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Ricardo Weder

Publications and source records attributed to Ricardo Weder.

At least 37 records · Page 2Linked to original sources
arXiv2018

Inverse scattering on the half line for the matrix Schrödinger equation

The matrix Schrödinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix potential is integrable, is selfadjoint, and has a finite first moment. The corresponding scattering data set is constructed, and such scattering data sets are characterized by providing a set of necessary and sufficient conditions assuring the existence and uniqueness of the one-to-one correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classic result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition.

math-ph↗
arXiv2017

The Inverse Scattering Problem for the Matrix Schrödinger Equation

The matrix Schrödinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix potential is integrable, is selfadjoint, and has a finite first moment. The corresponding scattering data set is constructed, and such scattering data sets are characterized by providing a set of necessary and sufficient conditions assuring the existence and uniqueness of the correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classical result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition. The theory presented is illustrated with various explicit examples.

math-ph↗
arXiv2016

High-Momenta Estimates for the Klein-Gordon Equation: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering

The study of obstacle scattering for the Klein-Gordon equation in the presence of long-range magnetic potentials is addressed. Previous results of the authors are extended to the long-range case and the results the authors previously proved for high-momenta long-range scattering for the Schrödinger equation are brought to the relativistic scenario. It is shown that there are important differences between relativistic and non-relativistic scattering concerning long-range. In particular, it is proved that the electric potential can be recovered without assuming the knowledge of the long-range part of the magnetic potential, which has to be supposed in the non-relativistic case. The electric potential and the magnetic field are recovered from the high momenta limit of the scattering operator, as well as fluxes modulo $2 π$ around handles of the obstacle. Moreover, it is proved that, for every $\hat{\mathbf v} \in \mathbb{S}^2$, $ A_\infty(\hat{\mathbf v}) + A_\infty(-\hat{\mathbf v})$ can be reconstructed, where $A_\infty$ is the long-range part of the magnetic potential. A a simple formula for the high momenta limit of the scattering operator is given, in terms of magnetic fluxes over handles of the obstacle and long-range magnetic fluxes at infinity, that are introduced in this paper. The appearance of these long-range magnetic fluxes is a new effect in scattering theory.

math-ph↗
arXiv2016

Hybrid resonance and long-time asymptotic of the solution to Maxwell's equations

We study the long-time asymptotic of the solutions to Maxwell's equation in the case of a upper-hybrid resonance in the cold plasma model. We base our analysis in the transfer to the time domain of the recent results of B. Després, L.M. Imbert-Gérard and R. Weder, J. Math. Pures Appl. {\bf 101} ( 2014) 623-659, where the singular solutions to Maxwell's equations in the frequency domain were constructed by means of a limiting absorption principle and a formula for the heating of the plasma in the limit of vanishing collision frequency was obtained. Currently there is considerable interest in these problems, in particular, because upper-hybrid resonances are a possible scenario for the heating of plasmas, and since they can be a model for the diagnostics involving wave scattering in plasmas.

physics.plasm-ph↗
arXiv2015

Time delay for the Dirac equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator $\int\limits_{0} ^{\infty}e^{iH_{0}t}ζ\left( \frac{\left\vert x\right\vert }{R}\right) e^{-iH_{0}t}dt,$ as $R\rightarrow\infty,$ is presented. Here $H_{0}$ is the free Dirac operator and $ζ\left( t\right) $ is such that $ζ\left( t\right) =1$ for $0\leq t\leq1$ and $ζ\left( t\right) =0$ for $t>1.$ This approach allows us to obtain the time delay operator $δ\mathcal{T}\left( f\right) $ for initial states $f$ in $\mathcal{H} _{2}^{3/2+\varepsilon}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) ,$ $\varepsilon>0,$ the Sobolev space of order $3/2+\varepsilon$ and weight $2.$ The relation between the time delay operator $δ\mathcal{T}\left( f\right) $ and the Eisenbud-Wigner time delay operator is given. Also, the relation between the averaged time delay and the spectral shift function is presented.

