, Volume 41, Issue 10, pp 1957-1970

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

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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, ( \({1 \mathord{\left/ {\vphantom {1 {\rho ^q }}} \right. \kern-0em} {\rho ^q }}\) )Z q+ℓ(ρ)Y (n) , 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.