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We rigorously solve the time-independent Schr\"odinger equation for theRosen-Morse type potential. By using the Nikiforov-Uvarov method, we obtain, ina systematic way, the complete solution of such equation, which includes theso-called bound states (square-integrable solutions) associated with thediscrete spectrum, as well as unbound states region (bounded but notnecessarily square-integrable solutions) related to the continuous part of thespectrum. The resolution of this problem is used to show that the kinks of thenon-linear Klein-Gordon equation with $\varphi^{2p+2}$ type potentials arestable. We also derive the orthogonality and completeness relations satisfiedby the set of eigenfunctions which are useful in the description of thedynamics of kinks under perturbations or interacting with antikinks.

The Schr\"odinger equation for the Rosen-Morse type potential revisited with applications