# Prove that the hydrogen continuum spectrum wavefunctions are distributions

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?

• What does "Identify the proper space of distributions" mean precisely? Does it mean "find a domain $\Omega$ such that all those functions are in $\mathcal{D}^{\prime}(\Omega)$"? But then "prove that $R_{kl}$ are vectors in it" doesn't make sense... Nov 9, 2017 at 23:27
• Yes, and if that domain is $\Omega = \left(0,\infty\right)$, then specify the concrete topology on this space and how it connects to the topology on $\mathcal D(0,\infty)$. Regarding the proof, this is simple: One has a statement (theorem): $\forall k, l; R_{kl}\in\mathcal{D}'\left(0,\infty\right)$. Proof: ... Nov 9, 2017 at 23:35
• So what is the space where we need to define a topology? Is it $\Omega$ or smth else? Because spaces $\mathcal{D}$ and $\mathcal{D}^{\prime}$ already have a defined topology. So far that's how it looks to me: $R_{kl}$ are in $\mathcal{D}^{\prime}$ because they are continuous hence locally integrable so they give rise to a distribution. They however are not in $\mathcal{S}^{\prime}$ since they grow very fast. Nov 9, 2017 at 23:57
• The topology needed is in $\mathcal D'$. How do I build its topology and convergence (weak, strong, seminorms? what you claim as "continuous") in this space? Nov 10, 2017 at 0:37
• @DanielC: Any continuous function $\phi(x)$ on an open set $\Omega$, e.g. the interval $(0,\infty)$ gives rise to a distribution in $\mathcal{D}'(\Omega)$ as the continuous linear form $f\mapsto \int_{\Omega} \phi(x)f(x)d^dx$. To check continuity you need the topology of the space of test functions $\mathcal{D}(\Omega)$. The easiest way to definie it is via the Horvath seminorms I mentioned in my answer to mathoverflow.net/questions/234025/… Nov 13, 2017 at 21:30