Scalar Waves in the Mixmaster Universe. I. The Helmholtz Equation in a Fixed Background

Solutions to the scalar wave equation in a fixed mixmaster background are presented. Separability conditions on the Laplace-Beltrami equation are related to the complete sets of invariant operators of the symmetry group S${\mathrm{O}}_3$. Solutions to the Helmholtz equation are equivalent to the quantum-mechanical problem of asymmetric rotors. The wave functions in the mixmaster space are the asymmetric-top wave functions. In Euler angle variables, they are expanded in terms of the symmetric-top wave functions (the wave functions in Taub space) possessing definite ($J, K, M$) symmetries, with a coupling in the intrinsic magnetic quantum number $K$; the case with $M=0\mathrm{}$ can be expressed as a product of the Lamé functions in elliptic coordinates. Invariance of the mixmaster space under the four - group characterizes the wave functions into four symmetry species and causes the factorization of the energy matrix into four submatrices. Eigenvalues and expansion coefficients for the wave functions are calculated for some low-lying levels.