# On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions

María Elena Luna-Elizarrarás, Marco Antonio Pérez De La Rosa, Ramõn M. Rodríguez-Dagnino, Michael Shapiro

Resultado de la investigación: Contribución a una revistaArtículo

12 Citas (Scopus)

### Resumen

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. Copyright © 2012 John Wiley & Sons, Ltd.
Idioma original Inglés estadounidense 1080-1094 970 Mathematical Methods in the Applied Sciences https://doi.org/10.1002/mma.2665 Publicada - 1 jun 2013

### Huella dactilar

Quaternionic Analysis
Mathieu Functions
Operator
Cauchy
Mathematical operators
Mathieu Equation
Clifford Analysis
Real variables
Complex Analysis
Integral Formula
Hermann Von Helmholtz
Complex Variables
Laplace Operator
Variable Coefficients
Analytic function
Analogue

### Citar esto

title = "On quaternionic analysis for the Schr{\"o}dinger operator with a particular potential and its relation with the Mathieu functions",
abstract = "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 Schr{\"o}dinger 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. Copyright {\circledC} 2012 John Wiley & Sons, Ltd.",
author = "Luna-Elizarrar{\'a}s, {Mar{\'i}a Elena} and Rosa, {Marco Antonio P{\'e}rez De La} and Rodr{\'i}guez-Dagnino, {Ram{\~o}n M.} and Michael Shapiro",
year = "2013",
month = "6",
day = "1",
doi = "10.1002/mma.2665",
language = "American English",
pages = "1080--1094",
journal = "Mathematical Methods in the Applied Sciences",
issn = "0170-4214",
publisher = "John Wiley and Sons Ltd",

}

On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions. / Luna-Elizarrarás, María Elena; Rosa, Marco Antonio Pérez De La; Rodríguez-Dagnino, Ramõn M.; Shapiro, Michael.

En: Mathematical Methods in the Applied Sciences, 01.06.2013, p. 1080-1094.

Resultado de la investigación: Contribución a una revistaArtículo

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 - 2013/6/1

Y1 - 2013/6/1

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. Copyright © 2012 John Wiley & Sons, Ltd.

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. Copyright © 2012 John Wiley & Sons, Ltd.

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U2 - 10.1002/mma.2665

DO - 10.1002/mma.2665

M3 - Article

SP - 1080

EP - 1094

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

ER -