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.     

在s=3/2情形下利用超对称量子力学方法解广义椭球函数

本文利用超对称量子力学的方法研究出广义椭球函数. 首先, 用超对称量子力学方法近似的算出前四阶超势W和相应本征值E, 然后递推出W_n的通式, 并利用数学归纳法来证明W_n通式的正确性, 从而得到了此时的广义椭球函数方程的基态波函数, 这对于它们的应用有很大的意义.     

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 Schrdinger-like form. This Schrdinger-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.     

Low frequency asymptotics for the spin-weighted spheroidal equation in the case of s=1/2

The spin-weighted spheroidal equation in the case of s=1/2 is thoroughly studied by using the perturbation method from the supersymmetric quantum mechanics.The first-five terms of the superpotential in the series of parameter β are given.The general form for the n-th term of the superpotential is also obtained,which could also be derived from the previous terms W k,k < n.From these results,it is easy to obtain the ground eigenfunction of the equation.Furthermore,the shape-invariance property in the series of parameter β is investigated and is proven to be kept.This nice property guarantees that the excited eigenfunctions in the series form can be obtained from the ground eigenfunction by using the method from the supersymmetric quantum mechanics.We show the perturbation method in supersymmetric quantum mechanics could completely solve the spin-weight spheroidal wave equations in the series form of the small parameter β.     

一维方势阱中束缚态粒子波函数和能级的求解

采用矩阵方程表述的方法解出了-维方势阱的波函数和能级,借助计算机软件图示了解的特征.     

量子激发态最陡下降微扰理论

本文发展了量子激发态能量与波函数的最陡下降微扰理论计算方法,该方法避免了普通微扰理论所需要的对于参考态的无限求和困难,并能通过逐步迭代计算逼近于体系精确的本征函数和本征值。只要保持激发态试探波函数正交于其对称性相同的低激发态或基态的波函数,避免计算过程中的变分坍陷,本文的方法能用于求精确的激发态能量和波函数。 关键词：     

A case of analytical solution of the Griffin-Hill-Wheeler equation: The ground state of the hydrogen atom with a Gaussian Generator function

It is shown that the Griffin-Hill-Wheeler equation for the ground state of the hydrogen atom can be solved analytically for a Gaussian trial function. Both the exact eigenfunction and eigenvalue can be generated.     

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 W 0 to W 4 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.     

二维各向同性变频率谐振子的超对称性及其不变量的超对称量子力学精确解

采用超对称量子力学与不变量相结合的方法讨论了二维各向同性变频率谐振子,给出了二维各向同性变频率谐振子的不变量,采用超对称量子力学方法精确求解了不变量的本征值和本征函数,并且给出了当频率恒定时,二维常频率谐振子的本征值和本征函数的精确解.最后对不变量的超对称性进行了讨论.     

Localization in Asymmetric Random Displacements Models with Infinite Range of Interaction

We consider a random displacements model in a Euclidean space with an infinite-range, polynomially decaying interaction potential, without assuming any symmetry properties of the latter, and give an elementary proof of eigenvalue concentration estimates without resorting to a more traditional Wegner-type analysis or explicit ground state energy minimization. In the strong disorder/semi-classical regime, we prove exponential spectral and dynamical localization, with sub-exponential decay of the eigenfunction correlators. Prior works focused on compactly supported interactions.     

Quantum model of the prolate spheroidal Thomson hydrogen atom

In a uniformly charged prolate spheroidal Thomson hydrogen atom the electron states have been investigated. It has been shown from the mathematical point of view that the problem is equivalent to a spheroidal hydrogen atom in a parabolic potential with the cylindrical symmetry. In the framework of adiabatic approximation, the energy of ground state has been calculated. Comparison with the case of uncharged spheroidal quantum dot has been made, and the analytical form of wave function of electron has been also obtained.     

On solving the Schrödinger equation for a complex deictic potential in one dimension

Making use of an ansatz for the eigenfunction, we investigate closed-form solutions of the Schrödinger equation for an even power complex deictic potential and its variant in one dimension. For this purpose, extended complex phase-space approach is utilized and nature of the eigenvalue and the corresponding eigenfunction is determined by the analyticity property of the eigenfunction. The imaginary part of the energy eigenvalue exists only if the potential parameters are complex, whereas it reduces to zero for real coupling parameters and the result coincides with those derived from the invariance of Hamiltonian under \(\mathcal {P}\mathcal {T}\) operations. Thus, a non-Hermitian Hamiltonian possesses real eigenvalue, if it is \(\mathcal {P}\mathcal {T}\) -symmetric.     

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.     

Spin-weighted spheroidal equation in the case of s=1

We present a series of studies to solve the spin-weighted spheroidal wave equation by using the method of super-symmetric 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 eigen- functions 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.     

Comparison of spheroidal and eigenfunction-expansion trial functions for a membrane in an infinite baffle

The on-axis far-field pressure response of a circular membrane in an infinite baffle when driven by a uniformly distributed electrostatic force is calculated using two different trial functions for the surface velocity distribution. The first is an expansion based upon a solution to the free space wave equation in oblate spheroidal coordinates, which has already been derived in a previous paper [J. Acoust. Soc. Am. 120(5), 2460-2477 (2006)], and the second is a membrane eigenfunction expansion (or Bessel series), which is rigorously derived in this letter. Although the latter can be used as a basis for calculating a number of different radiation characteristics such as the radiation impedance or directivity, etc., only the on-axis far-field sound pressure is considered here. The results are compared and discussed.     

二维耦合量子谐振子的本征值和本征函数

运用广义线性量子变换理论,给出一类二维耦合量子谐振子的能量本征值、本征函数、坐标和动量算符在能量表象中的矩阵元及演化算符.     

Explicit spheroidal wave functions of the hydrogen atom

A simple and straightforward calculating scheme is suggested for finding wave functions of the hydrogen atom in prolate spheroidal coordinates. The wave functions are found in an explicit form by the direct solution of appropriate one-dimensional equations. The suggested calculating scheme allows us to carry out simple calculations and to obtain spheroidal wave functions in principle for arbitrary eigenstates of the hydrogen atom. Expansions are found for the obtained spheroidal wave functions over a spherical basis.     

Understanding light scalar meson by color-magnetic wave function in QCD sum rule

In this paper, we study the 0⁺ nonet mesons as tetraquark states with interpolating currents inspired by the color-magnetic wave function. This wave function is the eigenfunction of an effective color-magnetic Hamiltonian with the lowest eigenvalue, meaning that the state depicted by this wave function is the most stable one and is the most probable to be observed in experiments. We perform an OPE calculation up to dimension-eight condensates and find that the best QCD sum rule is achieved when the current inspired by the color-magnetic wave function is a proper mixture of the tensor and pseudoscalar diquark-antidiquark bound states. Compared with previous results, to sigma(600) and kappa(800), our results appear better, due to larger pole contribution. The direct instanton contributions are also considered, which yields a consistent result with previous OPE results. Finally, we also discuss the h￠ \eta{^\prime} problem as a possible six-quark state.     

On the localization of eigenfunctions of the Laplace operator in rectangular domains

The localization problem is considered for eigenfunctions of the Laplace operator in a domain that consists of two rectangles linked by a small hole. The localization of the eigenfunction is proven in a subdomain. The velocity is estimated for the convergence of an eigenvalue of the original problem to a subdomain eigenvalue.