Refined asymptotics of the spectral gap for the Mathieu operatorAnahtarcı, 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 Full text not available from this repository. Official URL: http://dx.doi.org/10.1016/j.jmaa.2012.06.019 AbstractThe 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.
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