Refined asymptotics of the spectral gap for the Mathieu operator

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Anahtarcı, Berkay and Djakov, Plamen Borissov (2012) Refined asymptotics of the spectral gap for the Mathieu operator. Journal of Mathematical Analysis and Applications, 396 (1). pp. 243-255. ISSN 0022-247X

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Official URL: http://dx.doi.org/10.1016/j.jmaa.2012.06.019


The Mathieu operator L(y) = -y '' + 2a cos(2x)y, a is an element of C, a not equal 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 lambda(+)(n) - lambda(-)(n). For fixed a, we show that gimel(+)(n) - gimel(-)(n) = +/- 8(a/4)(n)/left perpendicular(n - 1)!right perpendicular(2) [1 - a(2)/4n(3) + o(1/n(4))], n -> infinity. This result extends the asymptotic formula of Harrell-Avron-Simon by providing more asymptotic terms.

Item Type:Article
Uncontrolled Keywords:Mathieu operator; Spectral gap asymptotics
Subjects:Q Science > QA Mathematics
ID Code:19871
Deposited By:Plamen Borissov Djakov
Deposited On:25 Oct 2012 21:29
Last Modified:31 Jul 2019 12:28

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