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Consider the Dirichlet Laplacian operator \(-\Delta ^D\) in a periodic waveguide \(\Omega \). Under the condition that \(\Omega \) is sufficiently thin, we show that its spectrum \(\sigma (-\Delta ^D)\) is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in \(\sigma (-\Delta ^D)\) and locate it.

Absolute Continuity and Band Gaps of the Spectrum of the Dirichlet Laplacian in Periodic Waveguides