Explicit vector spherical harmonics on the 3-sphere (Q2795585)

The scalar modes in the Hopf coordinates defined on the unit sphere \(S^3\) by \(x^1 =\sin\alpha\cos\varphi\), \(x^2 =\sin\alpha \sin \varphi\), \(x^3 =\cos\alpha\cos\theta\), \(x^4 =\cos\alpha \sin\theta\) and some of their properties are considered in the paper. The normalized scalar modes for the Laplace-de Rham operator \(\Delta=-(d\delta + \delta d)\) on \(S^3\) satisfy the eigenvalue equation \(\Delta\Phi_i=\lambda_i\Phi_i\), where \(\Phi_i\) stands for the modes corresponding to the eigenvalue \(\lambda_i=-L(L + 2)\), with \(L\in\mathbb N\), the index \(i\) is a shorthand for the indexes needed to label the modes. For instance, the modes are labeled by the three numbers \((L, m_+, m_-)\), where \(m_+\), \(m_-\) are such that \(|m_\pm|\leq L/2\), and \(L/2-m_\pm\in \mathbb N\), NEWLINE\[NEWLINE \Phi_i = C_{L, m_+, m_-}e^{i(S\varphi+D\theta)}(1-x)^{S/2} (1+x)^{D/2} P^{(S,D)}_{L/2-m_+}(x), NEWLINE\]NEWLINE in which \(P^{(a,b)}_n\) is a Jacobi polynomial, \(x=\cos2\alpha\), \(S=m_++m_-\), \(D=m_+-m_-\). Using the scalar modes consider the one-forms \(E_i =(L + 2)B_i + C_i\), \(E'_i=(L + 2)B'_i-C'_i\), where \(i=(L,m_+,m_-)\) (respectively, \(i=(L,m'_+,m'_-)\)), and \(A_i=d \Phi_i\) is an exact one-form while \(B_i=*d\Phi_i\widetilde\xi\), \(C_i=*dB_i\), \(B'_i=*d\Phi_i\widetilde{\xi'}\), and \(C'_i=*dB'_i\) are co-exact one-forms. Three properties of one-forms \(E_{L,m_+,m_-}\) and \(E'_{L,m_+,m_-}\) are established. For example NEWLINE\[NEWLINE \Delta E_{L,m_+,m_-}=-L^2E_{L,m_+,m_-},\quad \Delta E'_{L,m'_+,m'_-}=-L^2E'_{L,m'_+,m'_-}, NEWLINE\]NEWLINE for \(L\geq2\).