Gregory Eskin

INVERSE SCATTERING PROBLEM FOR THE SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL AT A FIXED ENERGY

In this article we consider the Schrödinger operator inRn,n≧3, with electric and magnetic potentials which decay exponentially as |x|→∞. We show that the scattering amplitude at fixed positive energy determines the electric potential and the magnetic field.

The inverse backscattering problem in three dimensions

This article is a study of the mapping from a potentialq(x) onR3 to the backscattering amplitude associated with the Hamiltonian −Δ+q(x). The backscattering amplitude is the restriction of the scattering amplitudea(θ, ω, k), (θ, ω, k)εS2×S2×ℝ+, toa(θ,−θ, k). We show that in suitable (complex) Banach spaces the map fromq(x) toa(x/|x|, −x/|x|, |x|) is usually a local diffeomorphism. Hence in contrast to the overdetermined problem of recoveringq from the full scattering amplitude the inverse backscattering problem is well posed.

On the inverse boundary value problem for linear isotropic elasticity

We derive three results on the inverse problem of determining the Lame parameters λ(x) and μ(x) for an isotropic elastic body from its Dirichlet-to-Neumann map.

A new approach to hyperbolic inverse problems

We present a modification of the BC method in inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper, we prove the crucial local step. The global step of the proof will be presented in a forthcoming paper.

Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov–Bohm effect

We consider the inverse boundary value problem for the Schrodinger operator with time-dependent electromagnetic potentials in domains with obstacles. We extend the resuls of the author’s works [Inverse Probl. 19, 49 (2003); 19, 985 (2003); 20, 1497 (2004)] to the case of time-dependent potentials. We relate our results to the Aharonov–Bohm effect caused by magnetic and electric fluxes.

Inverse Hyperbolic Problems with Time-Dependent Coefficients

We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R n with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.

Inverse boundary value problems in domains with several obstacles

We consider the inverse boundary value problem for the Schrodinger equations with electromagnetic potentials in domains with several obstacles. We show that the boundary data allow one to recover integrals of potentials along broken rays. Then we prove the uniqueness modulo a gauge transform of the recovery of electromagnetic potentials from the integrals over broken rays under some geometrical conditions on the obstacles.

Inverse Hyperbolic Problems and Optical Black Holes

In this paper we state a uniqueness theorem for the inverse hyperbolic problem in the case of a finite time interval. We apply this theorem to the inverse problem for the equation of the propagation of light in a moving medium (the Gordon equation). Then we study the existence of black and white holes for the general second order hyperbolic equation and for the Gordon equation and we discuss the impact of this phenomenon on the inverse problems.

Inverse boundary value problems and the Aharonov–Bohm effect

We consider the inverse boundary value problem for the Schr?dinger equation with electromagnetic or Yang?Mills potentials in multiconnected domains ? ? Rn, n ? 2. We prove that if the Dirichlet-to-Neumann operators on ?? are gauge equivalent then the corresponding potentials are gauge equivalent too. The multiconnectedness of ? leads to the Aharonov?Bohm effect.

Inverse problems for Schrodinger equations with Yang-Mills potentials in domains with obstacles and the Aharonov-Bohm effect

We study the inverse boundary value problems for the Schrodinger equations with Yang-Mills potentials in a bounded domain Ω0 ⊂ Rn containing finite number of smooth obstacles Ωj , 1 ≤ j ≤ r. We prove that the Dirichlet-to-Neumann operator on ∂Ω0 determines the gauge equivalence class of the Yang-Mills potentials. We also prove that the metric tensor can be recovered up to a diffeomorphism that is identity on ∂Ω0.