A general basis set algorithm for galactic haloes and discs

We present a unified approach to (bi-)orthogonal basis sets for gravitating systems. Central to our discussion is the notion of mutual gravitational energy, which gives rise to the self-energy inner product on mass densities. We consider a first-order differential operator that is self-adjoint with respect to this inner product, and prove a general theorem that gives the conditions under which a (bi-)orthogonal basis set arises by repeated application of this differential operator. We then show that these conditions are fulfilled by all the families of analytical basis sets with infinite extent that have been discovered to date. The new theoretical framework turns out to be closely connected to Fourier-Mellin transforms, and it is a powerful tool for constructing general basis sets. We demonstrate this by deriving a basis set for the isochrone model and demonstrating its numerical reliability by reproducing a known result concerning unstable radial modes.