The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eige...The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.展开更多
We present a series of studies to solve the spin-weighted spheroidal wave equation by using the method of supersymmetric quantum mechanics. We first obtain the first four terms of super-potential of the spin-weighted ...We present a series of studies to solve the spin-weighted spheroidal wave equation by using the method of supersymmetric quantum mechanics. We first obtain the first four terms of super-potential of the spin-weighted spheroidal wave equation in the case of s ： 1. These results may help summarize the general form for the n-th term of the super-potential, which is proved to be correct by means of induction. Then we compute the eigen-values and the eigenfunctions for the ground state. Finally, the shape-invariance property is proved and the eigen-values and eigen-functions for excited states are obtained. All the results may be of significance for studying the electromagnetic radiation processes near rotating black holes and computing the radiation reaction in curved space-time.展开更多
Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their an...Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their analytic ground eigenvalues and eigenfunctions are obtained by means of a series in low frequency. The ground eigenvalue and eigenfunction for small complex frequencies are numerically determined.展开更多
In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Sehwarzschild black hole with a straight st...In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Sehwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersymmetric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.展开更多
By using the super-symmetric quantum mechanics （SUSYQM） method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the gene...By using the super-symmetric quantum mechanics （SUSYQM） method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the general form for the n-th-order super-potential is summarized and is proved correct by mathematical induction. Hence the ground eigenvalue problem is completely solved. Particularly, the novel solutions of the excited state are investigated according to the shape-invariance property.展开更多
The spin-weighted spheroidal equation in the case of s = 1 is studied. By transforming the independent variables, we make it take the Schr6dinger-like form. This Schr6dinger-like equation is very interesting in itself...The spin-weighted spheroidal equation in the case of s = 1 is studied. By transforming the independent variables, we make it take the Schr6dinger-like form. This Schr6dinger-like equation is very interesting in itself. We investigate it by using super-symmetric quantum mechanics and obtain the ground eigenvalue and eigenfunction, which are consistent with the results previously obtained.展开更多
There are two kinds of recurrence relations for the spherical functions Pml. The first are those with the same m but different l. Thesecond are those with the same l but different m. The spheroidal functions are exten...There are two kinds of recurrence relations for the spherical functions Pml. The first are those with the same m but different l. Thesecond are those with the same l but different m. The spheroidal functions are extensions of the spherical functions. Recurrencerelations of the first kind are obtained for the spheroidal functions in recent studies. Using the shape invariance method in super-symmetric quantum mechanics, we investigate the second type of recurrence relations for the spheroidal functions. The resultsshow that the second kind of recurrence relation can not be extended to the spheroidal functions.展开更多
The integrable properties of the spheroidal equations are investigated. The shape-invariance property is proved to be retained for the spheroidal equations, for which the recurrence relations are obtained. This is the...The integrable properties of the spheroidal equations are investigated. The shape-invariance property is proved to be retained for the spheroidal equations, for which the recurrence relations are obtained. This is the extension of the recurrence relation of the Legendre polynomials.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10875018 and 10773002)
文摘The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10875018 and 10773002)
文摘We present a series of studies to solve the spin-weighted spheroidal wave equation by using the method of supersymmetric quantum mechanics. We first obtain the first four terms of super-potential of the spin-weighted spheroidal wave equation in the case of s ： 1. These results may help summarize the general form for the n-th term of the super-potential, which is proved to be correct by means of induction. Then we compute the eigen-values and the eigenfunctions for the ground state. Finally, the shape-invariance property is proved and the eigen-values and eigen-functions for excited states are obtained. All the results may be of significance for studying the electromagnetic radiation processes near rotating black holes and computing the radiation reaction in curved space-time.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10875018 and 10773002)
文摘Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their analytic ground eigenvalues and eigenfunctions are obtained by means of a series in low frequency. The ground eigenvalue and eigenfunction for small complex frequencies are numerically determined.
基金supported by the National Natural Science Foundation of China (Grant No. 10875018)
文摘In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Sehwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersymmetric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10875018 and 10773002)
文摘By using the super-symmetric quantum mechanics （SUSYQM） method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the general form for the n-th-order super-potential is summarized and is proved correct by mathematical induction. Hence the ground eigenvalue problem is completely solved. Particularly, the novel solutions of the excited state are investigated according to the shape-invariance property.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10875018 and 10773002)
文摘The spin-weighted spheroidal equation in the case of s = 1 is studied. By transforming the independent variables, we make it take the Schr6dinger-like form. This Schr6dinger-like equation is very interesting in itself. We investigate it by using super-symmetric quantum mechanics and obtain the ground eigenvalue and eigenfunction, which are consistent with the results previously obtained.
基金supported by the National Natural Science Foundation of China (Grant No. 10875018)the National Basic Research Program of China (Grant No. 2010CB923200)
文摘There are two kinds of recurrence relations for the spherical functions Pml. The first are those with the same m but different l. Thesecond are those with the same l but different m. The spheroidal functions are extensions of the spherical functions. Recurrencerelations of the first kind are obtained for the spheroidal functions in recent studies. Using the shape invariance method in super-symmetric quantum mechanics, we investigate the second type of recurrence relations for the spheroidal functions. The resultsshow that the second kind of recurrence relation can not be extended to the spheroidal functions.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10875018 and 10773002)
文摘The integrable properties of the spheroidal equations are investigated. The shape-invariance property is proved to be retained for the spheroidal equations, for which the recurrence relations are obtained. This is the extension of the recurrence relation of the Legendre polynomials.
