Edit May 24th 2022
Please note, I no longer think that
\begin{equation*} \int _0^\infty G_L(\eta,~k^\prime r) G_L(\eta,~kr) dr = N_{L,G} \delta(k-k^\prime) \end{equation*}
for some real number $N_{L,G}$. See also, my answer, to this question, ( dated May 10th 2021 ).
I have left this question, as it was though, apart from this edit.
End of edit
A one dimensional momentum eigenfunction $ e^{ikx}$, can be normalized in terms of the 'Dirac Delta Function', we can write that, for some value of $C$
\begin{equation*} \int_{-\infty}^\infty e^{-ik^\prime x} e^{ikx} dx = C \delta( k-k^\prime) \end{equation*}
$C$ can be evaluated, $C=2\pi$, see Reference 1,
I think that both the 'Regular Coulomb Wave Function', $F(\eta,~\rho)$ , and the 'Irregular Coulomb Wave Function' $G(\eta,~\rho)$, can also be normalised in terms of the delta function, with the numbers, $N_{L,F},~N_{L,G}$ each measuring the"ratios" of two infinities.
So, we should have, for some values of $N_{L,F}$ and $N_{L,G}$ \begin{equation*} \int _0^\infty F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr = N_{L,F} \delta(k-k^\prime) \end{equation*} and \begin{equation*} \int _0^\infty G_L(\eta,~k^\prime r) G_L(\eta,~kr) dr = N_{L,G} \delta(k-k^\prime) \end{equation*}
My understanding is, that $F_L(\eta,~ \rho)$ and $G_L(\eta,~ \rho)$, are real valued functions, so their complex conjugate functions, are themselves.
See 'Other Information', for the definitions of $F_L(\eta, ~\rho)$ and $G_L(\eta, ~\rho)$, the defining equations are quoted from Abramowitz and Stegun$^2$.
It can be argued, see 'Other Information', that for our purposes, the $F_L$ integral, behaves essentially the same as one, with $F_L$ replaced by it's asymptotic form $F_{L,a}$, see at the beginning of 'Other Information'.
So we have \begin{equation*} \int _0^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr = N_{L,F} \delta(k-k^\prime) \end{equation*}
I would like to do the same "trick" with the $G_L$ integral but it seems problematical. I think that there are maybe two infinities associated with the $G_L$ integral. One associated with the form of $G_L$, oscillating out to infinity, and possibly one to do with $G_L$, going to infinity at $r=0, ( or \rho = kr=0 )$
Could anyone help me with this difficulty? Perhaps you can tell me how to justify the following?
\begin{equation*} \int _0^\infty G_{L,a}(\eta,~k^\prime r) G_{L,a}(\eta,~kr) dr = N_{L,G} \delta(k-k^\prime) \end{equation*}
$G_{L,a}$ being the asymptotic form of $G_L$.
References
- See the materials at
within which it is proved, that $C$ has the value $2\pi$.
- See at
https://archive.org/details/AandS-mono600/page/n551/mode/2up
Other Information
Replacing $F_L$ by it's asymptotic form $F_{L,a}$, supporting argument.
NB Adding or subtracting a finite quantity, to or from respectively, one that is infinite, cannot really affect the infinite quantity. So the following should be OK when $k=k^\prime$.
\begin{equation*} \int _0^\infty F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr = \int _0^{r_a} F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr + \int _{r_a}^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr \end{equation*} where, $F_L$ has it's asymptotic form $F_{L,a}$, for any $r>r_a$.
What we are interested in, with evaluating $N_{L,F}$, is in finding the ratio of the infinity in '$\int _0^\infty F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr$', when $k^\prime=k$, to the infinity in '$ \delta(k-k^\prime)$'.
Introducing the symbol '$\doteq$' to mean 'for our purposes may be replaced by',
we may put,
\begin{equation*} \int _0^\infty F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr \doteq \int _{r_a}^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr \end{equation*}
\begin{align*} \int _{r_a}^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr &\doteq \int _{r_a}^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr+ \int _0^{r_a} F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr \\ &= \int _0^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr \end{align*}
Hence,
\begin{equation*} \int _0^\infty F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr \doteq \int _0^\infty F_{L,a}(\eta,~k^\prime r) F_{L,a}(\eta,~kr) dr \end{equation*}
when $k=k^\prime$.
WHEN $k\neq k^\prime$
Ignoring the possible orthogonality of eigenfunctions here, anyone with a comment?
Thinking in terms of continuity, when $k\approx k^\prime$ the infinity in $\int _0^\infty F_L(\eta,~k^\prime r) F_L(\eta,~kr) dr $ should still be a very large number, so the above "equations" of the argument should still be OK. Hence should be usefull in considering certain limiting behaviour.
For larger and larger $\lvert k-k^\prime \lvert$ the equations are probably more and more dubious.
Equations from Abramowitz and Stegun$^2$, see the provided link
$\mathbf{14.1.3}$ \begin{equation*} F_L(\eta,~ \rho)= C_L(\eta) \rho^{L+1} e^{ -i\rho }~M( L+1-i\eta,~2L+2,~2i\rho ) \end{equation*}
$\mathbf{14.1.14}$ \begin{equation*} G_L(\eta,~ \rho)=\frac{ 2\eta } { C_0^2(\eta) }~F_L(\eta,~ \rho) \left[ ~ln~2\rho+ \frac{q_L(\eta) }{ p_L(\eta } \right] +\theta_L(\eta,~\rho) \end{equation*}
To get on top of these definitions, various other results from Abramowitz and Stegun$^2$, need to be taken into account.
Related questions, see at
How do you work out, the 'Delta Function Normalization', of a 'Regular Coulomb Wave Function'?
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