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

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 |
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Uncontrolled Keywords: | Mathieu operator; Spectral gap asymptotics |

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences |

Depositing User: | Plamen Borissov Djakov |

Date Deposited: | 25 Oct 2012 21:29 |

Last Modified: | 31 Jul 2019 12:28 |

URI: | https://research.sabanciuniv.edu/id/eprint/19871 |