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Ricardo Weder

Publications and source records attributed to Ricardo Weder.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic stability of Bernstein-Greene-Kruskal (BGK) waves, Landau Damping, and scattering theory

We consider the two species (electrons and positive charged ions) Vlasov-Poisson system linearized around BGK waves. We formulate the problem as an equivalent Vlasov-Amp\`ere system, that we write as a system of Schr\"odinger type in an appropriate Hilbert space, and with a selfadjoint Vlasov-Amp\`ere operator as a Hamiltonian. We develop a complete stationary scattering theory. We identify the absolutely continuous spectrum and the singular spectrum of the Vlasov-Amp\`ere operator, we construct the generalized Fourier maps, we prove that the wave operators exist, are complete, satisfy Birman's invariance principle, and that the stationary formulae hold. Using these results we prove that the BGK waves are asymptotically stable. We obtain a precise description of the large time behaviour of the solutions to the Vlasov-Amp\`ere system Namely,for large times the phase-space densities of electrons and ions are asymptotic to the phase-space densities of solutions of the unperturbed Vlasov-Amp\`ere system. This implies that they follow the trajectories of solutions to Newton's equations for electrons and ions with the potential of the BGK wave, in the sense that they are transported along these trajectories. Furthermore, we prove that Landau damping holds, that is to say, the electric field tends to zero in pointwise sense for large times.

math.AP↗
arXiv2025

Factorization for the matrix-valued general Jacobi system on the full-line lattice

The Jacobi system with matrix-valued coefficients and with the spectral parameter depending on a matrix-valued weight factor is considered on the full-line lattice. The scattering from the full-line lattice is expressed in terms of the scattering from the fragments of the whole lattice by developing a factorization formula for the corresponding transition matrices. In particular, the matrix-valued transmission and reflection coefficients for the full-line lattice are explicitly expressed in terms of the scattering coefficients for the left and right lattice fragments. Since the matrix-valued scattering coefficients are easier to determine for the fragments than for the full-line lattice, the factorization formula presented provides a method to determine the scattering coefficients for full-line lattices. The theory presented is illustrated with various explicit examples, including an example demonstrating that the matrix-valued left transmission coefficient in general is not equal to the matrix-valued right transmission coefficient for a lattice.

math-ph↗
arXiv2025

Changing the discrete spectrum of the half-line matrix Schrödinger operator

We consider the matrix-valued Schrödinger operator on the half line with the general selfadjoint boundary condition. When the discrete spectrum is changed without changing the continuous spectrum, we present a review of the transformations of the relevant quantities including the regular solution, the Jost solution, the Jost matrix, the scattering matrix, and the boundary matrices used to describe the selfadjoint boundary condition. The changes in the discrete spectrum are considered when an existing bound state is removed, a new bound state is added, and the multiplicity of a bound state is decreased or increased without removing the bound state. We provide various explicit examples to illustrate the theoretical resultspresented.

math-ph↗
arXiv2025

Galaxy dynamics, gravitational Vlasov-Poisson system, Landau damping, and scattering theory

We consider the gravitational Vlasov-Poisson system linearized around steady states that are extensively used to study the dynamics of galaxies, or of clusters of galaxies. Namely, polytropes and King steady states. We develop a complete stationary scattering theory for the selfadjoint, strictly positive, Antonov operator that governs the plane-symmetric linearized dynamics. We identify the absolutely continuous spectrum of the Antonov operator. Moreover, we prove that the part of the singular spectrum of the Antonov operator that is embedded in its absolutely continuous spectrum is contained in a closed set of measure zero, that we characterize. We construct the generalized Fourier maps, and we prove that the wave operators exist and are complete. Moreover, we obtain stationary formulae for the wave operators, and we prove that Birman's invariance principle holds. Using these results we obtain a precise description of the dynamics of the stars in the galaxies, or of the galaxies in the clusters of galaxies, for large times. Namely, we prove that the distribution function of the solutions to the linearized gravitational Vlasov-Poisson system with initial data in the absolutely continuous subspace of the Antonov operator are asymptotic, for large times, to the solutions to the unperturbed linearized gravitational Vlasov-Poisson system. This implies that they are asymptotic to the trajectories of the solutions to Newton's equation with the gravitational potential of the steady state, in the sense that they are transported along these trajectories. Moreover, for these initial states the gravitational Landau damping holds. Namely, we prove that the gravitational force and its time derivative, as well as the gravitational potential and its time derivative, tend to zero for large times.

