Physlib.QuantumMechanics.DDimensions.Hydrogen.Basic

For a $d$-dimensional hydrogen atom system $H$ with mass $m$, it holds that $m \neq 0$.

For a $d$-dimensional hydrogen atom characterized by mass $m$ and a potential constant $k$, the regularized Hamiltonian $\mathbf{H}(\epsilon)$ for a non-zero parameter $\epsilon \in \mathbb{R}^\times$ is a continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ defined as: $$\mathbf{H}(\epsilon) = \frac{1}{2m} \mathbf{p}^2 - k \mathbf{r}_\epsilon^{-1}$$ where $\mathbf{p}^2$ is the momentum-squared operator and $\mathbf{r}_\epsilon^{-1}$ is the regularized radius operator that acts via pointwise multiplication by the function $x \mapsto (\|x\|^2 + \epsilon^2)^{-1/2}$.