An interesting limit property of zeros of Bessel functions?

Question

I was trying to prove the orthogonality relations

\begin{equation} \int\limits_0^\infty k r \cdot J_n[k r] \cdot J_n[k^{'} r] dr = \delta(k-k^{'}) \tag{1} \end{equation}

for the Bessel functions, as quoted here for example. I followed the Sturm-Liouville approach as described here for example. Starting from the ODE being satisfied by the functions in question it is fairly easy to show that the following identity holds true:

\begin{equation} \int\limits_0^M r \cdot J_n[ \frac{\theta_n^{(p)}}{M} r] \cdot J_n[ \frac{\theta_n^{(q)}}{M} r] dr = \delta_{p,q} \cdot \frac{M^2}{2} (-1) J_{n-1}[\theta_n^{(p)}] J_{n+1}[ \theta_n^{(p)}] \tag{2} \end{equation}

where $M >0$ and $\left( \theta_n^{(p)} \right)_{p=1}^\infty $ are zeros of the Bessel function $J_n()$.

If we now multiply $(2)$ by $\theta_n^{(p)}/M$ and then by $(\theta_n^{(p)} - \theta_n^{(p-1)})/M$ and then take the limit $M \rightarrow \infty $ then, if $(1)$ is true, we should be getting the following:

\begin{equation} \lim\limits_{p \rightarrow \infty}\frac{1}{2} (\theta_n^{(p)} - \theta_n^{(p-1)}) (-\theta_n^{(p)}) \cdot J_{n-1}[\theta_n^{(p)}] J_{n+1}[ \theta_n^{(p)}] = 1 \tag{3} \end{equation}

Now we have tested $(3)$ numerically using the code below:

n = RandomReal[{1/10, 5}]; mm = 50;
ths = Table[BesselJZero[n, k], {k, 1, mm}];

ListPlot[Table[(ths[[p]] - ths[[p - 1]])/2 (-ths[[p]]) BesselJ[n - 1, 
    ths[[p]]] BesselJ[n + 1, ths[[p]]], {p, 2, mm}]]

  

As you can see truncating the left hand side of $(3)$ at $p \le 10$ gives already the right hand side to three digits of precision.

Therefore my question would be twofold. Firstly, how do you go about proving $(1)$. Secondly, can we prove that odd conjecture $(3)$ without using $(1)$?

Answer 1

The meaning of the expression is that the Hankel transform followed by the inverse Hankel transform (which is the same as the Hankel transform) gives you back the original function $f$: $$ f(r)=\int_0^\infty J_{\nu}(kr)\left(\int_0^\infty f(\rho)J_\nu(k\rho)\rho d\rho\right) kdk $$ Without regard to issues of interchanging order of integration, you can rewrite this expression as $$ f(r)=\int_0^{\infty}\left(\int_0^\infty J_{\nu}(kr)J_{\nu}(k\rho)k dk\right)f(\rho)\rho d\rho $$ So the inner integral appears to be $$ \int_0^{\infty}J_{\nu}(kr)J_{\nu}(k\rho)kdk = \delta(r-\rho). $$ You really need to deal with the truncated integral and allow the upper limit on this integral to tend to $\infty$. Then things can be made rigorous. The first integral in your post does not converge, and neither does my last integral. However, this does make sense under reasonable Fourier-type conditions on $f$: $$ \lim_{R\rightarrow\infty}\int_0^\infty\left(\int_0^{R}J_{\nu}(kr)J_{\nu}(k\rho)kdk \right)f(\rho)\rho d\rho = f(r). $$