  We study fermionic zero modes in the domain wall background. The fermions
have Dirac and left- and right-handed Majorana mass terms. The source of the
Dirac mass term is the coupling to a scalar field $\Phi$. The source of the
Majorana mass terms could also be the coupling to a scalar field $\Phi$ or a
vacuum expectation value of some other field acquired in a phase transition
well above the phase transition of the field $\Phi$. We derive the fermionic
equations of motion and find the necessary and sufficient conditions for a zero
mode to exist. We also find the solutions numerically. In the absence of the
Majorana mass terms, the equations are solvable analytically. In the case of
massless fermions a zero energy solution exists and we show that although this
mode is not discretely normalizable it is Dirac delta function normalizable and
should be viewed as part of a continuum spectrum rather than as an isolated
zero mode.
