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arXiv · 1603.04924

WKB and Resurgence in the Mathieu Equation

Gerald V. Dunne·Mithat Unsal
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Abstract

In this paper, based on lectures by the authors at the May 2015 workshop {\it Resurgence, Physics and Numbers}, at the Centro di Ricerca Matematica Ennio De Giorgio of the Scuola Normale Superiore in Pisa, we explain the origin of resurgent trans-series in the Mathieu equation spectral problem, using uniform WKB and all-orders (exact) WKB. Exact quantization conditions naturally arise, and their expansion in the weak coupling regime produces resurgent trans-series expressions which exhibit precise relations between different instanton sectors. Indeed, the perturbative expansion encodes all orders of the multi-instanton expansion, an explicit realization of the general concept of "resurgence". We also discuss the transition from weak to strong coupling, an explicit realization of "instanton condensation".

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Gerald V. Dunne, Mithat Unsal. 2016-03-16. WKB and Resurgence in the Mathieu Equation. https://arxiv.org/abs/1603.04924

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