论文标题
关于具有周期系数的线性差分 - 代数方程的光谱理论
On the Spectral Theory of Linear Differential-Algebraic Equations with Periodic Coefficients
论文作者
论文摘要
在本文中，我们考虑了针对规范形式的周期性DAE的线性差分 - 代数方程（DAE）的光谱理论，即 j \ frac {df} {dt}+hf =λwf，\ end {equation*}，其中$ j $是不可逆转的常数偏斜的偏斜$ n \ times n $矩阵，$ h = h（t）$ w = w（t）$ w = w（t）$ w = w（t）$ d $ d $ d $ - perimitian $ nmite $ nmite nmite nmite nmitian $ nmmite nmitian $ nmmitian $ nmmitian $ nmmitian $ nmmitian $ nmmitian $ nmmitian $ nmmitian $ n c。作为条目的功能，$ w（t）$是阳性的半菲尼特，对于A.E. $ t \ in \ mathbb {r} $（即，几乎到处都是lebesgue）。根据$ h $和$ w $的其他一些假设，称为本地索引1假设，我们研究了最大和最小操作员$ l $和$ l_0' $，与差分级别的操作员$ \ Mathcal {l} = w^{ - 1}（j \ frac} $ frac {dt} dt Hilbert Space中的运营商$ l^2（\ Mathbb {r}; W）加权方形矢量值函数的$。我们证明了以下内容：（i）最小操作员$ l_0' $是一个密集且可封闭的操作员； （ii）最大操作员$ l $是$ l_0' $的关闭； （iii）$ l $是$ l^2（\ mathbb {r}; w）$上的自动伴侣运算符，没有有限的多重性能，但可能具有无限多重性的特征值。作为一个重要的应用，我们表明，对于具有无源损失介质的一维光子晶体，麦克斯韦的电磁场方程成为变量的分离，以典型形式的周期性DAE满足我们的假设，使我们的光谱理论适用于它们（本文的主要动机）。
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=λWf, \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).
