Adgyebraichyeskoye rassmotryeniye nyeryelyativistskikh i ryelyativistskikh kvantovykh uravnyenii{cyrillic, short} i ikh svyaz{cyrillic small soft sign} s tyeoriyei{cyrillic, short} diffyeryentsial{cyrillic small soft sign}nykh uravnyenii{cyrillic, short}.

The algebraic treatment of the eigenvalue equations for quantum systems, based on the introduction of the spectrum-generating algebra SO _2.1, in the sense of a previous work, is generalized by allowing energy-dependent realizations of the algebra. A basic differential equation (3.10) is derived, which expresses in a concise way the conditions that the Hamiltonian of the system must satisfy in order that an algebraic treatment, within our framework, be possible. The formalism is such that it allows a straightforward and unified rederivation of all results previously obtained by algebraic methods both for the nonrelativistic and relativistic cases, and also, to some extent, a discussion of the Dirac equation with an electrostatic potential, which has not been solved algebraically up to now. Moreover, the obtained equation (3.10) yields in a natural way the link between the algebraic treatment of quantum systems based on the SO _2.1 algebra and the theory of differential equations. It is shown that the conditions that the system must satisfy in order to have SO _2.1 as spectrum-generating algebra in our sense lead to the same condition as the one obtained by imposing that the Schrödinger equation can be reduced, by means of a functional transformation, to the confluent hypergeometric equation. This result is particularly interesting since all quantum-mechanical systems that have been solved algebraically up to now possess SO _2.1 as spectrum-generating algebra.