math-ph↗
arXiv2015

Equations of a Moving Mirror and the Electromagnetic Field

We consider a system composed of a mobile slab and the electromagnetic field. We assume that the slab is made of a material that has the following properties when it is at rest: it is linear, isotropic, non-magnetizable, and ohmic with zero free charge density. Using instantaneous Lorentz transformations, we deduce the set of self-consistent equations governing the dynamics of the system and we obtain approximate equations to first order in the velocity and the acceleration of the slab. As a consequence of the motion of the slab, the field must satisfy a wave equation with damping and slowly varying coefficients plus terms that are small when the time-scale of the evolution of the mirror is much smaller than that of the field. Also, the motion of the slab and its interaction with the field introduce two effects in the slab's equation of motion. The first one is a position- and time-dependent mass related to the $\textit{effective mass}$ taken in phenomenological treatments of this type of systems. The second one is a velocity-dependent force that can give rise to friction and that is related to the much sought $\textit{cooling}$ of mechanical objects.

physics.optics↗
arXiv2015

Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the Klein-Gordon Equation

We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, $Λ\subset \mathbb{R}^3$, of a compact obstacle $K \subset \mathbb{R}^3$. The connected components of the obstacle are handlebodies. The particles interact with an electro-magnetic field in $Λ$ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov-Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: We give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes' theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo $2 π$. We additionally give a simple formula for the high-momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo $π$ and the simple expression of the high-momenta limit of the scattering operator does not hold true.

math-ph↗
arXiv2014

Classical Dynamics of a Thin Moving Mirror Interacting with a Laser

We analyze the classical dynamics of a system composed of a one-dimensional cavity with a perfect, fixed mirror and a movable mirror with non-zero transparency interacting with a monochromatic laser. The movable mirror can deviate far from an equilibrium position, it is assumed to be thin so that it is modelled by a delta function, and we use the exact modes of the complete system. The transparency and the mirror-position dependent cavity resonance frequencies are built into the modes and this allows us to deduce that the radiation pressure force comes from a periodic potential with period half the wavelength of the field. The exact modes and the radiation pressure potential allow us to give intuitive physical interpretations of the dynamics of the system and to obtain approximate analytic solutions for the motion of the mirror. Three regimes are identified depending on the intensity of the field and it is found that the dynamics can be qualitatively very different in each of them with some regimes being very sensitive to the values of the parameters. Moreover, we determine conditions when the Maxwell-Newton equations used to describe the system constitute accurate approximations of the exact equations governing the dynamics.

quant-ph↗
arXiv2014

High-Velocity Estimates for Schrödinger Operators in Two Dimensions: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering

We introduce a general class of long-range magnetic potentials and derive high velocity limits for the scattering operators in quantum mechanics, in the case of two dimensions. We analyze the high velocity limits in the presence of an obstacle and we uniquely reconstruct from them the electric potential and the magnetic field outside the obstacle. We also reconstruct the inaccessible magnetic fluxes produced by fields inside the obstacle modulo $2 π$. For every magnetic potential $A$ in our class we prove that its behavior at infinity ($A_\infty(\hat{\mathbf v}), \hat{\mathbf v} \in \mathbb{S}^1$) can be characterized in a natural way. Under very general assumptions we prove that $A_\infty(\hat{\mathbf v}) + A_\infty(- \hat{\mathbf v})$ can be uniquely reconstructed for every $\hat{\mathbf v} \in \mathbb{S}^1$. We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct $A_\infty(\hat{\mathbf v})$ either for all $ \hat{\mathbf v} \in \mathbb{S}^1$ or for $\hat{\mathbf v}$ in a subset of $\mathbb{S}^1$. We also give a wide class of magnetic fields outside the obstacle allowing us to uniquely reconstruct the total magnetic flux (and $A_\infty(\hat{\mathbf v})$ for all $ \hat{\mathbf v} \in \mathbb{S}^1$). This is relevant because, as it is well-known, in general the scattering operator (even if is known for all velocities or energies) does not define uniquely the total magnetic flux (and $A_\infty(\hat{\mathbf v})$ ). We analyze additionally injectivity (i.e., uniqueness without giving a method for reconstruction) of the high velocity limits of the scattering operator with respect to $A_\infty(\hat{\mathbf v})$. Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid.

