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Absence of Absolutely Continuous Spectrum for One-Dimensional Capacitance Operators with Critical Long-Range Interactions

by H. Ammari and J. Qiu

(Report number 2026-35)

Abstract
In this paper, we establish the absence of absolutely continuous spectrum for scalar one-dimensional Laurent operators with multiplicative disorder and critical long-range hoppings. In particular, the requirement of at least quartic off-diagonal decay rate in the existing theory is bypassed by exploiting the finite-difference bound of the convolution kernel, which is available even when the interaction decays only as \(|n|^{-1}\log^{-\beta}|n|\) with \(\beta>1\). We then apply the abstract main result to a concrete physical model, the capacitance operator in subwavelength physics. For a chain of three-dimensional resonators with one resonator per period, we verify the required decay and finite-difference estimates and conclude the absence of absolutely continuous spectrum. These results leave open the distinction between pure point and singular continuous spectrum, but constitute a key step in proving Anderson localization in wave systems with critical long-range interactions.

Keywords: Capacitance operator, absolute continuous spectrum, long-range hopping, Bragg scattering, Anderson localization.

@Techreport{AQ26_1190,
  author = {H. Ammari and J. Qiu},
  title = {Absence of Absolutely Continuous Spectrum for One-Dimensional Capacitance Operators with Critical Long-Range Interactions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-35},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-35.pdf },
  year = {2026}
}

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