On the Spectral Theory of Linear Differential-Algebraic Equations with Periodic Coefficients

TL;DR

This paper develops a spectral theory for periodic linear differential-algebraic equations, establishing operator properties and applying the results to Maxwell's equations in photonic crystals with passive media.

Contribution

It introduces a spectral framework for periodic DAEs in canonical form and applies it to electromagnetic problems in photonic crystals, extending the understanding of their spectral properties.

Findings

The minimal operator is densely defined and closable.

The maximal operator is the closure of the minimal operator.

The maximal operator is self-adjoint with no finite-multiplicity eigenvalues.

Abstract

In this paper, we consider the spectral theory of linear differential-algebraic equations (DAEs) for periodic DAEs in canonical form, i.e., \begin{equation*} J \frac{df}{dt}+Hf=\lambda Wf, \end{equation*} where $J$ is a constant skew-Hermitian $n\times n$ matrix that is not invertible, both $H=H(t)$ and $W=W(t)$ are $d$-periodic Hermitian $n\times n$-matrices with Lebesgue measurable functions as entries, and $W(t)$ is positive semidefinite and invertible for a.e. $t\in \mathbb{R}$ (i.e., Lebesgue almost everywhere). Under some additional hypotheses on $H$ and $W$, called the local index-1 hypotheses, we study the maximal and the minimal operators $L$ and $L_0'$, respectively, associated with the differential-algebraic operator $\mathcal{L}=W^{-1}(J\frac{d}{dt}+H)$, both treated as an unbounded operators in a Hilbert space $L^2(\mathbb{R};W)$ of weighted square-integrable vector-valued functions. We prove the following: (i) the minimal operator $L_0'$ is a densely defined and closable operator; (ii) the maximal operator $L$ is the closure of $L_0'$; (iii) $L$ is a self-adjoint operator on $L^2(\mathbb{R};W)$ with no eigenvalues of finite multiplicity, but may have eigenvalues of infinite multiplicity. As an important application, we show that for 1D photonic crystals with passive lossless media, Maxwell's equations for the electromagnetic fields become, under separation of variables, periodic DAEs in canonical form satisfying our hypotheses so that our spectral theory applies to them (a primary motivation for this paper).