相关论文: A new basis for eigenmodes on the Sphere
Standard spectral codes for full sphere dynamics utilize a combination of spherical harmonics and a suitableradial basis to represent fluid variables. These basis functions have a rotational invariance not present ingeophysical flows.…
Let $M\subset S^{n+1}$ be the hypersurface generated by rotating a hypersurface $M_0$ contained in the interior of the unit ball of $\mathbb{R}^{n-k+1}$. More precisely, $M=\{(\sqrt{1-|m|^2}\, y, m):y\in S^k, m\in M_0\}$. We derive the…
In this paper we propose an approach of obtaining of N-dimensional spherical harmonics based exclusively on the methods of solutions of differential equations and the use of the special functions properties. We deduce the Laplace-Beltrami…
In this article, we prove that for an embedded minimal hypersurface $\Sigma^{m}$ in $S^{m+1}$, the first eigenvalue $\lambda_1$ of the Laplacian operator on $\Sigma$ satisfies: $$\lambda_1> \frac{m}{2}+G(m, |A|_{\max}, |A|_{\min} ) ,$$…
This paper presents recent results obtained by the authors (partly in collaboration with A. Dabrowska) concerning expansions of zonal functions on Euclidean spheres into spherical harmonics and some applications of such expansions for…
We study \alpha-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for \alpha\in(1,2). We describe the corresponding Hardy spaces and prove the Fatou…
This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We…
The global rotational degrees of freedom in the Schr\"{o}dinger equation for an $N$-body system are completely separated from the internal ones. After removing the motion of center of mass, we find a complete set of $(2\ell+1)$ independent…
We demonstrate that quantum Hamiltonian operator for a free transverse field within the framework of the second quantization reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based upon…
In this paper the classical theory of spherical harmonics in R^m is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of…
This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented.…
We generalize the spherical harmonics for l=1 and give the differential equation that the generalized forms satisfy. The new forms have an obvious interpretation in the context of quantum mechanics.
Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a remarkable connection between critical points of this functional…
Multipolar expansions are a foundational tool for describing basis functions in quantum mechanics, many-body polarization, and other distributions on the unit sphere. Progress on these topics is often held back by complicated and competing…
A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi…
We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The…
In this paper, we study the first eigenvalue of the Laplace--Beltrami operator on the Lawson minimal surfaces $\xi_{m,k}$ embedded in the unit three-sphere $\mathbb{S}^3$. Motivated by Yau's conjecture on the first eigenvalue of closed…
We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential $V(r)=(a^2\omega^2/4)sinh(r/a)^2$ where $a$ is the curvature radius and $r$ is the geodesic distance from a fixed center. Thus the potential is…
An infinite dimensional algebra which is a non-decomposable reducible representation of $su(2)$ is given. This algebra is defined with respect to two real parameters. If one of these parameters is zero the algebra is the commutative algebra…
In [M. R\"osler and M. Voit. Integral Representation and Uniform Limits for Some Heckman-Opdam Hypergeometric Functions of type BC, Transactions of the American Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], R\"osler and Voit…