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.,

$$\begin{aligned} J \frac{df}{dt}+Hf=\lambda Wf, \end{aligned}$$

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.

Appendix A: Notation and auxiliary theorems

For any $z\in {\mathbb {C}}$, its complex conjugate and norm are ${\overline{z}}$ and $\vert z \vert =({\overline{z}}z)^{1/2},$ respectively. For any interval $I\subseteq {\mathbb {R}}$, the complex vector space of Lebesgue measurable functions (with equality in the sense of equal a.e. on I) is denoted by ${\mathcal {M}}(I)$. For each $p\in [1,\infty ]$, the subspace $L^p(I)$ of ${\mathcal {M}}(I)$, defined by

$$\begin{aligned} L^p(I)&=\left\{ f\in {\mathcal {M}}(I):\int _{I}\vert f(t)\vert ^p dt<\infty \right\} ,\text { if } p\not =\infty ,\\ L^{\infty }(I)&=\left\{ f\in {\mathcal {M}}(I):\text {ess}\,\text {sup}_{t\in I}\vert f(t)\vert <\infty \right\} \end{aligned}$$

is a Banach space with norm

$$\begin{aligned} f\in L^p(I),\; \Vert f\Vert _p= \left\{ \begin{array}{cc} \left( \int _{I}\vert f(t)\vert ^p dt \right) ^{1/p}, &{} \text { if } p\not =\infty , \\ \text {ess}\,\text {sup}_{t\in I}\vert f(t)\vert , &{} \text { if } p=\infty \end{array} \right. \end{aligned}$$

and, in the case $p=2$, is a Hilbert space with inner product

$$\begin{aligned} \langle f,g \rangle _2=\int _{I}\overline{f(t)}g(t) dt,\;\;f,g\in L^2(I). \end{aligned}$$

For any compact interval $I=[a,b]$, we denote the Banach space of all complex-valued absolutely continuous functions on the interval I by AC(I) with norm

$$\begin{aligned} \Vert f\Vert _{AC(I)} = \vert f(a)\vert +\int _{I}\left| \frac{df}{dt}(\tau )\right| d\tau ,\;\;f\in AC(I). \end{aligned}$$

For any interval I, the subspace $L_{loc}^p(I)$ of ${\mathcal {M}}(I)$ and the complex vector space $AC_{loc}(I)$ are defined by

$$\begin{aligned} L_{loc}^p(I)=\{f\in {\mathcal {M}}(I):f\in L^p([a,b]),\text { for every compact interval }[a,b]\subseteq I\},\\ AC_{loc}(I)=\{f:I\rightarrow {\mathbb {C}}\;\vert \;f\in AC([a,b]),\text { for every compact interval }[a,b]\subseteq I\}, \end{aligned}$$

respectively. The subspace $W^{1,p}(I)$ of $L^p(I)$, defined by

$$\begin{aligned} W^{1,p}(I)=\left\{ f\in L^p(I):f\in AC_{loc}(I), \frac{df}{dt}\in L^p(I)\right\} , \end{aligned}$$

is a Banach space with norm

$$\begin{aligned} \Vert f\Vert _{1,p}=\Vert f\Vert _p+\left\| \frac{df}{dt}\right\| _p,\;\;f\in W^{1,p}(I). \end{aligned}$$

The subspace $W^{1,p}_{loc}(I)$ of $L^p_{loc}(I)$ is defined by

$$\begin{aligned} W^{1,p}_{loc}(I)=\{f\in {\mathcal {M}}(I):f\in W^{1,p}([a,b]),\text { for every compact interval }[a,b]\subseteq I\}. \end{aligned}$$

If $f\in W_{loc}^{1,p}(I)$ then there is a unique $g\in AC_{loc}(I)$ such that $f(t)=g(t)$ for a.e. $t\in I$, and as such, we will always use this representative of f when we evaluate f at a point, i.e., for each $t_0\in I$ will define $f(t_0):=g(t_0)$. Similarly for $f\in W^{1,p}(I)$.

