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Gregory Eskin

Publications and source records attributed to Gregory Eskin.

At least 37 records · Page 2Linked to original sources
arXiv2013

Gauge Equivalence and the Inverse Spectral Problem for the Magnetic Schrödinger Operator on the Torus

We study the inverse spectral problem for the Schrödinger operator $H$ on the two-dimensional torus with even magnetic field $B(x)$ and even electric potential $V(x)$. V.Guillemin [11] proved that the spectrum of $H$ determines $B(x)$ and $V(x)$. A simple proof of Guillemin's results was given by the authors in [3]. In the present paper we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of $H$ determine the magnetic field and the electric potential. We also show that generically the spectrum and the magnetic field determine the "extended" gauge equivalence class of the magnetic potential. The proof is a modification of the proof in [3] with some corrections and clarifications.

math.SP↗
arXiv2013

The Aharonov-Bohm effect in spectral asymptotics of the magnetic Schrödinger operator

We show that in the absence of a magnetic field the spectrum of the magnetic Schrödinger operator in an annulus depends on the cosine of the flux associated with the magnetic potential. This result follows from an analysis of a singularity in the wave trace for this Schrödinger operator, and hence shows that even in the absence of a magnetic field the magnetic potential can change the asymptotics of the Schrödinger spectrum, i.e. the Aharonov-Bohm effect takes place. We also study the Aharonov-Bohm effect for the magnetic Schrödinger operator on a torus.

math-ph↗
arXiv2011

Remarks on magnetic and electric Aharonov-Bohm effects

We give a direct proof of the magnetic Aharonov-Bohm effects without using the scattering theory and the theory of inverse boundary value problems. This proof can serve as a framework for a physical experiment to confirm the magnetic AB effect. We prove also the electric AB effect and we suggest a physical experiment to demonstrate the electric AB effect. In addition, we consider a combined electric and magnetic AB effect and we propose a new inverse problem for the time-dependent Schrödinger equations. Finally we study the gravitational AB effect.

math-ph↗
arXiv2011

Artificial black holes

We study black holes for the linear hyperbolic equations describing the wave propagation in the moving medium. Such black holes are called artificial since the Lorentz metric associated with the hyperbolic equation does not necessary satisfies the Einstein equations. Artificial black holes also arise when we consider perturbations of the Einstein equations. In this paper we review author's results of [E2] and [E3] on the existence and the stability of black holes for the stationary wave equations in two space dimensions, and in the axisymmetric case.

math.AP↗
arXiv2011

Gauge equivalence and inverse scattering for long-range magnetic potentials

For Schroedinger operators with long-range magnetic vector potentials and short-range electric scalar potentials in an exterior domain $Ω$ in ${\bf R}^n$ with $n \geq 2$, we show that there is a one-to-one correspondence between the gauge equivalent classes of Hamiltonians and those of S-matrices if $Ω$ is exterior to a bounded convex obstacle.

math-ph↗
arXiv2010

Gauge Equivalence and Inverse Scattering for Aharonov-Bohm Effect

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.

math-ph↗
arXiv2009

On the determination of Moving Boundaries for Hyperbolic Equations

We consider wave equations in domains with time-dependent boundaries (moving obstacles) contained in a fixed cylinder for all time. We give sufficient conditions for the determination of the moving boundary from the Cauchy data on part of the boundary of the cylinder. We also study the related problem of accessibility of the moving boundary by time-like curves from the boundary of the cylinder.

math-ph↗
arXiv2009

Perturbations of the Kerr black hole in the class of axisymmetric artificial black holes

Artificial black holes (called also acoustic or optical black holes) are the black holes for the linear wave equation describing the wave propagation in a moving medium. They attracted a considerable interest of physicists who study them to better understand the black holes in general relativity. We consider the case of stationary axisymmetric metrics and we show that the Kerr black hole is not stable under perturbations in the class of all axisymmetric metrics. We describe families of axisymmetric metrics having black holes that are the perturbations of the Kerr black hole. We also show that the ergosphere can be determined by boundary measurements and we prove the uniform boundness of the solution in the exterior of the black hole when the event horizon coincides with the ergosphere.

math-ph↗
arXiv2008

Inverse hyperbolic problems and optical black holes

In this paper we give a more geometrical formulation of the main theorem in [E1] on the inverse problem for the second order hyperbolic equation of general form with coefficients independent of the time variable. 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 hyperbolic and for the Gordon equation and we discuss the impact of this phenomenon on the inverse problems.

math-ph↗
arXiv2007

Optical Aharonov-Bohm effect: an inverse hyperbolic problems approach

We describe the general setting for the optical Aharonov-Bohm effect based on the inverse problem of the identification of the coefficients of the governing hyperbolic equation by the boundary measurements. We interpret the inverse problem result as a possibility in principle to detect the optical Aharonov-Bohm effect by the boundary measurements.

math-ph↗
arXiv2007

Inverse Scattering for Gratings and Wave Guides

We consider the problem of unique identification of dielectric coefficients for gratings and sound speeds for wave guides from scattering data. We prove that the "propagating modes" given for all frequencies uniquely determine these coefficients. The gratings may contain conductors as well as dielectrics and the boundaries of the conductors are also determined by the propagating modes.

math.AP↗
arXiv2007

A new approach to hyperbolic inverse problems II (Global step)

We study the inverse problem for the second order self-adjoint hyperbolic equation with the boundary data given on a part of the boundary. This paper is the continuation of the author's paper [E]. In [E] we presented the crucial local step of the proof. In this paper we prove the global step. Our method is a modification of the BC-method with some new ideas. In particular, the way of the determination of the metric is new.

math.AP↗
arXiv2006

Inverse spectral problems in rectangular domains

We consider the Schrodinger operator in n-dimensional rectangular domains with either Dirichlet or Neumann boundary conditions on the faces and study the constraints on the potential imposed by fixing the spectrum of the operator.We study also the asymptotics of the heat kernel when t tends to 0.

math.AP↗
arXiv2006

A new approach to hyperbolic inverse problems

We present a modification of the BC-method in the 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 the forthcoming paper.

math.AP↗
arXiv2006

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.

math.AP↗
arXiv2005

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 Schrödinger equations with Yang-Mills potentials in a bounded domain $Ω_0\subset\R^n$ containing finite number of smooth obstacles $Ω_j,1\leq j \leq r$. We prove that the Dirichlet-to-Neumann operator on $\partialΩ_0$ 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 $\partialΩ_0$.

math.AP↗