Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ ⁿ

Abstract

Three problems related to the spherical quantum billiard in $\mathbb{R}^n$are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for “diagonal eigenstates” of “spherical-like” quantum billiards in $\mathbb{R}^n$. The second topic addresses the “first-excited-state theorem” for the spherical quantum billiard in $\mathbb{R}^n$. Wavefunctions for this system are given by the product form, (${1 \mathord{\left/ {\vphantom {1 {\rho ^q }}} \right. \kern-\nulldelimiterspace} {\rho ^q }}$)Z _q+ℓ(ρ)Y ⁽ⁿ⁾_ℓ , where ρ is dimensionless displacement, $\ell$is angular-momentum number, qis an integer function of dimension, Z(ρ) is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and θrepresents (n− 1) independent angular components. Generalized spherical harmonics are written $Y_\ell ^{(n)}(\theta )$. It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in $\mathbb{R}^n$is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in $\mathbb{R}^n$. In a third study, an expression is derived for the dimension of the ℓth irreducible representation (“irrep”) of the rotation group O(n) in $\mathbb{R}^n$by enumerating independent degenerate product eigenstates of the Laplacian.