Reconstructing Charge Distribution from Multipole Expansion

Question

Let $\rho$ be a static, discrete or continuous charge distribution, and $\phi(\mathbf{r})$ the corresponding electric potential. We may expand $\phi$ in a multipole series, $$ \phi(\mathbf{r}) = \frac{1}{r}\sum_{n \geq 0} \int \text{d}^3r' \rho(\mathbf{r}')\left(\frac{r'}{r}\right)^nP_n(\cos\theta).$$ My question is, if given the multipole expansion, one can determine the source $\rho$.

Evidently, this is possible if given the multipole expansion of two equally and oppositely charged point sources a distance $a$ apart. From the dipole moment, one can deduce $a$, and thus reconstruct the charge distribution. But is this true in general?

Answer 1

But is this true in general?

No.

For a simple counterexample, consider two uniform solid spheres of charge, of different radii but with identical total charge $Q$: they will have vanishing multipole moments for all $l\geq 1$, and identical monopole moment, but the distributions are different.

There are equivalent examples for all multipole moments ─ the multipole distribution is basically blind to the radial aspects of the distribution, given suitable rescaling of the charges ─ but they're all basically copies of that monopole example.