Exact solutions to the angular Teukolsky equation with s≠0  

作　　者：Chang-Yuan Chen Xiao-Hua Wang Yuan You Dong-Sheng Sun Fa-Lin Lu Shi-Hai Dong 

机构地区：[1]School of Physics and Electronic Engineering,Yancheng Teachers University,Yancheng,224007,China [2]Research Center for Quantum Physics,Huzhou University,Huzhou,313000,China [3]Laboratorio de Información Cuántica,CIDETEC,Instituto Politécnico Nacional,UPALM,C.P 07700,CDMX,Mexico

出　　处：《Communications in Theoretical Physics》2022年第11期1-15,共15页理论物理通讯（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant No.11975196);partially by 20220355-SIP,IPN。

摘　　要：We first convert the angular Teukolsky equation under the special condition ofτ≠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 whenτis 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 numberτ,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 numberτ,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.Whenτis 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∣.

关 键 词：angular Teukolsky equation linearly dependent Wronskian determinant 

分 类 号：O175[理学—数学]