This is a truly difficult question, because the answer I am looking for is strictly mathematical. In the textbook by L. Landau & E. Lifschitz, "Quantum Mechanics", the Schrödinger wavefunction generically denoted by $R_{kl}$ of the H-atom corresponding to the continuous $E\in[0,\infty)$ spectrum of the dummy particle is given by formula (36.18), where F is a Kummer (confluent) hypergeometric function.
The question: Identify the proper space of distributions and prove that $R_{kl}$ are vectors in it.
I know a bit about distribution spaces, and found this discussion here Do tempered distributions form a topological subspace of the space of distributions? both illuminating and dissapointing (in the sense that I was truly hoping for a proof that the inclusion $\mathcal{S}'(\Omega)\subset \mathcal{D}'(\Omega)(\Omega\subset\mathbb{R}^n, \text{open}$) was also topological), and I cannot say for sure that the TVS sought in the question is $\mathcal{D}'\left(\left(0,\infty),~dr\right)\right)$ or a particular subset of it under the right topology.
So can you help me with the answer and a rigorous proof?
