Ricardo Weder

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 point transformed electromagnetic 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 cloaking boundary conditions that the solutions of Maxwell equations have to satisfy at the outside, ∂K+, and at the inside, ∂K−, of the boundary of the cloaked object K in the case where the permittivity and the permeability are bounded below and above in K. Namely, that the tangential components of the electric and the magnetic fields have to vanish at ∂K+—which is always true—and that the normal components of the curl of the electric and the magnetic fields have to vanish at ∂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 ∂K we verify by an explicit calculation that our 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 passive and active devices contained in general anisotropic media, in particular to objects with passive and active devices contained inside general crystals. Our results suggest a method to enhance cloaking in the approximate transformation media that are used in practice. Namely, to coat the boundary of the cloaked object (the inner boundary of the cloak) with a material that imposes the boundary conditions above. As these boundary conditions have to be satisfied for exact transformation media, adding a lining that enforces them in the case of approximate transformation media will improve the performance of approximate cloaks.

The Aharonov–Bohm effect and time-dependent inverse scattering theory

We study the Aharonov–Bohm effect from the point of view of time-dependent inverse scattering theory. As this three-dimensional problem is invariant under translations along the vertical axis, it reduces to a problem in 2. We first consider an unshielded magnetic field that has a singular part produced by a tiny solenoid and a regular part. The wavefunction is zero at the location of the solenoid. We then consider the case where the singular part of the magnetic field is shielded inside a cylinder whose transverse section is a compact set K, and there is also a regular magnetic field. In this case the magnetic field inside K is quite general. In fact, the only condition is that the magnetic flux across K has to be finite. Moreover, the wavefunction is defined in Ω : = 2 K and it is zero on ∂K. Assuming that K is convex, we prove that in the unshielded case the scattering operator determines uniquely the regular magnetic field and that in the shielded case it determines uniquely the magnetic field in Ω. Moreover, in the unshielded case the scattering operator determines the magnetic flux of the solenoid modulo 2 and in the shielded case it determines the magnetic flux across K modulo 2. Our results follow from a reconstruction formula with an error term.

Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation

The Schrodinger equation on the half-line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the 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 extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectrum.

Inverse scattering for the nonlinear Schrödinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case

We solve the inverse scattering problem for the nonlinear Schrodinger equation on Rn, n ≥ 3: i ∂ ∂t u(t, x) = −∆u(t, x) + V0(x)u(t, x) + ∞ ∑ j=1 Vj(x)|u|0u(t, x). We prove that the small-amplitude limit of the scattering operator uniquely determines Vj , j = 0, 1, · · · . Our proof gives a method for the reconstruction of the potentials Vj , j = 0, 1, · · · . The results of this paper extend our previous results for the problem on the line.

Reconstruction of a potential on the line that is a priori known on the half line

In this paper the authors study the inverse Schrodinger scattering on the real line. A method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies. In particular no information on the forming constants is required. The method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the Schrodinger equation. The Deift–Trubowitz approach to inverse scattering is followed. The main new point is the way in which bound states are treated. In addition to its mathematical interest, the case when the potential is a priori known on the half line is particularly interesting in many applications. One can consider for example a potential that has compact support or that it is zero on a half line.

Scattering theory for the Klein-Gordon equation

Abstract We develop the scattering theory for the Klein-Gordon equation. We follow the usual procedure of considering an equivalent equation, which is first order in time, in the Hilbert space of vector valued functions which have a finite energy norm. We prove existence and completeness of the wave operators, the intertwining relations, and the invariance principle as well. This is done for a large class of potentials. In particular, the magnetic potential may even be divergent at infinity. Electric and scalar potentials that behave at infinity as ¦x¦ −ϵ − l , ϵ > 0 are contained in our class.

On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity

We study the inverse scattering problem for electric potentials and magnetic fields in , that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at some positive energy.

Characterization of the scattering data in multidimensional inverse scattering theory

The author considers the characterization of the scattering data of the inverse problem for the Schrodinger equation in n dimensions, n>or=2. The author obtains a solution of this problem by means of the generalized limiting absorption principle and the nonlinear delta -equation.

A theorem on the existence of dyon solutions

Abstract We prove the existence of finite energy dyon solutions to Yang-Mills-Higgs equations satisfying the Julia-Zee ansatz, and the generalization to SU(N) gauge groups. This rigorously establishes the existence of a model for the particles having electric and magnetic charge conjectured by Schwinger. We also prove that the solutions are real analytic on (0, ∞) and C∞ at r = 0. To establish our result we prove a new abstract theorem that allows one to study singular constrained minimization problems without the introduction of Lagrange multipliers.

The forced non-linear Schrödinger equation with a potential on the half-line

In this paper we prove that the initial-boundary value problem for the forced non-linear Schrodinger equation with a potential on the half-line is locally and (under stronger conditions) globally well posed, i.e. that there is a unique solution that depends continuously on the force at the boundary and on the initial data. We allow for a large class of unbounded potentials. Actually, for local solutions we have no restriction on the growth at infinity of the positive part of the potential, and for global solutions very mild assumptions that allow, for example, for exponential growth. Copyright