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Ricardo Weder

Publications and source records attributed to Ricardo Weder.

At least 55 records · Page 3Linked to original sources
arXiv2012

Entanglement Creation in Low-Energy Scattering

We study the entanglement creation in the low-energy scattering of two particles in three dimensions, for a general class of interaction potentials that are not required to be spherically symmetric. The incoming asymptotic state, before the collision, is a product of two normalized Gaussian states. After the scattering the particles are entangled. We take as a measure of the entanglement the purity of one of them. We provide a rigorous explicit computation, with error bound, of the leading order of the purity at low-energy. The entanglement depends strongly in the difference of the masses. It takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses. It is quite remarkable that the anisotropy of the potential gives no contribution to the leading order of the purity, on spite of the fact that entanglement is a second order effect.

quant-ph↗
arXiv2011

The Electric Aharonov-Bohm Effect

In their seminal paper Aharonov and Bohm (1959) claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate. They proposed two experiments to verify their theoretical conclusions. The magnetic effect, that has been extensively studied, and the electric effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue, after more than fifty years. The existence of electric Aharonov-Bohm effect, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an Ansatz for the solution to the Schroedinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this Ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schroedinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will estimulate the experimental research on the electric Aharonov-Bohm effect.

math-ph↗
arXiv2010

Aharonov-Bohm Effect and High-Velocity Estimates of Solutions to the Schrödinger Equation

The Aharonov-Bohm effect is a fundamental and controversial issue in physics. At stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. in 1982 and 1986 with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. in 2007 that shows that the results of these experiments can not be explained by a force. The Aharonov-Bohm Ansatz of 1959 predicts the results of the experiments of Tonomura et al. and of Caprez et al. In 2009 we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al.. In this paper we prove that our results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handle bodies. We formulate the Aharonov-bohm Ansatz that is appropriate to this general case and we prove that the exact solution to the Schroedinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by $v^ρ, 0 < ρ<1$, with $v$ the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature.

math-ph↗
arXiv2010

The Aharonov-Bohm Effect and Tonomura et al. Experiments. Rigorous Results

We study the Aharonov-Bohm effect under the conditions of the Tonomura et al. experiments, that gave a strong evidence of the physical existence of the Aharonov-Bohm effect, and we give the first rigorous proof that the classical Ansatz of Aharonov and Bohm is a good approximation to the exact solution of the Schroedinger equation. We provide a rigorous, quantitative, error bound for the difference in norm between the exact solution and the approximate solution given by the Aharonov-Bohm Ansatz. Our error bound is uniform in time. Using the experimental data, we rigorously prove that the results of the Tonomura et al. experiments, that were predicted by Aharonov and Bohm, actually follow from quantum mechanics. Furthermore, our results show that it would be quite interesting to perform experiments for intermediate size electron wave packets (smaller than the ones used in the Tonomura et al. experiments, that were much larger than the magnet) whose variance satisfies appropriate lower and upper bounds that we provide. One could as well take a larger magnet. In this case, the interaction of the electron wave packet with the magnet is negligible -the probability that the electron wave packet interacts with the magnet is smaller than $10^{-199}$- and, moreover, quantum mechanics predicts the results observed by Tonomura et al. with an error bound smaller than $10^{-99}$, in norm. Our error bound has a physical interpretation. For small variances it is due to Heisenberg's uncertainty principle and for large variances to the interaction with the magnet.

math-ph↗
arXiv2009

Analysis of Acoustic Wave Propagation in a Thin Moving Fluid

We study the propagation of acoustic waves in a fluid that is contained in a thin two-dimensional tube, and that it is moving with a velocity profile that only depends on the transversal coordinate of the tube. The governing equations are the Galbrun equations, or, equivalently, the linearized Euler equations. We analyze the approximate model that was recently derived by Bonnet-Bendhia, Duruflé and Joly to describe the propagation of the acoustic waves in the limit when the width of the tube goes to zero. We study this model for strictly monotonic stable velocity profiles. We prove that the equations of the model of Bonnet-Bendhia, Duruflé and Joly are well posed, i.e., that there is a unique global solution, and that the solution depends continuously on the initial data. Moreover, we prove that for smooth profiles the solution grows at most as $t^3$ as $t \to \infty$, and that for piecewise linear profiles it grows at most as $t^4$. This establishes the stability of the model in a weak sense. These results are obtained constructing a quasi-explicit representation of the solution. Our quasi-explicit representation gives a physical interpretation of the propagation of acoustic waves in the fluid and it provides an efficient way to compute numerically the solution.

