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

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).