On the Instability Intervals of the Mathieu–Hill Operator

Kıraç, Alp Arslan

doi:10.1007/s11005-007-0215-6

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“…Let us mention that the paper [15] claims to provide "the formula which states the isolated terms of arbitrary number in the asymptotics of the sequence γ n ." However, this claim is false due to a unavoidable technical mistake (in [15], formula (5) does not imply (4) for m = k + 1 since, by Stirling's formula, the remainder in (5) is much larger than the main term in (4)).…”

Section: Introductionmentioning

confidence: 99%

Refined asymptotics of the spectral gap for the Mathieu operator

Anahtarcı

Djakov

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tThe Mathieu operator L(y) = −y ′′ + 2a cos(2x)y, a ∈ C, a ̸ = 0, considered with periodic or anti-periodic boundary conditions has, close to n 2 for large enough n, two periodic (if n is even) or anti-periodic (if n is odd) eigenvalues λ − n , λ + n . For fixed a, we show thatThis result extends the asymptotic formula of Harrell-Avron-Simon by providing more asymptotic terms.

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