TY - JOUR
T1 - On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions
AU - Luna-Elizarrarás, María Elena
AU - Rosa, Marco Antonio Pérez De La
AU - Rodríguez-Dagnino, Ramõn M.
AU - Shapiro, Michael
PY - 2013/6
Y1 - 2013/6
N2 - 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 α-hyperholomorphic functions and the Helmholtz operator.
AB - 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 α-hyperholomorphic functions and the Helmholtz operator.
KW - Mathieu functions
KW - Schrödinger operator
KW - quaternionic analysis
UR - http://www.scopus.com/inward/record.url?scp=84878011307&partnerID=8YFLogxK
U2 - 10.1002/mma.2665
DO - 10.1002/mma.2665
M3 - Artículo
SN - 0170-4214
VL - 36
SP - 1080
EP - 1094
JO - Mathematical Methods in the Applied Sciences
JF - Mathematical Methods in the Applied Sciences
IS - 9
ER -