TY - JOUR
T1 - Exact solutions to the angular Teukolsky equation with s ≠ 0
AU - Chen, Chang Yuan
AU - Wang, Xiao Hua
AU - You, Yuan
AU - Sun, Dong Sheng
AU - Lu, Fa Lin
AU - Dong, Shi Hai
N1 - Publisher Copyright:
© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing.
PY - 2022/11/1
Y1 - 2022/11/1
N2 - We first convert the angular Teukolsky equation under the special condition of τ ≠ 0, s ≠ 0, m = 0 into a confluent Heun differential equation (CHDE) by taking different function transformation and variable substitution. And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function (CHF), we find two linearly dependent solutions corresponding to the same eigenstate, from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant. After that, we are able to localize the positions of the eigenvalues on the real axis or on the complex plane when τ is a real number, a pure imaginary number, and a complex number, respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l = ∣s∣+ n, n = 0, 1, 2···. The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple. The features of the angular probability distribution (APD) and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed. We find that for a real number τ, the eigenvalue is a real number and the eigenfunction is a real function, and the eigenfunction system is an orthogonal complete system, and the APD is asymmetric in the northern and southern hemispheres. For a pure imaginary number τ, the eigenvalue is still a real number and the eigenfunction is a complex function, but the APD is symmetric in the northern and southern hemispheres. When τ is a complex number, the eigenvalue is a complex number, the eigenfunction is still a complex function, and the APD in the northern and southern hemispheres is also asymmetric. Finally, an approximate expression of complex eigenvalues is obtained when n is greater than ∣s∣.
AB - We first convert the angular Teukolsky equation under the special condition of τ ≠ 0, s ≠ 0, m = 0 into a confluent Heun differential equation (CHDE) by taking different function transformation and variable substitution. And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function (CHF), we find two linearly dependent solutions corresponding to the same eigenstate, from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant. After that, we are able to localize the positions of the eigenvalues on the real axis or on the complex plane when τ is a real number, a pure imaginary number, and a complex number, respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l = ∣s∣+ n, n = 0, 1, 2···. The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple. The features of the angular probability distribution (APD) and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed. We find that for a real number τ, the eigenvalue is a real number and the eigenfunction is a real function, and the eigenfunction system is an orthogonal complete system, and the APD is asymmetric in the northern and southern hemispheres. For a pure imaginary number τ, the eigenvalue is still a real number and the eigenfunction is a complex function, but the APD is symmetric in the northern and southern hemispheres. When τ is a complex number, the eigenvalue is a complex number, the eigenfunction is still a complex function, and the APD in the northern and southern hemispheres is also asymmetric. Finally, an approximate expression of complex eigenvalues is obtained when n is greater than ∣s∣.
KW - Wronskian determinant
KW - angular Teukolsky equation
KW - linearly dependent
UR - http://www.scopus.com/inward/record.url?scp=85140897419&partnerID=8YFLogxK
U2 - 10.1088/1572-9494/ac85d8
DO - 10.1088/1572-9494/ac85d8
M3 - Artículo
AN - SCOPUS:85140897419
SN - 0253-6102
VL - 74
JO - Communications in Theoretical Physics
JF - Communications in Theoretical Physics
IS - 11
M1 - 115001
ER -