Let $m,n\in {\mathbb {N}}$, I an interval, $p\in [1,\infty ]$, and

$$\begin{aligned} {\mathcal {V}}\in \{{\mathbb {C}},\;{\mathcal {M}}(I),\; L_{loc}^p(I),\; W^{1,p}_{loc}(I),\; AC_{loc}(I)\}. \end{aligned}$$

We denote the set of all $n\times m$ matrices with entries in ${\mathcal {V}}$ by $M_{n,m}({\mathcal {V}})$, and define

$$\begin{aligned} {\mathcal {V}}^n=M_{n,1}({\mathcal {V}}),\;\;M_{n}({\mathcal {V}})=M_{n,n}({\mathcal {V}}) \end{aligned}$$

and identify ${\mathcal {V}}$ with ${\mathcal {V}}^1.$ If ${\mathcal {V}}$ is a Banach space (Hilbert space) with norm $\Vert \cdot \Vert _{{\mathcal {V}}}$ then $M_{n,m}({\mathcal {V}})$ will denote the Banach space (Hilbert space) with norm

$$\begin{aligned} \Vert [a_{ij}]\Vert =\left( \sum _{i=1}^n\sum _{j=1}^m\Vert a_{ij}\Vert _{{\mathcal {V}}}^2\right) ^{1/2},\;\;[a_{ij}]\in M_{n,m}({\mathcal {V}}). \end{aligned}$$

In particular, for the Hilbert spaces ${\mathbb {C}}^n$ and $(L^2(I))^n$, their inner products $\langle \cdot , \cdot \rangle$ and $\langle \cdot ,\cdot \rangle _2$, respectively, are defined as

$$\begin{aligned}{} & {} \langle x, y \rangle =x^*y,\;\;x,y\in {\mathbb {C}}^n,\\{} & {} \langle f,g\rangle _2=\int _I\langle f(t),g(t)\rangle dt,\;\;f,g\in (L^2(I))^n, \end{aligned}$$

where $*$ denotes conjugate-transpose.

If ${\mathcal {V}}$ is a functional space in which integration $\int _{U}(\cdot )\; dt$ over an interval $U\subseteq I$ or differentiation $\frac{d}{dt}$ (either in classical or weak sense) is well-defined then for any $[a_{ij}]\in M_{n,m}({\mathcal {V}})$ we define, respectively,

$$\begin{aligned} \int _{U}[a_{ij}](t)dt=\left[ \int _{U}a_{ij}(t)dt\right] ,\;\; \frac{d[a_{ij}]}{dt}=\left[ \frac{da_{ij}}{dt}\right] . \end{aligned}$$

We need the following from [45, Theorem 1.2.1], [46, Theorem 3.2] (resp.):

Theorem 36

Let I be any interval and $m,n\in {\mathbb {N}}$. If

$$\begin{aligned} A\in M_n(L^1_{loc}(I)),\;\;F\in M_{n,m}(L^1_{loc}(I)) \end{aligned}$$

(A1)

then every initial-value problem (IVP)

$$\begin{aligned} \frac{dX}{dt}+AX=F,\;\;X(t_0)=C,\;\;t_0\in I,\;\;C\in M_{n,m}({\mathbb {C}}) \end{aligned}$$

on I, has a unique solution

$$\begin{aligned} X\in M_{n,m}(W^{1,1}_{loc}(I)). \end{aligned}$$

(A2)

Similarly, in the case in which the interval I is bounded, the statement is true if the “loc" is dropped in the hypotheses (A1) and in the conclusion (A2).

Theorem 37

If H, K are Hilbert spaces with inner products $\langle \cdot , \cdot \rangle _H$, $\langle \cdot , \cdot \rangle _K$, respectively, D(A) and D(B) are subspaces of H and K, respectively, and $A:D(A)\rightarrow K$, $B:D(B)\rightarrow H$ are linear operators satisfying

$$\begin{aligned} \langle Ax, y \rangle _K=\langle x, By\rangle _H,\;\;\text {for all }x\in D(A), y\in D(B),\\ \ker A +{\text {ran}}B=H,\;\ker B +{\text {ran}}A=K, \end{aligned}$$

then A and B are densely defined closed operators with closed ranges and are adjoints of each other, i.e., $A^*=B,\;\;B^*=A.$