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Weder Ricardo | Weder Benjamin | Weder D. | Weder R. | Weder Silvan

Ricardo Manuel | Ricardo Luis Giraldo González | Ricardo L. G. González | Ricardo Fraiman | Ricardo Guilherme I. | Ricardo J. | Ricardo José Luis Serrano | Ricardo Salazar

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

, Weder Ricardo
06 Aug 2020 hep-th math-ph math.MP arxiv.org/abs/2008.02842

We have previously studied -in part I- 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 part I, 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.

Quantum field theory with dynamical boundary conditions and the Casimir effect

, Weder Ricardo
12 Apr 2020 hep-th math-ph math.MP quant-ph arxiv.org/abs/2004.05646

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.

The Vlasov-Amp\`ere system and the Bernstein-Landau paradox

, , Rege Alexandre, Weder Ricardo
26 Feb 2020 physics.plasm-ph math-ph math.MP math.SP arxiv.org/abs/2002.11380

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 Vlasov-Poisson system, as it is usually done, we study the linear Vlasov-Amp`ere system. We formulate the Vlasov-Amp`ere system as a Schr\"odinger equation with a selfadjoint Vlasov-Amp`ere operator in the Hilbert space of states with finite energy. The Vlasov-Amp`ere operator has a complete set of orthonormal eigenfunctions, that include the Bernstein modes. The expansion of the solution of the Vlasov-Amp`ere 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 Vlasov-Amp`ere 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. Furthermore, we present numerical simulations that illustrate the Bernstein-Landau paradox.

The $L^{p}$ boundedness of the wave operators for matrix Schr\"{o}dinger equations

Weder Ricardo
29 Dec 2019 math-ph math.MP arxiv.org/abs/1912.12793

We prove that the wave operators for $n \times n$ matrix Schr\"odinger 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 $\int_{0}^{\infty}\, (1+x) |V(x)|\, dx < \infty,$ both in the generic and in the exceptional cases. We also prove that the wave operators for $n\times n$ matrix Schr\"odinger 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 $\int_{-^{\infty}}^{\infty}\,(1+|x|)\, |\mathcal V(x)|\, dx < \infty.$ We obtain our results for $n\times n$ matrix Schr\"odinger equations on the line from the results for $2n\times 2n$ matrix Schr\"odinger equations on the half line.

$L^{p}-L^{p^{\prime}}$ estimates for matrix Schr\"{o}dinger equations

Naumkin Ivan, Weder Ricardo
18 Jun 2019 math-ph math.MP arxiv.org/abs/1906.07846

This paper is devoted to the study of dispersive estimates for matrix Schr\"odinger 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.

Trace maps for rough hypersurfaces in Sobolev spaces

Weder Ricardo
14 Oct 2018 math.AP arxiv.org/abs/1810.06094

In this paper we revisit the problem of bounded trace maps on hypersurfaces, for Sobolev spaces. We look to this problem from a point of view that is fundamentally different from the one in the classical theory of trace maps. In this way, we construct bounded trace maps on rough hypersurfaces, for Sobolev spaces, under weak assumptions on the regularity of the hypersurfaces. Furthermore, using our trace maps we prove a coarea formula where the level sets are rough hypersurfaces that satisfy weak regularity assumptions.

Inverse scattering on the half line for the matrix Schr\"odinger equation

Aktosun Tuncay, Weder Ricardo
02 Jun 2018 math-ph math.MP arxiv.org/abs/1806.01644

The matrix Schr\"odinger 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.

The Inverse Scattering Problem for the Matrix Schr\"odinger Equation

Aktosun Tuncay, Weder Ricardo
12 Aug 2017 math-ph math.MP arxiv.org/abs/1708.03837

The matrix Schr\"odinger 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.

The number of eigenvalues of the matrix Schr\"odinger operator on the half line with general boundary conditions

Weder Ricardo
08 May 2017 math-ph math.MP arxiv.org/abs/1705.03157

We prove a bound, of Bargmann- Birman-Schwinger type, on the number of eigenvalues of the matrix Schr\"odinger operator on the half line, with the most general self adjoint boundary condition at the origin, and with selfadjoint matrix potentials that are integrable and have a finite first moment.

Trace Identities for the matrix Schr\"odinger operator on the half line with general boundary conditions

Weder Ricardo
30 Mar 2016 math-ph math.MP arxiv.org/abs/1603.09432

We prove Buslaev-Faddeev trace identities for the matrix Schr\"odinger operator on the half line, with general boundary conditions at the origin, and with selfadjoint matrix potentials.