PreprintarXiv licencearXiv
The Spectrum of the Discrete Mathieu Operator with Nonreal Coupling
Abstract
We study the discrete Mathieu operator with nonreal coupling in the large-period limit, using complex WKB methods for analytic finite-difference equations. We obtain quantization formulas in different regions of the complex energy plane. Away from the corners of the image of the classical symbol, these formulas involve cosines of classical action integrals with small analytic corrections to the actions. Near each corner, we derive a uniform quantization formula involving the reciprocal Gamma function. The quantization formulas constrain spectral accumulation to curves determined by the competing exponential terms. Numerical computations illustrate the resulting spectral geometry.
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BibTeX
@misc{klopp2026spectrum,
title = {{The Spectrum of the Discrete Mathieu Operator with Nonreal Coupling}},
author = {Frédéric Klopp and Izak Oltman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27487},
eprint = {2610.27487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27487},
}
RIS
TY - UNPB
TI - The Spectrum of the Discrete Mathieu Operator with Nonreal Coupling
AU - Frédéric Klopp
AU - Izak Oltman
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27487
UR - https://arxiv.org/abs/2610.27487
AB - We study the discrete Mathieu operator with nonreal coupling in the large-period limit, using complex WKB methods for analytic finite-difference equations. We obtain quantization formulas in different regions of the complex energy plane. Away from the corners of the image of the classical symbol, these formulas involve cosines of classical action integrals with small analytic corrections to the actions. Near each corner, we derive a uniform quantization formula involving the reciprocal Gamma function. The quantization formulas constrain spectral accumulation to curves determined by the competing exponential terms. Numerical computations illustrate the resulting spectral geometry.
ER -