2022/11/03 by Bader Alshammari, Alshammari, Bader, Aaron Welters +1
Physics and Astronomy · #34A09 #34L05 #46N20 #47A75 #47B25 #47B38 #47B93 #47N20 #47N50 #78M22 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Photonic Crystals and Applications #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2211.02134
openalex publication_date 2022/11/03 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28
In this paper, we consider the spectral theory of linear differential-algebraic equations (DAEs) for periodic DAEs in canonical form, i.e., J (df)/(dt)+Hf=λWf, where J is a constant skew-Hermitian n× n matrix that is not invertible, both H=H(t) and W=W(t) are d-periodic Hermitian n× n-matrices with Lebesgue measurable functions as entries, and W(t) is positive semidefinite and invertible for a.e. t∈ 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 L0', respectively, associated with the differential-algebraic operator L=W-1(J(d)/(dt)+H), both treated as an unbounded operators in a Hilbert space L2(ℝ;W) of weighted square-integrable vector-valued functions. We prove the following: (i) the minimal operator L0' is a densely defined and closable operator; (ii) the maximal operator L is the closure of L0'; (iii) L is a self-adjoint operator on L2(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).