Combinatorial approach to Mathieu and Lam´eequations Wei He1, a) 1Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China 2Instituto de F´ısica Te´orica, Universidade Estadual Paulista, Barra Funda, 01140-070, S˜ao Paulo, SP, Brazil Based on some recent progress on a relation between four dimensional super Yang- Mills gauge theory and quantum integrable system, we study the asymptotic spec- trum of the quantum mechanical problems described by the Mathieu equation and the Lam´eequation. 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, we confirm this fact by performing the WKB analysis in each asymptotic expansion region. The results presented here give some new perspective on the Floquet theory about periodic differential equation. PACS numbers: 12.60.Jv, 02.30.Hq, 02.30.Mv Keywords: Seiberg-Witten theory, instanton partition function, Mathieu equation, Lam´eequation, WKB method arXiv:1108.0300v6 [math-ph] 27 Jul 2015 a)Electronic mail:
[email protected] 1 CONTENTS I. Introduction 3 II. Differential equations and gauge theory 4 A. The Mathieu equation 4 B. The Lam´eequation 5 C. The spectrum of Schr¨odinger operator and quantum gauge theory 8 III. Spectrum of the Mathieu equation 9 A. Combinatorial evaluating in the electric region 10 B. Extension to other regions 11 IV. Combinatorial approach to the Lam´eeigenvalue 13 A. Identify parameters 14 B.