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Eric Stachura

Publications and source records attributed to Eric Stachura.

7 recordsLinked to original sources
arXiv2026

Refraction laws in spatio-temporal media

We study the time-dependent Maxwell system, formulated in the sense of distributions, for electromagnetic waves propagating through media with temporal and spatial material interfaces. Under explicit trace and regularity assumptions on the permittivity and permeability, we derive the jump conditions produced by temporal discontinuities and by subsequent spatial interfaces. These conditions are used to obtain generalized Snell laws for the wave vectors generated by temporal splitting, reflection, and transmission. We also derive the associated amplitude equations and organize them as a finite-dimensional linear system for a space-time slab geometry. Finally, we provide an explicit oblique-incidence example. Our approach does not impose a smooth-field ansatz and allows material parameters that need not be constant away from the interfaces.

math.AP↗
arXiv2025

Refraction laws in temporal media

We consider the time dependent Maxwell system in the sense of distributions in the context of temporal interfaces. Just as with spatial interfaces, electromagnetic waves at temporal interfaces scatter and create a transmitted and reflected wave. We provide a rigorous derivation of boundary conditions for the electric and magnetic fields at temporal interfaces with precise assumptions on the material parameters. In turn, we use this to obtain a general Snell's Law at such interfaces. From this, we obtain explicit formulas for the reflection and transmission coefficients. Unlike previous works, we do not make any simplifying ansatz on the solution to the Maxwell system, nor do we assume that the fields are smooth. We also consider material parameters which are not necessarily constant on either side of the temporal interface.

math.AP↗
arXiv2020

Quantitative Trace Estimates for the Maxwell system in Lipschitz Domains

We develop various quantitative estimates for the anisotropic Maxwell system in Lipschitz domains, with a focus on how the estimates precisely depend on the Lipschitz character of the domain. We pay special attention to trace operators and extension operators over certain Sobolev spaces. Finally, we provide a weak formulation of the interior scattering problem in terms of the exterior Calderón operator, and provide explicit bounds for the solution of the interior problem in terms of the incident fields and the Lipschitz character of the domain.

math.AP↗
arXiv2017

Existence of Propagators for Time Dependent Coulomb-like Potentials

We prove existence of propagators for a time dependent Schrödinger equation with a new class of softened Coulomb potentials, which we allow to be time dependent, in the context of time dependent density functional theory. We compute explicitly the Fourier transform of these new potentials, and provide an alternative proof for the Fourier transform of the Coulomb potential using distribution theory. Finally we show the new potentials are dilatation analytic, and so the spectrum of the corresponding Hamiltonian can be fully characterized.

math-ph↗
arXiv2017

General Refraction Problems with Phase Discontinuity

This paper provides a mathematical approach to study metasurfaces in non flat geometries. Analytical conditions between the curvature of the surface and the set of refracted directions are introduced to guarantee the existence of phase discontinuities. The approach contains both the near and far field cases. A starting point is the formulation of a vector Snell law in presence of abrupt discontinuities on the interfaces.

physics.optics↗
arXiv2015

Uniform Refraction in Negative Refractive Index Materials

We study the problem of constructing an optical surface separating two homogeneous, isotropic media, one of which has a negative refractive index. In doing so, we develop a vector form of Snell's law, which is used to study surfaces possessing a certain uniform refraction property, both in the near and far field cases. In the near field problem, unlike the case when both materials have positive refractive index, we show that the resulting surfaces can be neither convex nor concave.

physics.optics↗