Research Article Received 18 November 2011 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2665 MOS subject classification: 30G35; 35J10; 33E10; 31A10; 35J15; 35J05 On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions María Elena Luna-Elizarrarás a , Marco Antonio Pérez-de la Rosa a , Ramón M. Rodríguez-Dagnino b * † and Michael Shapiro a Communicated by S. Georgiev It has been found recently that there exists a theory of functions with quaternionic values and in two real variables, which is determined by a Cauchy–Riemann-type operator with quaternionic variable coefficients, and that is intimately related to the so-called Mathieu equations. In this work, it is all explained as well as some basic facts of the arising quaternionic function theory. We establish analogues of the basic integral formulas of complex analysis such as Borel–Pompeiu’s, Cauchy’s, and so on, for this version of quaternionic function theory. This theory turns out to be in the same relation with the Schrödinger operator with special potential as the usual holomorphic functions in one complex variable, or quaternionic hyperholomorphic functions, or functions of Clifford analysis, are with the corresponding Laplace operator. Moreover, it is similar to that of ˛-hyperholomorphic functions and the Helmholtz operator. Copyright © 2012 John Wiley & Sons, Ltd. Keywords: Mathieu functions; Schrödinger operator; quaternionic analysis 1. Introduction The angular and radial Mathieu functions, or elliptic-cylinder functions, are solutions of the two ordinary differential equations of the second order with variable coefficients that were originally proposed by È. Mathieu in 1868 for finding the modes in an elliptic membrane (see [1]). Since then, they have found numerous and important applications to diverse problems in physics and engineering science, thus generating a vast research literature about them. An important compilation of these functions, their main properties, and their applications have been elaborated by McLachlan in his monograph [2]; see also [3, 4]. These functions are the solutions of the following two equations: d 2 R./ d 2  .a  2L cosh 2/R./ D 0, (1) d 2 ˆ./ d 2 C .a  2L cos 2/ˆ./ D 0. (2) The second one is called the angular or ordinary Mathieu equation, and the first is the radial Mathieu equation; the solutions are called the Mathieu functions of the second and first types. As a matter of fact, because a and L are constants, one has two families of ordinary differential equations parametrized by the same parameters a and L. a ESFM-IPN , Mexico City, Mexico b DIEC, ITESM , Campus Monterrey, Monterrey, Nuevo León, Mexico *Correspondence to: Ramón M. Rodríguez-Dagnino, DIEC, ITESM, Campus Monterrey, Monterrey, Nuevo León, Mexico. † E-mail: rmrodrig@itesm.mx Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012