Group-theoretical analysis shows that SO(2,1) is an underlying dynamical symmetry for all Hamiltonians that are compatible with the Jacobi matrix (J-matrix) formalism. The class of central potentials with this property is obtained including, but not limited to, the oscillator, Coulomb and Morse potentials. The L² bases and J-matrix elements for these potentials are found. SO(2,1)-invariant transformation of the solutions of the recursion relation for one potential gives those for another potential in the class. Phase-shift and resonance calculations for a single-channel potential are carried out in the oscillator basis to illustrate the use of our results.