phaseless problem from the point of view of interior transmission problem.
The inverse electromagnetic scattering problem for anisotropic media in general does not have a unique solution. A possible approach to this problem is through the use of appropriate "target signatures," i.e. eigenvalues associated with the direct scattering problem that are accessible to measurement from a knowledge of the scattering data. In this paper we shall consider three different sets of eigenvalues that can be used as target signatures: 1) eigenvalues of the electric far field operator, 2) transmission eigenvalues and 3) Stekloff eigenvalues.
This paper is concerned with the inverse scattering and the transmission eigenvalues for anisotropic periodic layers. For the inverse scattering problem, we study the Factorization method for shape reconstruction of the periodic layers from near field scattering data. This method provides a fast numerical algorithm as well as a unique determination for the shape reconstruction of the scatterer. We present a rigorous justification and numerical examples for the factorization method. The transmission eigenvalue problem in scattering have recently attracted a lot of attentions. Transmission eigen…
We discuss a time-harmonic inverse scattering problem for a nonlinear Helmholtz equation with compactly supported inhomogeneous scattering objects that are described by a nonlinear refractive index in unbounded free space. Assuming the knowledge of a nonlinear far field operator, which maps Herglotz incident waves to the far field patterns of corresponding solutions of the nonlinear scattering problem, we show that the nonlinear index of refraction is uniquely determined. We also generalize two reconstruction methods, a factorization method and a monotonicity method, to recover the support of …
In this paper, we study the inverse acoustic medium scattering problem to reconstruct the unknown inhomogeneous medium from far field patterns of scattered waves. We propose the reconstruction scheme based on the Kalman filter, which becomes possible to sequentially estimate the inhomogeneous medium. We also show that in the linear inverse problem, the estimation for the Kalman filter is equivalent to that for the Tikhonov regularization. Finally, we give numerical examples to demonstrate our proposed method.
The inverse scattering theory of Kadomtsev-Petviashvili II equations
该研究概述KP II方程的逆散射理论，聚焦受扰多线孤子逆问题，证明不同背景下的逆散射理论具有一致性。
Determination of an anisotropic perturbation in elastic inverse scattering
本文研究弹性波线性反散射问题，证明了通过单次散射波可唯一确定弹性参数的各向异性扰动，并给出了各向异性扰动的稳定性估计及刚性结论。
We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that, in the case of a radial potential supported sufficiently near the boundary, infinite resolution can be achieved from measurements of the near-field operator in the monotone case.
These are my lecture notes from a minicourse I gave at the Universite de Nantes about many-body scattering. I also discuss the relationship between many-body scattering and higher rank symmetric spaces, whose description is the result of joint work with Rafe Mazzeo.
We study the direct and inverse scattering problem for the semilinear Schrödinger equation $Δu+a(x,u)+k^2u=0$ in $\mathbb{R}^d$. We show well-posedness in the direct problem for small solutions based on the Banach fixed point theorem, and the solution has the certain asymptotic behavior at infinity. We also show the inverse problem that the semilinear function $a(x,z)$ is uniquely determined from the scattering data. The idea is certain linearization that by using sources with several parameters we differentiate the nonlinear equation with respect to these parameter in order to get the linear …
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.
The monotonicity method for the inverse acoustic scattering problem is to understand the inclusion relation between an unknown object and artificial one by comparing the far field operator with artificial operator. This paper introduces the development of this method to the inverse crack scattering problem. Our aim is to give the following two indicators: One (Theorem 1.1) is to determine whether an artificial small arc is contained in the unknown arc. The other one (Theorem 1.2) is whether an artificial large domain contain the unknown one. Finally, numerical examples based on Theorem 1.1 are…
We prove a removable singularities theorem for stationary Einstein equations, with useful implications for constructions of stationary solutions using soliton methods.
This paper is concerned with an inverse source problem for the three-dimensional Helmholtz equation by a single boundary measurement at a fixed frequency. We show the Lipschitz stability under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains..
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