I have a question on eigenvalue and eigenfunction in spherical coordinates.
Let $0 \leq u \leq 1$, and a function $j_0(u)$ (zero order spherical Bessel first kind):
$$ j_0(u) = \frac{\sin(u)}{u} $$
Let us define the roots $v_n > 0$ of $j_1(u) = - \partial_u j_0(u)$ as:
$$ j_1(v_n) = \frac{\sin(v_n) - v_n \cos(v_n)}{v_n^2} = 0 $$
Then we have the roots defined by:
$$ v_n = \tan(v_n) $$
The question:
I am looking for a function $f(u v_n)$ which obeys to:
$$ \langle j_0(u v_n) \vert f(u v_m) \rangle = \int_0^1 u^2 j_0(u v_n) f(u v_m) d u \propto \delta_{n,m} $$
Do you have an idea ? I tried $f(u v_n) = j_0(u v_n)$ but it does not work, I started to try $f(u v_n) = j_1(u v_n)$ but that leads to complicated functions... maybe there is a simple rule? I know the orthogonality relation for:
$$ \langle j_0(u \lambda_n) \vert j_0(u \lambda_m) \rangle \propto \delta_{n,m} $$
with $j_0(\lambda_n) = 0$, but that's it...