math-ph↗
arXiv2008

The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks

In this paper we study point transformed electromagnetic invisibility cloaks in transformation media that are obtained by transformation from general anisotropic media. We assume that there are several cloaks located in different points in space. Our results apply in particular to the first order invisibility cloaks introduced by Pendry et al. and to the high order invisibility cloaks introduced by Hendi et al. and by Cai et al.. We identify the appropriate {\it cloaking boundary conditions} that the solutions of Maxwell equations have to satisfy at the outside, $\partial K_+$, and at the inside, $\partial K_-$, of the boundary of the cloaked object $K$. Namely, that the tangential components of the electric and the magnetic fields have to vanish at $\partial K_+$ -what is always true- and that the normal components of the curl of the electric and the magnetic fields have to vanish at $\partial K_-$. These results are proven requiring that energy be conserved. In the case of one spherical cloak with a spherically stratified $K$ and a radial current at $\partial K$ we verify by an explicit calculation that our {\it cloaking boundary conditions} are satisfied and that cloaking of active devices holds even if the current is at the boundary of the cloaked object. As we prove our results for media that are obtained by transformation from general anisotropic media, our results apply to the cloaking of objects with active and passive devices contained in general anisotropic media, in particular to objects with active and passive devices contained inside general crystals.

physics.optics↗
arXiv2008

A rigorous analysis of high order electromagnetic invisibility cloaks

There is currently a great deal of interest in the invisibility cloaks recently proposed by Pendry et al. that are based in the transformation approach. They obtained their results using first order transformations. In recent papers Hendi et al. and Cai et al. considered invisibility cloaks with high order transformations. In this paper we study high order electromagnetic invisibility cloaks in transformation media obtained by high order transformations from general anisotropic media. We consider the case where there is a finite number of spherical cloaks located in different points in space. We prove that for any incident plane wave, at any frequency, the scattered wave is identically zero. We also consider the scattering of finite energy wave packets. We prove that the scattering matrix is the identity, i.e., that for any incoming wave packet the outgoing wave packet is the same as the incoming one. This proves that the invisibility cloaks can not be detected in any scattering experiment with electromagnetic waves in high order transformation media, and in particular in the first order transformation media of Pendry et al. We also prove that the high order invisibility cloaks, as well as the first order ones, cloak passive and active devices. The cloaked objects completely decouple from the exterior. Actually, the cloaking outside is independent of what is inside the cloaked objects. The electromagnetic waves inside the cloaked objects can not leave the concealed regions and viceversa, the electromagnetic waves outside the cloaked objects can not go inside the concealed regions. As we prove our results for media that are obtained by transformation from general anisotropic materials, we prove that it is possible to cloak objects inside general crystals.

physics.optics↗
arXiv2008

A Rigorous Time-Domain Analysis of Full--Wave Electromagnetic Cloaking (Invisibility)

There is currently a great deal of interest in the theoretical and practical possibility of cloaking objects from the observation by electromagnetic waves. The basic idea of these invisibility devices \cite{glu1, glu2, le},\cite{pss1} is to use anisotropic {\it transformation media} whose permittivity and permeability $\var^{λν}, μ^{λν}$, are obtained from the ones, $\var_0^{λν}, μ^{λν}_0$, of isotropic media, by singular transformations of coordinates. In this paper we study electromagnetic cloaking in the time-domain using the formalism of time-dependent scattering theory. This formalism allows us to settle in an unambiguous way the mathematical problems posed by the singularities of the inverse of the permittivity and the permeability of the {\it transformation media} on the boundary of the cloaked objects. We write Maxwell's equations in Schrödinger form with the electromagnetic propagator playing the role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked objects is essentially self-adjoint. Moreover, the unique self-adjoint extension is unitarily equivalent to the electromagnetic propagator in the medium $\var_0^{λν}, μ^{λν}_0$. Using this fact, and since the coordinate transformation is the identity outside of a ball, we prove that the scattering operator is the identity. Our results give a rigorous proof that the construction of \cite{glu1, glu2, le}, \cite{pss1} perfectly cloaks passive and active devices from observation by electromagnetic waves. Furthermore, we prove cloaking for general anisotropic materials. In particular, our results prove that it is possible to cloak objects inside general crystals.

physics.optics↗
arXiv2007

High-Velocity Estimates for the Scattering Operator and Aharonov-Bohm Effect in Three Dimensions

