Extend global absence of absolutely continuous spectrum beyond reflection symmetry

Extend the global absence-of-absolutely-continuous-spectrum result for multiplicatively perturbed one-dimensional capacitance operators to resonator geometries without the stated reflection symmetry by determining the additional properties of the capacitance Floquet symbol needed to establish global monotonicity.

Background

For a one-resonator-per-period chain, the paper establishes absence of absolutely continuous spectrum in a low-energy acoustic window for general resonator geometries. A global result is obtained only when the reference resonator is invariant under reflection across the transverse cell midpoint, because that symmetry enables a Dirichlet-principle argument proving that the scalar capacitance Floquet symbol is strictly increasing on the full Brillouin half-zone.

Without reflection symmetry, the paper does not establish the corresponding global monotonicity of the Floquet symbol. Consequently, extending the global spectral conclusion to nonsymmetric geometries remains unresolved and is explicitly identified as requiring further information about the symbol.