  The four-dimensional Dirac-Schr\"odinger equation satisfied by
quark-antiquark bound states is derived from Quantum Chromodynamics. Different
from the Bethe-Salpeter equation, the equation derived is a kind of first-order
differential equations of Schr\"odinger-type in the position space. Especially,
the interaction kernel in the equation is given by two different closed
expressions. One expression which contains only a few types of Green's
functions is derived with the aid of the equations of motion satisfied by some
kinds of Green's functions. Another expression which is represented in terms of
the quark, antiquark and gluon propagators and some kinds of proper vertices is
derived by means of the technique of irreducible decomposition of Green's
functions. The kernel derived not only can easily be calculated by the
perturbation method, but also provides a suitable basis for nonperturbative
investigations. Furthermore, it is shown that the four-dimensinal
Dirac-Schr\"odinger equation and its kernel can directly be reduced to rigorous
three-dimensional forms in the equal-time Lorentz frame and the
Dirac-Schr\"odinger equation can be reduced to an equivalent
Pauli-Schr\"odinger equation which is represented in the Pauli spinor space. To
show the applicability of the closed expressions derived and to demonstrate the
equivalence between the two different expressions of the kernel, the t-channel
and s-channel one gluon exchange kernels are chosen as an example to show how
they are derived from the closed expressions. In addition, the connection of
the Dirac-Schr\"odinger equation with the Bethe-Salpeter equation is discussed.
