Symmetry Regularization of 1D Generalized Coulomb Problems

Abstract

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 ${\hat{\hat{I}}}_{\pm }^1$, the quantum analogs of the classical symmetry regularization maps ${\hat{I}}_{\pm }^1$ of \cite{MMX2025}, that unitarily identify each $H_{\hat{I}}º$-energy-definite portion of the Hilbert space $L^2({\mathbb{R}}_{>0},\mathrm{d}q)$ with a unitary lowest-weight representation of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$.

14 pages