The wave function \( \Psi_{\lambda} \) is sought in the Hilbert space of either symmetric or anti-symmetric \( N \)-body functions, namely $$ \Psi_{\lambda}\in {\cal H}_A:= {\cal H}_1\oplus{\cal H}_1\oplus\dots\oplus{\cal H}_1, $$ where the single-particle Hilbert space \( \hat{H}_1 \) is the space of square integrable functions over \( \in {\mathbb{R}}^{d}\oplus (\sigma)\oplus (\tau) \) resulting in $$ {\cal H}_1:= L^2(\mathbb{R}^{d}\oplus (\sigma)\oplus (\tau)). $$