Contributions to Modified Spherical Harmonics in Four Dimensions
- PDF / 354,698 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 48 Downloads / 200 Views
Complex Analysis and Operator Theory
Contributions to Modified Spherical Harmonics in Four Dimensions Heinz Leutwiler1 Received: 26 November 2019 / Accepted: 25 August 2020 © Springer Nature Switzerland AG 2020
Abstract A modification of the classical theory of spherical harmonics in four dimensions is presented. The space R4 = {(x, y, t, s)} is replaced by the upper half space R4+ = {(x, y, t, s), s > 0}, and the unit sphere S in R4 by the unit half sphere S+ = (x, y, t, s) : x 2 + y 2 + t 2 + s 2 = 1, s > 0 . Instead of the Laplace equation h = 0 we shall consider the Weinstein equation su + k ∂u ∂s = 0, for k ∈ N. The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on S+ . It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In addition we shall deduct—with respect to this non-Euclidean scalar product—an orthonormal system of homogeneous polynomials, which satisfies the above Weinstein equation. Keywords Spherical harmonics · Generalized axially symmetric potentials · Modified spherical harmonics Mathematics Subject Classification 30G35 · 33A45
1 Introduction The main purpose of this paper is to show that the results of our paper [12] can be extended to the full Weinstein equation
Communicated by Uwe Kähler. “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.
B 1
Heinz Leutwiler [email protected] Department of Mathematics, Friedrich-Alexander-University Erlangen-Nuremberg, Cauerstrasse 11, 91058 Erlangen, Germany 0123456789().: V,-vol
67
Page 2 of 19
H. Leutwiler
su + k
∂u = 0, k ∈ N, ∂s
(1.1)
considered in R4 = {(x, y, t, s)}, where :=
∂2 ∂2 ∂2 ∂2 + + + ∂x2 ∂ y2 ∂t 2 ∂s 2
denotes the Laplace operator. In [12] we just considered the special case k = 2. The solutions of Eq. (1.1) are often called generalized axially symmetric potentials for the following reason: The function u = u(x, y, t, s), defined in a subdomain of the half space R4+ = {(x, y, t, s), s > 0}, is a solution of (1.1) if and only if the function 2 2 w(ξ1 , ξ2 , . . . , ξk+4 ) := u ξ1 , ξ2 , ξ3 , ξ4 + · · · + ξk+4 is harmonic—in the classical sense—in the corresponding domain in (k + 4)— dimensional space. By some authors (see, e.g., [5] or [6]) solutions of (1.1) are also called k—hyperbolic harmonic functions. But we shall not use these notations here. Instead we introduce the following definition: Definition 1 Solutions of the Eq. (1.1) will be called k-modified harmonic functions. The Eq. (1.1) is the Laplace–Beltrami equation associated with the Riemannian metric dω2 = s k d x 2 + dy 2 + dt 2 + ds 2 , since the corresponding Laplace–Beltrami operator L B is given by L B
1 := k s
k ∂ + . s ∂s
The Eq. (1.1) has already been considered by several authors. A good reference is Brelot’s article in [4]. We also refer to [2,8,9] and [15] for further investigations. A glance at Eq. (1.1) shows that if u = u(x, y, t, s) is a
Data Loading...
