Dirac equation in Euclidean Newman–Penrose formalism with applications to instanton metrics

2012

Mathematical Physics

Most of the theoretical physics known today is described using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2010. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. An integral transform solution using simpler functions also is not obtainable. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and few times updated version of the conference talk by the same author, published in

Section: Some Examples Of the Heun Equation In Physical Applicationsmentioning

confidence: 98%

Heun Functions and Their Uses in Physics

2012

Mathematical Physics

“…We can give the Eguchi-Hanson case as an example. The wave equation for the scalar particle in the background of the Eguchi-Hanson metric [3] in four dimensions has hypergeometric functions as solutions [27], whereas the Nutku helicoid [28,29] metric, the next higher one, gives us Mathieu functions [30], a member of the Heun function set, if the method of separation of variables is used to get a solution. We also find that the scalar particle, in the background of the Eguchi-Hanson metric, trivially extended to five dimensions gives Heun type solutions.…”

Section: Some Examples Of the Heun Equation In Physical Applicationsmentioning

confidence: 99%

Heun Functions and Some of Their Applications in Physics

Advances in High Energy Physics

2018

Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and many times updated version of the conference talk by the same author, published in

“…These equations have simple solutions, 24 which can also be expanded in terms of products of radial and angular Mathieu functions. 28,25 A problem arises when these solutions are restricted to the boundary.…”

Section: A Singularitiesmentioning

confidence: 99%

“…One can also write the Dirac equation and obtain the solutions in terms of Mathieu functions. [24][25][26] One needs to study the eigenvalue problem on the boundary to impose the boundary conditions in this problem. The same, problem in the bulk, using partial differential equations, can be solved in terms of known functions.…”

Section: Introductionmentioning

confidence: 99%

Comment on “Dirac equation in the background of the Nutku helicoid metric” [J. Math. Phys. 48, 092301 (2007)]

Birkandan

Journal of Mathematical Physics

2008

The Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an Appendix which gives a new example wherein one encounters a form of the Heun equation.