Ground Eigenvalue Research Articles

Analytic solutions of the ground and excited states of the spin-weighted spheroidal equation in the case of s = 2

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.

Verification of the spin-weighted spheroidal equation in the case of s = 1

The spin-weighted spheroidal equation in the case of s = 1 is studied. By transforming the independent variables, we make it take the Schrödinger-like form. This Schrödinger-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.

Solving the spheroidal wave equation with s=3/2 by super-symmetric quantum mechanic

In the paper, we use the method of super-symmetric quantum mechanics to study the spin-weighted spheroidal wave equation in the case of s=3/2. We first change the equation into Schrdinger's form, then calculate and derive the first four terms concerning E and super-potential W. we summarize the general formula of super-potential Wn and use mathematical induction to prove its correctness. In turn, this completely gives all the information about the ground eigenvalue and eigenfunction.

Ground eigenvalue and eigenfunction of a spin-weighted spheroidal wave equation in low frequencies

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.

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

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.

Ground State Eigenfunction of Spheroidal Wave Functions

We study the spin-weighted spheroidal wave functions in the case of s = m = 0. Their eigenvalue problem is investigated by the perturbation method in supersymmetric quantum mechanics. In the first three terms of parameter α = a2w2, the ground eigenvalue and eigenfunction are obtained. The obtained ground eigenfunction is elegantly in closed forms. These results are new and very useful for the application of the spheroidal wave functions.

Schrödinger Equation Solutions for the Central Field Power Potential Energy I. V ( r ) = V 0 ( r / a 0 ) 2ν−2 , ν ≥ 1

The solution of a generalized non-relativistic Schrodinger equation with radial potential energy V(r)=V 0(r/a 0)2ν−2 is presented. After reviewing the general properties of the radial ordinary differential equation, power series solutions are developed. The Green's function is constructed, its trace and the trace of its first iteration are calculated, and the ability of the traces to provide upper and lower bounds for the ground eigenvalue is examined. In addition, WKB-like solutions for the eigenvalues and eigenfunctions are derived. The approximation method yields valid eigenvalues for large quantum numbers (Rydberg states).

An extremum property of the n-dimensional sphere

Using arguments by E. Krahn (1925), it is shown that the n-dimensional sphere possesses the following minimum property: among all connected regions of the given volume it yields the smallest ground eigenvalue of the Laplace operator with Dirichlet's boundary condition. This result is important in establishing an adiabatic invariant for quantum mechanical systems.