  The aim of this work is to find exact solutions of the one-dimensional Dirac
equation that do not belong to the already known conventional class. We write
the spinor wavefunction as a bounded infinite sum in a complete basis set,
which is chosen such that the matrix representation of the Dirac wave operator
becomes tridiagonal and symmetric. This makes the wave equation equivalent to a
symmetric three-term recursion relation for the expansion coefficients of the
wavefunction. We solve the recursion relation and obtain the relativistic
energy spectrum and corresponding state functions. Restriction to the diagonal
representation results in the conventional class of solutions. As illustration,
we consider the square well with half-sine potential bottom in the vector and
scalar coupling under spin symmetry. We obtain the relativistic energy spectrum
and show that the nonrelativistic limit agrees with results found elsewhere.