We obtain high-velocity estimates with error bounds for the scattering operator of the Schrödinger equation in three dimensions with electromagnetic potentials in the exterior of bounded obstacles that are handlebodies. A particular case is a finite number of tori. We prove our results with time-dependent methods. We consider high-velocity estimates where the direction of the velocity of the incoming electrons is kept fixed as its absolute value goes to infinity. In the case of one torus our results give a rigorous proof that quantum mechanics predicts the interference patterns observed in the fundamental experiments of Tonomura et al. that gave a conclusive evidence of the existence of the Aharonov-Bohm effect using a toroidal magnet. We give a method for the reconstruction of the flux of the magnetic field over a cross-section of the torus modulo $2π$. Equivalently, we determine modulo $2π$ the difference in phase for two electrons that travel to infinity, when one goes inside the hole and the other outside it. For this purpose we only need the high-velocity limit of the scattering operator for one direction of the velocity of the incoming electrons. When there are several tori -or more generally handlebodies- the information that we obtain in the fluxes, and on the difference of phases, depends on the relative position of the tori and on the direction of the velocities when we take the high-velocity limit of the incoming electrons. For some locations of the tori we can determine all the fluxes modulo 2$π$ by taking the high-velocity limit in only one direction. We also give a method for the unique reconstruction of the electric potential and the magnetic field outside the handlebodies from the high-velocity limit of the scattering operator.

math-ph↗
arXiv2007

On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{ö}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.

math-ph↗
arXiv2006

Inverse Scattering at a Fixed Energy for Long-Range Potentials

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.

math-ph↗
arXiv2005

The Borg-Marchenko Theorem with a Continuous Spectrum

The Schrödinger equation is considered on the half line with a selfadjoint boundary condition when the potential is real valued, integrable, and has a finite first moment. It is proved that the potential and the two boundary conditions are uniquely determined by a set of spectral data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result provides a generalization of the celebrated uniqueness theorem of Borg and Marchenko using two sets of discrete spectra to the case where there is also a continuous spectrum. The proof employed yields a method to recover the potential and the two boundary conditions, and it also constructs data sets used in various inversion methods. A comparison is made with the uniqueness result of Gesztesy and Simon using Krein's spectral shift function as the inversion data.

math-ph↗
arXiv2005

The Initial Value Problem, Scattering and Inverse Scattering, for Non-Linear Schrödinger Equations with a Potential and a Non-Local Non-Linearity

We consider non-linear Schrödinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail the initial value problem for these equations. In particular, existence and uniqueness of local and global solutions, continuous dependence on the initial data and regularity. We allow for a large class of unbounded potentials. We have no restriction on the growth at infinity of the positive part of the potential. We also construct the scattering operator in the case of potentials that go to zero at infinity. Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator. In the case of the quantum capacitor, our method allows us to uniquely reconstruct all the physical parameters from the small amplitude limit of the scattering operator.

math-ph↗
arXiv2005

Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy

We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials $(V,A)$ where $V(x)=O(|x|^{-ρ}), A(x)= O(|x|^{-ρ}), |x| \to \infty, ρ>1,$ are dense in the set of all solutions to the Schrödinger equation that are in $L^2(K)$ where $K$ is any connected bounded open set in $\ere^n,n\geq 2,$ with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials $(V_1,A_1)$ and $(V_2,A_2)$ in $\ere^n, n\geq 3,$ have the same scattering matrix at a fixed positive energy and if the electric potentials $V_j$ and the magnetic fields $ F_j:={\rm curl} A_j, j=1,2,$ coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.

math-ph↗
arXiv2004

Multidimensional Borg-Levinson Theorem

We consider the inverse problem of the reconstruction of a Schrödinger operator on a unknown Riemannian manifold or a domain of Euclidean space. The data used is a part of the boundary $Γ$ and the eigenvalues corresponding to a set of impedances in the Robin boundary condition which vary on $Γ$. The proof is based on the analysis of the behaviour of the eigenfunctions on the boundary as well as in perturbation theory of eigenvalues. This reduces the problem to an inverse boundary spectral problem solved by the boundary control method.

math.AP↗
arXiv2004

Inverse Scattering at a Fixed Quasi-Energy for Potentials Periodic in Time

We prove that the scattering matrix at a fixed quasi--energy determines uniquely a time--periodic potential that decays exponentially at infinity. We consider potentials that for each fixed time belong to $L^{3/2}$ in space. The exponent 3/2 is critical for the singularities of the potential in space. For this singular class of potentials the result is new even in the time--independent case, where it was only known for bounded exponentially decreasing potentials.

math-ph↗