On the Spectral Theory of Linear Differential-Algebraic Operators with Periodic Coefficients
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In this thesis, the spectral theory of linear differential algebraic equations (DAEs) is considered in detail and extended to treat the weighted spectral theory which generalizes the classical theory, i.e., we develop the spectral theory for the most general DAEs: J df dt + Hf = λWf, (0.0.1) where J is a constant nonzero skew-Hermitian n×n-matrix, both H and W are dperiodic Hermitian n×n-matrices with Lebesgue measurable functions as entries, and W is positive semidefinite and invertible for a.e. t ∈ R (i.e., Lebesgue almost everywhere). Under weakest hypotheses on H and W currently known, 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 L := W−1(J d dt + H) treated as an unbounded operator in a Hilbert space L2(R;W) of weighted square-integrable vector-valued functions. We first prove that the minimal operator L′ 0 is a densely defined and closable operator. Then, we show that the maximal operator L is a densely defined closed operator whose adjoint is the closure of the minimal operator L′ 0. Finally, under the local index-1 hypotheses, we prove that the maximal operator L is self-adjoint operator and has no eigenvalues of finite multiplicity. As an important application, we show that for 1D photonic crystals with passive lossless media, Maxwell’s equations become DAEs of the form (0.0.1) satisfying all our hypotheses and hence our spectral theory applies to them (a primary motivation for this thesis).