  I study the midgap spectrum of the fermion-vortex system in two spatial
dimensions. The existence of bound states, in addition to the zero modes found
by Jackiw and Rossi, is established. For a singly quantized vortex, I present
complete analytical solutions in terms of generalized Laguerre polynomials in
the opposite limits of vanishing and large vortex core size. There is an
infinite number of such bound states, with a spectrum that is, when squared,
given by, respectively, the Coulomb potential and the isotropic harmonic
oscillator. Possible experimental signatures of this spectrum in
condensed-matter realizations of the system are pointed out.
