In this paper, we consider the spectral theory of linear differential-algebraic equations (DAEs) for periodic DAEs in canonical form, i.e.,

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. Finally, 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.

在本文中，我们考虑规范形式的周期 DAE 的线性微分代数方程 (DAE) 的谱理论，即

其中J是不可逆的常数偏斜埃尔米特矩阵\(n\times n\) ， \(H=H(t)\)和\(W=W(t)\)都是d周期埃尔米特矩阵\ (n\times n\)矩阵，以勒贝格可测函数为条目，W ( t ) 是正半定且对于 ae \(t\in {\mathbb {R}}\)可逆（即，勒贝格几乎无处不在）。在H和W的一些附加假设（称为局部索引 1 假设）下，我们分别研究与微分代数运算符\({\mathcal {L }}=W^{-1}(J\frac{d}{dt}+H)\)，两者都被视为希尔伯特空间中的无界运算符\(L^2({\mathbb {R}}; W )\)加权平方可积向量值函数。我们证明： (i) 最小算子\(L_0'\)是一个密集定义且可闭算子；(ii) 最大算子L是\(L_0'\)的闭包；(iii) L是\(L^2({\mathbb {R}}; W)\)上的自伴算子，没有有限重数的特征值，但可能有无限重数的特征值。最后，我们证明，对于具有无源无损介质的一维光子晶体，电磁场的麦克斯韦方程在变量分离下变成满足我们假设的规范形式的周期性 DAE，以便我们的谱理论适用于它们。