The prolate spheroidal phenomena and bispectrality

具有长球形性质与双谱性的矩阵值离散-连续函数

Matrix valued discrete-continuous functions with the prolate spheroidal property and bispectrality

本文建立了离散-连续双谱函数与长球形现象的内在联系，在矩阵值函数框架下，利用非交换矩阵值 Fourier 代数，构造性地证明了自伴 Darboux 变换可生成具有交换微分算子和平移算子的积分算子，并给出了阶数与带宽的显式上界。

Black-hole perturbation theory: The asymptotic spectrum of the prolate spin-weighted spheroidal harmonics

Prolate spin-weighted spheroidal harmonics play a key role in black-hole perturbation theory. In particular, the highly damped quasinormal resonances of rotating Kerr black holes are closely related to the asymptotic eigenvalues of these important functions. We here present a novel and compact derivation of the asymptotic eigenvalues of the prolate spin-weighted spheroidal harmonics. Our analysis is based on a simple trick which transforms the corresponding spin-weighted spheroidal angular equation into a Schrödinger-like wave equation which is amenable to a standard WKB analysis. Our analytic…

Algebraic structure underlying spherical, parabolic and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problem

The nonrelativistic motion of a charged particle around a dyon in (9+1) spacetime is known as the nine-dimensional MICZ-Kepler problem. This problem has been solved exactly by the variables-separation method in three different coordinate systems, spherical, parabolic, and prolate spheroidal. In the present study, we establish a relationship between the variable separation and the algebraic structure of SO(10) symmetry. Each of the spherical, parabolic, or prolate spheroidal bases is proved to be a set of eigenfunctions of a corresponding nonuplet of algebraically-independent integrals of motio…

Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability

We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H$_2^+$ molecular ion. The symmetry operator is constructed explicitly. We give the details of the Hamiltonian reduction of the 3D system to a 2D system with modified potential that is separable in elliptic coordinates. The potentials for which there is double-periodicity of the Schrödinger operator in the space of prolate spheroidal coordinates, including one for the H$_2^+$ molecular ion, are indicated. We study possible potentials…

Monodromy in Prolate Spheroidal Harmonics

We show that spheroidal wave functions viewed as the essential part of the joint eigenfunction of two commuting operators of $L_2(S^2)$ has a defect in the joint spectrum that makes a global labelling of the joint eigenfunctions by quantum numbers impossible. To our knowledge this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analogue of the Laplace-Runge-Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of …

长椭球波函数与谱分析的精度和维数

Prolate Spheroidal Wave Functions and the Accuracy and Dimensionality of Spectral Analysis

本研究提出高效高精度的有限时长信号频率确定方案，以PSWF为最优基，扩展其集中特性并改进采样截断估计，建立低维子空间谱分析近似理论，还给出算子对易关系的新几何视角。