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Title: Combinatorial approach to Mathieu and Lamé equations

Based on some recent progress on a relation between four dimensional super Yang-Mills gauge theory and quantum integrable system, we study the asymptotic spectrum of the quantum mechanical problems described by the Mathieu equation and the Lamé equation. The large momentum asymptotic expansion of the eigenvalue is related to the instanton partition function of supersymmetric gauge theories which can be evaluated by a combinatorial method. The electro-magnetic duality of gauge theory indicates that in the parameter space, there are three asymptotic expansions for the eigenvalue, and we confirm this fact by performing the Wentzel–Kramers–Brillouin (WKB) analysis in each asymptotic expansion region. The results presented here give some new perspective on the Floquet theory about periodic differential equation.
Authors:
 [1]
  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China and Instituto de Física Teórica, Universidade Estadual Paulista, Barra Funda 01140-070, São Paulo, SP (Brazil)
Publication Date:
OSTI Identifier:
22479579
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; DUALITY; EIGENVALUES; GAUGE INVARIANCE; MATHIEU EQUATION; PARTITION FUNCTIONS; QUANTUM MECHANICS; SPACE; SUPERSYMMETRY; YANG-MILLS THEORY