math.AP↗
arXiv2024

Membrane-in-the-middle optomechanical system and structural frequencies

We consider a one-dimensional membrane-in-the-middle model for a cavity that consists of two fixed, perfect mirrors and a mobile dielectric membrane between them that has a constant electric susceptibility. We present a sequence of exact cavity angular frequencies that we call structural angular frequencies and that have the remarkable property that they are independent of the position of the membrane inside the cavity. Furthermore, the case of a thin membrane is considered and simple, approximate formulae for the angular frequencies and for the modes of the cavity are obtained. Finally, the cavity electromagnetic potential is numerically calculated and it is found that the potential is accurately described by a multiple scales solution.

quant-ph↗
arXiv2024

The transformations to remove or add bound states for the half-line matrix Schrödinger operator

We present the transformations to remove or add bound states or to decrease or increase the multiplicities of any existing bound states for the half-line matrix-valued Schrödinger operator with the general selfadjoint boundary condition, without changing the continuous spectrum of the operator. When the matrix-valued potential is selfadjoint, is integrable, and has a finite first moment, the relevant transformations are constructed through the development of the Gel'fand-Levitan method for the corresponding Schrödinger operator. In particular, the bound-state normalization matrices are constructed at each bound state for any multiplicity. The transformations are obtained for all relevant quantities, including the matrix potential, the Jost solution, the regular solution, the Jost matrix, the scattering matrix, and the boundary condition. For each bound state, the corresponding dependency matrix is introduced by connecting the normalization matrix used in the Gel'fand-Levitan method and the normalization matrix used in the Marchenko method of inverse scattering. Various estimates are provided to describe the large spacial asymptotics for the change in the potential when the bound states are removed or added or their multiplicities are modified. An explicit example is provided showing that an asymptotic estimate available in the literature in the scalar case for the potential increment is incorrect.

math-ph↗
arXiv2023

Factorization for the full-line matrix Schrödinger equation and a unitary transformation to the half-line scattering

The scattering matrix for the full-line matrix Schrödinger equation is analyzed when the corresponding matrix-valued potential is selfadjoint, integrable, and has a finite first moment. The matrix-valued potential is decomposed into a finite number of fragments, and a factorization formula is presented expressing the matrix-valued scattering coefficients in terms of the matrix-valued scattering coefficients for the fragments. Using the factorization formula, some explicit examples are provided illustrating that in general the left and right matrix-valued transmission coefficients are unequal. A unitary transformation is established between the full-line matrix Schrödinger operator and the half-line matrix Schrödinger operator with a particular selfadjoint boundary condition and by relating the full-line and half-line potentials appropriately. Using that unitary transformation, the relations are established between the full-line and the half-line quantities such as the Jost solutions, the physical solutions, and the scattering matrices. Exploiting the connection between the corresponding full-line and half-line scattering matrices, Levinson's theorem on the full line is proved and is related to Levinson's theorem on the half line.

math-ph↗
arXiv2022

The Matrix Nonlinear Schrödinger Equation with a Potential

This paper is devoted to the study of the large-time asymptotics of the small solutions to the matrix nonlinear Schrödinger equation with a potential on the half-line and with general selfadjoint boundary condition, and on the line with a potential and a general point interaction, in the whole supercritical regime. We prove that the small solutions are scattering solutions that asymptotically in time, $t \to\pm\infty, $ behave as solutions to the associated linear matrix Schrödinger equation with the potential identically zero. The potential can be either generic or exceptional. Our approach is based on detailed results on the spectral and scattering theory for the associated linear matrix Schrödinger equation with a potential, and in a factorization technique that allows us to control the large-time behaviour of the solutions in appropriate norms.

