Chang-Yuan Chen,Xiao-Hua Wang,Yuan You,Dong-Sheng Sun,Fa-Lin Lu and Shi-Hai Dong

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

Abstract 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|.

Keywords:angular Teukolsky equation,linearly dependent,Wronskian determinant

1.Introduction

The general form of the angular Teukolsky equation also named as the spin-weighed spheroidal wave equation has played an important role in the study of black holes with the gravitational self-force[1–4],quasi-normal modes[5–8],etc.The equation is given explicitly as[9,10]

This equation has extremely wide applications in many fields such as electromagnetic field theory[17–20],and atomic and molecular physics[21–25].Up to now,many authors have used different methods to approximately calculate the eigenvalues and the eigenfunctions.The main methods include the expansion of associated Legendre functions[16,17]and the numerical methods[26–31].Recently,we have proposed a scheme to construct the Wronskian determinant by finding two linearly dependent solutions with respect to the same eigenstate to obtain accurate eigenvalues and the analytical normalized eigenfunction expressed by a confluent Heun function(CHF)[32,33].We find that for a real or pure imaginary τ(c2=-τ2is real),the eigenvalues are real.For a complex τ,the eigenvalues are complex.Although the angular probability distribution(APD)has obvious directionality,the northern and southern hemispheres are always symmetric.The advantage of this method is obvious,that is,it not only allows us to obtain accurate eigenvalues(the accuracy of numerical calculation depends on the calculation accuracy of Maple software for CHF and its first derivative)but also can obtain normalized eigenfunction.The analytical wave function makes it possible to discuss properties such as APD.In addition,we have recently obtained the exact solutions of the Stark effect for 3D rigid rotors[34],2D planar rotors[35],and rigid symmetric top molecules[36]using this method,respectively.The exact solution of the bound state for the Mathieu potential[37]and a kind of hyperbolic potential wells[38]fully demonstrates the merit of this scheme.

In this paper,we mainly study the exact solution of the angular Teukolsky equation for the other special case m=0,s≠0 andτ≠0.This will reveal the quantum characteristics of the system under this special condition and provides some ideas for the accurate solution of the general angular Teukolsky equation.When m=0 buts≠ 0 andτ≠0,equation(1)can be simplified as

which is different from equation(2).Thus,its eigenvalues and eigenfunctions have different characteristics from the former(2),which is the main reason why we must study them.

This paper is organized as follows.In section 2 we use different forms of function transformation and variable substitution to convert equation(3)into a confluent Heun differential equation(CHDE),and then according to the characteristics of both the CHDE and its analytical solution expressed by the CHF,two linearly dependent solutions corresponding to the same eigenstate are obtained,and the Wronskian determinant which is constructed by obtained two linearly dependent solutions can be used to obtain the exact eigenvalues.Next,in sections 3–5,we discuss the position of the eigenvalues on the real axis or complex plane when τ is a real number,a pure imaginary number and a complex number,respectively,and determine the relationship between the quantum number l and the spin weight quantum number s,and present the calculation of exact eigenvalues,and the linear dependence of eigenfunctions as well as 2D and 3D graphics of APDs.Finally,we summarize the conclusions in section 6.

2.Construction of the Wronskian determinant

Considering the behaviors of the wave function S(x)with the natural boundary conditions atx→±1,i.e.zero or finite,we take it as the following form

3.Real number τ cases

4.Imaginary number τ cases

It can be seen from the above formulae that the real parts of S(1)and S(2)are symmetric to the vertical axis,and the imaginary parts of S(1)and S(2)are symmetric to the origin,so their modulus squares is an even function.Figure 5 displays the real and imaginary parts and modulus squares of S(1)and S(2)for τ=1.5i,s=1/2,n=0,l=1/2 and τ=5.5i,s=2,n=3,l=5,respectively.It can be seen from figure 5 that when τ is a pure imaginary number,for a certain eigenstate(l,s),the number of nodes of the real and imaginary parts of the eigenfunction will not be equal to n given in equation(26).Figure 6 shows APDs in 2D and 3D for τ = 1.5i,s=1/2,n=0,l=1/2 and τ=5.5i,s=2,n=3,l=5.It can be seen that when τ is a pure imaginary number,the APD is characterized by obvious directionality,but the northern and southern hemispheres are always symmetric.

5.Complex number τ cases

6.Concluding remarks

In this work,we first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠ 0,m=0 into CHDE by informal function transformation and variable substitution,and then according to the characteristics of CHDE and its analytical solution CHF,two solutions describing the linear correlation of the same eigenstate are found,and the precise energy spectrum equation is obtained by constructing the Wronskian determinant.Next,in sections 3–5,we discuss the position of the eigenvalues on the real axis or the complex plane,the calculation of the exact eigenvalues,and the linear correlation of the eigenfunctions when τ is respectively a real number,an imaginary number,and a complex number,and the characteristics of APD in 2D and 3D graphics.The main conclusions are as follows:

1.Whenτ=τRis a real number,the corresponding operator is a Hermitian operator,so the eigenvalue is a real number,the eigenfunction is a real function,and the eigenfunction system is an orthonormal complete system.Determine the n value in the quantum number relation l=|s| + n is the number of nodes of the eigenfunction.The APD is clearly directional,and the northern and southern hemispheres are asymmetric.

2.Whenτ= iτIis a pure imaginary number,the corresponding operator is a non-Hermitian operator,but the eigenvalue is still a real number,the eigenfunction is a complex function,and the eigenfunction system is not an orthonormal complete system.The number of nodes in the real or imaginary part of the complex eigenfunction may not be equal to the value of n in the quantum number relation l=|s|+n.The APD has obvious directionality,and the northern and southern hemispheres are always symmetrical.

3.Whenτ=τR+ iτIis a complex number,the corresponding operator is not a Hermitian operator,the eigenvalue is a complex number,the eigenfunction is also a complex function,and the eigenfunction cannot make an orthonormal complete system.The number of nodes of the real part or the imaginary part of the complex eigenfunction may also be unequal to the n in equation(26).The APD has obvious directionality,and the northern and southern hemispheres are also asymmetric.

4.Equations(39)and(40)expressing the linear dependence between S(1)and S(2)are also applicable to the case where τ is a real number or a pure imaginary number.The complex function of the pure imaginary number calculated from this is the same as that of S(2),but it is obviously not as clear as the physical meaning of formula(31).When τ is a real number,both S(1)and S(2)are real functions,so the imaginary part is 0,and from equation(40),we getCI=0,respectively.Substitute into(39)to getand this is nothing but equation(27).

5.When m=0 and n is much larger than |s|,we can summarize and generalize the complex eigenvalue with the method provided in this paper.The approximate calculation formula is as follows

Tables 1–3 list the calculation results of low-energy states,which fully reflect the quantum properties of this system.

Table 1.The real eigenvalues A for real numbers τ.

Table 2.The real eigenvalues A for pure imaginary numbers τ.

Table 3.The complex eigenvalues A for complex numbers τ.

Finally,it should be pointed out that the formula(20)obtained in this paper is an accurate energy spectrum equation,but the accuracy of numerical calculation depends on the calculation accuracy of Maple software for the CHF and its first derivative.In the case of large parameters,the calculation results are not satisfactory,and we expect this to be improved as Maple versions are updated.

Acknowledgments

We would like to thank the referees for making invaluable suggestions and criticisms which have improved the manuscript.This work is supported by the National Natural Science Foundation of China(Grant No.11975196)and partially by 20220355-SIP,IPN.Prof.Dong is on leave of IPN due to permission of research stay at Huzhou University,China.