Trapping neutral fermions with kink-like potentials

The intrinsically relativistic problem of neutral fermions subject to kink-like potentials (∼tanhγx) is investigated and the exact bound-state solutions are found. Apart from the lonely hump solutions for E=±mc2, the problem is mapped into the exactly solvable Sturm-Liouville problem with a modified Pöschl-Teller potential. An apparent paradox concerning the uncertainty principle is solved by resorting to the concepts of effective mass and effective Compton wavelength.