math.AP↗
arXiv2022

A short review of the Casimir effect with emphasis on dynamical boundary conditions

We give a short review on the static and dynamical Casimir effects, recalling their historical prediction, as well as their more recent experimental verification. We emphasise on the central role played by so-called {\it dynamical boundary conditions} (for which the boundary condition depends on a second time derivative of the field) in the experimental verification of the dynamical Casimir effect by Wilson et al. We then go on to review our previous work on the static Casimir effect with dynamical boundary conditions, providing an overview on how to compute the so-called local Casimir energy, the total Casimir energy and the Casimir force. We give as a future perspective the direction in which this work should be generalised to put the theoretical predictions of the dynamical Casimir effect experiments on a rigorous footing.

hep-th↗
arXiv2021

The Magnetized Vlasov-Ampère system and the Bernstein-Landau paradox

We study the Bernstein-Landau paradox in the collisionless motion of an electrostatic plasma in the presence of a constant external magnetic field. The Bernstein-Landau paradox consists in that in the presence of the magnetic field, the electric field and the charge density fluctuation have an oscillatory behavior in time. This is radically different from Landau damping, in the case without magnetic field, where the electric field tends to zero for large times. We consider this problem from a new point of view. Instead of analyzing the linear magnetized Vlasov-Poisson system, as it is usually done, we study the linear magnetized Vlasov-Ampère system. We formulate the magnetized Vlasov-Ampère system as a Schrödinger equation with a selfadjoint magnetized Vlasov-Ampère operator in the Hilbert space of states with finite energy. The magnetized Vlasov-Ampère operator has a complete set of orthonormal eigenfunctions, that include the Bernstein modes. The expansion of the solution of the magnetized Vlasov-Ampère system in the eigenfunctions shows the oscillatory behavior in time. We prove the convergence of the expansion under optimal conditions, assuming only that the initial state has finite energy. This solves a problem that was recently posed in the literature. The Bernstein modes are not complete. To have a complete system it is necessary to add eigenfunctions that are associated with eigenvalues at all the integer multiples of the cyclotron frequency. These special plasma oscillations actually exist on their own, without the excitation of the other modes. In the limit when the magnetic fields goes to zero the spectrum of the magnetized Vlasov-Ampère operator changes drastically from pure point to absolutely continuous in the orthogonal complement to its kernel, due to a sharp change on its domain. This explains the Bernstein-Landau paradox.

physics.plasm-ph↗
arXiv2021

Trace maps under weak regularity assumptions

We study bounded trace maps on hypersurfaces for Sobolev spaces from a point of view that is fundamentally different from the one in the classical theory. This allows us to construct bounded trace maps under weak regularity assumptions on the hypersurfaces. In the case of bounded domains in $\mathbf R^n$ we only require the continuity of the boundary. For hypersurfaces in the whole space $\mathbf R^n$ we only assume that the hypersurfaces are Lebesgue measurable. As an application of our trace maps we consider the Dirichlet problem and we prove a coarea formula where the level sets are only assumed to be Lebesgue measurable hypersurfaces.

math.AP↗
arXiv2021

The $L^{p}$ boundedness of the wave operators for matrix Schrödinger equations

We prove that the wave operators for $n \times n$ matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, $ for slowly decaying selfadjoint matrix potentials, $V, $ that satisfy $\int_{0}^{\infty }\, (1+x) |V(x)|\, dx < \infty.$ Moreover, assuming that $\int_{0}^{\infty }\, (1+x^γ) |V(x)|\, dx < \infty, γ> \frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(\mathbb R^+, \mathbb C^n),$ and in $L^\infty(\mathbb R^+, \mathbb C^n).$ We also prove that the wave operators for $n\times n$ matrix Schrödinger equations on the line are bounded in the spaces $L^p(\mathbb R, \mathbb C^n), 1 < p < \infty, $ assuming that the perturbation consists of a point interaction at the origin and of a potential, $\mathcal V,$ that satisfies the condition $\int_{-^{\infty}}^{\infty}\, (1+|x|)\, |\mathcal V(x)|\, dx < \infty.$ Further, assuming that $\int_{-\infty}^{\infty }\, (1+|x|^γ) |\mathcal V(x)|\,dx < \infty, γ> \frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(\mathbb R, \mathbb C^n),$ and in $L^\infty(\mathbb R, \mathbb C^n).$ We obtain our results for $n\times n$ matrix Schrödinger equations on the line from the results for $2n\times 2n$ matrix Schrödinger equations on the half line.

