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High-Index Asymptotics of Mathieu Spectral Gaps and Splittings
Abstract
Consider the Mathieu operator L=−d2/dx2+2acos(2x) for fixed a∈ℂ∖{0}. For large enough n, the periodic (n even) or anti-periodic (n odd) problem on [0,π] has exactly two algebraically simple eigenvalues λn-,λn+ in the disc |λ−n2|<1. We prove that, as n→∞, λn+−λn-=±8(a/4)n/([(n−1)!]2)exp[−(a2(n+2))/(4(n2−1)2)−15a4/128n7+O(n−8)], where the sign depends on the labelling. For real a with λn-<λn+, this difference is the length of the nth spectral gap. Expanding the exponential determines the relative corrections of orders n−4 to n−7, beyond the correction −a2/(4n3) obtained by Anahtarci and Djakov (2012) with a relative O(n−4) remainder. By isospectrality, the same formula holds for the potential ce−2ix+de2ix with a2=cd≠0.
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BibTeX
@misc{anahtarci2026highindex,
title = {{High-Index Asymptotics of Mathieu Spectral Gaps and Splittings}},
author = {Berkay Anahtarci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12751},
eprint = {2610.12751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12751},
}
RIS
TY - UNPB
TI - High-Index Asymptotics of Mathieu Spectral Gaps and Splittings
AU - Berkay Anahtarci
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12751
UR - https://arxiv.org/abs/2610.12751
AB - Consider the Mathieu operator L=−d^2/dx^2+2acos(2x) for fixed a∈ℂ∖{0}. For large enough n, the periodic (n even) or anti-periodic (n odd) problem on [0,π] has exactly two algebraically simple eigenvalues λ_n^-,λ_n^+ in the disc |λ−n^2|<1. We prove that, as n→∞, λ_n^+−λ_n^-=±8(a/4)^n/([(n−1)!]^2)exp[−(a^2(n+2))/(4(n^2−1)^2)−15a^4/128n^7+O(n^−8)], where the sign depends on the labelling. For real a with λ_n^-<λ_n^+, this difference is the length of the nth spectral gap. Expanding the exponential determines the relative corrections of orders n^−4 to n^−7, beyond the correction −a^2/(4n^3) obtained by Anahtarci and Djakov (2012) with a relative O(n^−4) remainder. By isospectrality, the same formula holds for the potential ce^−2ix+de^2ix with a^2=cd≠0.
ER -