Darboux transformation in black hole perturbation theory

Darboux transformation in black hole perturbation theory The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schrödinger equation with specific spectral properties. In this paper we show that the same transformation appears in black hole theory, relating, for instance, the Zerilli and Regge-Wheeler equations for axial and polar Schwarzschild perturbations. The transformation reveals these two equations to be isospectral, a well known result whose method has been repeatedly reintroduced under different names. We highlight the key role that the so-called algebraically special solutions play in the black hole Darboux theory and show that a similar relation exists between the Chandrasekhar-Detweiler equations for Kerr perturbations. Finally, we discuss the limitations of the method when dealing with long-range potentials and explore the possibilities offered by a generalized Darboux transformation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review D American Physical Society (APS)

Darboux transformation in black hole perturbation theory

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Darboux transformation in black hole perturbation theory

Abstract

The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schrödinger equation with specific spectral properties. In this paper we show that the same transformation appears in black hole theory, relating, for instance, the Zerilli and Regge-Wheeler equations for axial and polar Schwarzschild perturbations. The transformation reveals these two equations to be isospectral, a well known result whose method has been repeatedly reintroduced under different names. We highlight the key role that the so-called algebraically special solutions play in the black hole Darboux theory and show that a similar relation exists between the Chandrasekhar-Detweiler equations for Kerr perturbations. Finally, we discuss the limitations of the method when dealing with long-range potentials and explore the possibilities offered by a generalized Darboux transformation.
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Publisher
American Physical Society (APS)
Copyright
Copyright © © 2017 American Physical Society
ISSN
1550-7998
eISSN
1550-2368
D.O.I.
10.1103/PhysRevD.96.024036
Publisher site
See Article on Publisher Site

Abstract

The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schrödinger equation with specific spectral properties. In this paper we show that the same transformation appears in black hole theory, relating, for instance, the Zerilli and Regge-Wheeler equations for axial and polar Schwarzschild perturbations. The transformation reveals these two equations to be isospectral, a well known result whose method has been repeatedly reintroduced under different names. We highlight the key role that the so-called algebraically special solutions play in the black hole Darboux theory and show that a similar relation exists between the Chandrasekhar-Detweiler equations for Kerr perturbations. Finally, we discuss the limitations of the method when dealing with long-range potentials and explore the possibilities offered by a generalized Darboux transformation.

Journal

Physical Review DAmerican Physical Society (APS)

Published: Jul 15, 2017

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