The Schr\"odinger-type formalism of the Klein-Gordon quantum mechanics is
adapted for the case of the SL(2,R) spacetime. The free particle case is
solved, the results of a recent work are reproduced while all the other,
topologically nontrivial solutions and the antiparticle modes are also found,
and a deeper insight into the physical content of the theory is given.Comment: version to appear in J. Math. Phys, 12 pages, LaTe