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Malcolm Brown

Publications and source records attributed to Malcolm Brown.

8 recordsLinked to original sources
arXiv2022

Spectrum of the Maxwell Equations for a Flat Interface between Homogeneous Dispersive Media

The paper determines and classifies the spectrum of a non-self-adjoint operator pencil generated by the time-harmonic Maxwell problem with a nonlinear dependence on the frequency for the case of two homogeneous materials joined at a planar interface. We study spatially one-dimensional and two-dimensional reductions in the whole space $\mathbb{R}$ and $\mathbb{R}^2$. The dependence on the spectral parameter, i.e. the frequency, is in the dielectric function and we make no assumptions on its form. These function values determine the spectral sets. In order to allow also for non-conservative media, the dielectric function is allowed to be complex, yielding a non-self-adjoint problem. The whole spectrum consists of eigenvalues and the essential spectrum, but the various standard types of essential spectra do not coincide in all cases. The main tool for determining the essential spectra are Weyl sequences.

math.SP↗
arXiv2018

Essential spectrum for Maxwell's equations

We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity $\varepsilon$, permeability $\mu$ and conductivity $\sigma$, on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil $\mathrm{div}((\omega\varepsilon + i \sigma) \nabla\,\cdot\,)$, and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwell's and related systems.

math.FA↗
arXiv2015

Gap localization of TE-Modes by arbitrarily weak defects

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator.

math-ph↗
arXiv2015

Uniqueness for an inverse problem in electromagnetism with partial data

A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is proven. No special geometrical condition is imposed on the inaccessible part of the boundary of the domain, apart from imposing that the boundary of the domain is $C^{1,1}$. The coefficients are assumed to coincide on a neighbourhood of the boundary, a natural property in applications.

math.AP↗
arXiv2012

On the spectrum of waveguides in planar photonic bandgap structures

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, guided mode spectrum can be created by arbitrarily "small" perturbations. Secondly we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum $k_x$ the perturbation generates at most finitely many new eigenvalues inside the gap.

math-ph↗
arXiv2009

The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -- which together indicate that the abstract results are probably best possible.

math.SP↗
arXiv2007

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map.

math.SP↗