How do you work out, the 'Delta Function Normalization', of a 'Regular Coulomb Wave Function'?

Question

A one dimensional momentum eigenfunction $ e^{ikx}$, can be normalized in terms of the 'Dirac Delta Function', because 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, see the materials at

https://sites.google.com/site/rwhawksworthexamples/home/math/dirac-delta-function-various-pages/delta-function-normalization/momentum-eigenfunctions

within which it is proved, that $C$ has the value $2\pi$.

I think that a 'Regular Coulomb Wave Function', $F_L(\eta,~\rho)$, can also be normalised in terms of the delta function, as \begin{equation*} \int _0^\infty F_L(\eta^\prime,~k^\prime r) F_L(\eta,~kr) dr = N_L \delta(k-k^\prime) \end{equation*}

NB: we could put \begin{equation*} kr=\rho, ~~~~~~ k^\prime r = \rho^\prime \end{equation*} and also, my understanding is, that $F_L(\eta,~ \rho)$ is a real valued function, so it's complex conjugate function, is itself.

See $\mathbf{14.1.3}$ in 'Other Information', for the definition of $F_L(\eta, ~\rho)$, this equation is quoted from, Abramowitz and Stegun, see at

https://archive.org/details/AandS-mono600/page/n551/mode/2up

Could anyone provide a proof of what $N_L$ should be? I think a good guess is $N_L=1$ or $\frac{1}{k}$.

I have started a proof of what $N_L$ should be, but it looks as if it's going to be a very long one, I may never finish it.

Other Information

Some equations are quoted here, from Abramowitz and Stegun see the provided link

$\mathbf{14.1.1}$ \begin{equation*} \frac{ d^2 \omega }{ d \rho^2 } + \left[ 1- \frac{ 2 \eta }{\rho } - \frac{ L(L+1) }{ \rho^2 } \right] \omega = 0 \end{equation*} $(\rho >0, -\infty < \eta < \infty$, L a non-negative integer$)$

$\mathbf{14.1.2}$ \begin{equation*} ~~~~~~~~~~~~~~~~~~~~~~~\omega = C_1F_L(\eta, ~\rho)+C_2G_L(\eta,~ \rho)~~~~~~~~~(C_1, ~~C_2~~constants) \end{equation*} where $F_L(\eta, ~\rho)$ is the regular Coulomb wave function and $G_L(\eta,~ \rho)$ is the irregular (logarithmic) Coulomb wave function.

$\mathbf{14.1.3}$

\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*}

$\mathbf{14.1.7}$

\begin{equation*} C_L(\eta)= ~\frac{ 2^L~e{ \frac{ -\pi\eta}{ 2 } } ~ |~\Gamma(L+1+i\eta~) | } { \Gamma(2L+2) }~~~~~~~~~~~~~~~~~~~ \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 the definition of the irregular solution, quite a number of other results need to be taken into account.

Related Questions,

A problem with analysing the 'Delta Function Normalization', of an 'Irregular Coulomb Wave Function'.

Is this the way to 'Delta Function Normalise' a 'Continuum Wave Function'?

About the idea that, the "Normalization" of a "scattering wavefunction", being insensitive to the functions form near to the scattering centre.

https://physics.stackexchange.com/questions/635101/finding-two-remarks-in-diracs-the-principles-of-quantum-mechanics

Answer 1

Please note the link to a paper on arxiv.org, given in 'Other Information'. The paper linked to may be really useful.

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I have a link to what I thought was a proof, in a very restricted sense, of what $N_L$ is.

Please see at

https://sites.google.com/site/rwhawksworthexamples/home/math/dirac-delta-function-various-pages/delta-function-normalization/regular-coulomb-wave-function

Unfortunately my analysis of one of the limits involved, the one called $L_{2,3}$, is flawed.

Various results are incorrect for the special case, k= k’ ( essentially, “0 divided by 0” possibilities, escaped my attention ).

I might, in the future, bring into the analysis of $L_{2,3}$, the idea of ‘the limit of a function’ and material on arg ( Gamma(z) ).

I thought I had proved that

\begin{equation*} N_L = \frac{\pi}{2} \end{equation*}

So that we would have had \begin{equation*} \int _0^\infty F_L(\eta,~kr)~F_L(\eta^\prime,~k^\prime r) ~ dr = \frac{\pi}{2} \delta(k-k^\prime) \end{equation*}

NB: $N_{L,F}$ is used in the PDF found at the linked to web page, not $N_L$.

Originally, I thought there would be a lot more integrals, and hence limits, to be looked at in the proof.

NB: I think this value for $N_L$, can still be justified by a correct proof.

NB: I did not have to perform any integrals containing the '$ln$' function.

Other Information

Those interested in the Dirac delta function normalisation of functions used in scattering theory, might benefit from reading,

Y. T. Li, R. Wong, Integral and Series Representations of the Dirac Delta Function https://arxiv.org/abs/1303.1943.

Initially, please note the abstract, the text immediately before (1.5), (1.5), (1.6), (1.7) and the text just after (1.7). The authors discuss the Coulomb wave function, the Bessel function of the first kind, and also the Airy function, Ai(x). Also note (3.7) and (4.8).

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See at

How can you justify the delta function equation $\int_0^\infty~\delta(r)~dr = 1$?

and linked to questions, for the ideas of

The One-Sided Definition of the Dirac Delta Function

The Strong Definition of the Dirac Delta Function

See at

The Dirac Delta Function: How do you prove the $f(0)$ property using rigorous mathematics?

for how to prove, using rigorous mathematics, the f(0) property of the Dirac delta function for real valued f(x).