math-ph↗
arXiv2021

Quantum field theory with dynamical boundary conditions and the Casimir effect: Coherent states

We have studied in a previous work the quantization of a mixed bulk-boundary system describing the coupled dynamics between a bulk quantum field confined to a spacetime with finite space slice and with timelike boundary, and a boundary observable defined on the boundary. Our bulk system is a quantum field in a spacetime with timelike boundary and a dynamical boundary condition-the boundary observable's equation of motion. Owing to important physical motivations, in such previous work we have computed the renormalized local state polarization and local Casimir energy for both the bulk quantum field and the boundary observable in the ground state and in a Gibbs state at finite, positive temperature. In this work, we introduce an appropriate notion of coherent and thermal coherent states for this mixed bulk-boundary system, and extend our previous study of the renormalized local state polarization and local Casimir energy to coherent and thermal coherent states. We also present numerical results for the integrated Casimir energy and for the Casimir force.

hep-th↗
arXiv2020

$L^{p}-L^{p^{\prime}}$ estimates for matrix Schrödinger equations

This paper is devoted to the study of dispersive estimates for matrix Schrödinger equations on the half-line with general boundary condition, and on the line. We prove $L^{p}-L^{p^{\prime}}$ estimates on the half-line for slowly decaying selfadjoint matrix potentials that satisfy $\int_{0}^{\infty }\, (1+x) |V(x)|\, dx < \infty$ both in the generic and in the exceptional cases. We obtain our $L^{p}-L^{p^{\prime}}$ estimate on the line for a $n \times n$ system, under the condition that $\int_{-^{\infty}}^{\infty}\, (1+|x|)\, |V(x)|\, dx < \infty,$ from the $L^{p}-L^{p^{\prime}}$ estimate for a $2n\times2n$ system on the half-line. With our $L^{p}-L^{p^{\prime}}$ estimates we prove Strichartz estimates.

math-ph↗
arXiv2020

Quantum field theory with dynamical boundary conditions and the Casimir effect

We study a coupled system that describes the interacting dynamics between a bulk field, confined to a finite region with timelike boundary, and a boundary observable. In our system the dynamics of the boundary observable prescribes dynamical boundary conditions for the bulk field. We cast our classical system in the form of an abstract linear Klein-Gordon equation, in an enlarged Hilbert space for the bulk field and the boundary observable. This makes it possible to apply to our coupled system the general methods of quantization. In particular, we implement the Fock quantization in full detail. Using this quantization we study the Casimir effect in our coupled system. Specifically, we compute the renormalized local state polarization and the local Casimir energy, which we can define for both the bulk field and the boundary observable of our system. Numerical examples in which the integrated Casimir energy is positive or negative are presented.

hep-th↗
arXiv2018

Scattering Theory for the matrix Schrödinger operator on the half line with general boundary conditions

We study the stationary scattering theory for the matrix Schrödinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption principle, we construct the generalized Fourier maps, and we prove that they are partially isometric with initial space the subspace of absolute continuity of the matrix Schrödinger operator and final space $L^2((0, \infty))$. We prove the existence and the completeness of the wave operators and we establish that they are given by the stationary formulae. We also construct the spectral shift function and we give its high-energy asymptotics. Furthermore, assuming that the potential also has a finite first moment, we prove a Levinson's theorem for the spectral shift function.

math-ph↗