title   
  

Refined asymptotics of the spectral gap for the Mathieu operator

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

Abstract

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:30 Oct 2012 22:51

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