Symmetry Regularization of 1D Generalized Coulomb Problems

For the 1D generalized Coulomb problems–a family that includes the quantizations of the 1D generalized Kepler problems of Ma-Meng-Xiao \cite{MMX2025}–we construct two explicit unitary intertwiners ${\widehat{\iota }}_{\pm }$, the quantum analogs of the classical symmetry regularization maps ${\iota }_{\pm }$ of \cite{MMX2025}, that unitarily identify each $H_{\kappa }$-energy-definite portion of the Hilbert space $L^2({\mathbb{R}}_{>0},\mathrm{d}q)$ with a unitary lowest-weight representation of $\widetilde{\mathrm{S}\mathrm{L}}(2,\mathbb{R})$.