Fermionic zero modes on domain walls

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 the scalar field $\Phi .$ The source of the Majorana mass terms could also be the coupling to the scalar field $\Phi$ or the 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.