math-ph↗
arXiv2014

Small-Energy Analysis for the Selfadjoint Matrix Schroedinger Operator on the Half Line

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the corresponding Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.

math-ph↗
arXiv2014

High-Energy analysis and Levinson's theorem for the selfadjoint matrix Schroedinger operator on the half line

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are established for the related Jost matrix, the inverse of the Jost matrix, and the scattering matrix. Under the additional assumption that the matrix potential has a first moment, Levinson's theorem is derived, relating the number of bound states to the change in the argument of the determinant of the scattering matrix.

math-ph↗
arXiv2014

Small-energy analysis for the selfadjoint matrix Schroedinger operator on the half line. II

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a second moment, it is shown that the corresponding scattering matrix is differentiable at zero energy. An explicit formula is provided for the derivative of the scattering matrix at zero energy. The previously established results when the potential has only the first moment are improved when the second moment exists, by presenting the small-energy asymptotics for the related Jost matrix, its inverse, and various other quantities relevantc to the corresponding direct and inverse scattering problems.

math-ph↗
arXiv2014

Hybrid resonance of Maxwell's equations in slab geometry

Hybrid resonance is a physical mechanism for the heating of a magnetic plasma. In our context hybrid resonance is a solution of the time harmonic Maxwell's equations with smooth coefficients, where the dielectric tensor is a non diagonal hermitian matrix. The main part of this work is dedicated to the construction and analysis of a mathematical solution of the hybrid resonance with the limit absorption principle. We prove that the limit solution is singular: it is constituted of a Dirac mass at the origin plus a principle value and a smooth square integrable function. The formula obtained for the plasma heating is directly related to the singularity.

physics.plasm-ph↗
arXiv2013

High-energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and applications

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short-range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high-energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy $E.$ Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in $L^{2}\left( Ω\right) ,$ where $Ω$ is any connected bounded open set in $\mathbb{R}^{3}$ with smooth boundary, and we show that if we know an electric potential and a magnetic field for $\mathbb{R}^{3}\setminusΩ$, then the scattering amplitude, given for some energy $E$, uniquely determines these electric potential and magnetic field everywhere in $\mathbb{R}^{3}$. Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some $E$, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field, respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge-conjugation and time-reversal transformations for the Dirac operator.

math-ph↗
arXiv2013

On Open Scattering Channels for Manifolds with Ends

In the framework of time-dependent geometric scattering theory, we study the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension $n$. The smallness condition for the perturbation is expressed (intrinsically and coordinate free) in purely geometric terms using the harmonic radius; therefore, the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the Ricci-curvature. As an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the Riemannian metric. This stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations.

math-ph↗
arXiv2013

Universality of Entanglement Creation in Low-Energy Two-Dimensional Scattering

We prove that the entanglement created in the low-energy scattering of two particles in two dimensions is given by a universal coefficient that is independent of the interaction potential. This is strikingly different from the three dimensional case, where it is proportional to the total scattering cross section. Before the collision the state is a product of two normalized Gaussians. We take the purity as the measure of the entanglement after the scattering. We give a rigorous computation, with error bound, of the leading order of the purity at low-energy. For a large class of potentials, that are not assumed to be spherically symmetric, we prove that the low-energy behaviour of the purity, $\mathcal P$, is universal. It is given by $\mathcal P= 1- \frac{1}{(\ln (σ/\hbar))^2} \mathcal E$, where $σ$ is the variance of the Gaussians and the entanglement coefficient, $\mathcal E$, depends only on the masses of the particles and not on the interaction potential. 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.

quant-ph↗