is the projection of $\Psi(x^{\mu},y)$ on the 3-brane and $\zeta^n(y)$ is the extra dimensional component of 5D spinor. Left ($\chi_L$) and right ($\chi_R$) states are constructed by $\chi^n_{L,R} = \frac{1}{2}(1 \mp \gamma^{5})\chi^n$. Thus the KK mode expansion can be written in the following way : $$\Psi(x^{\mu},y) = \sum[\chi_L^{n}(x^{\mu})\zeta_L^{n}(y) + \chi_R^{n}(x^{\mu})\zeta_R^{n}(y)]
   \label{kk mode expansion}$$ Substituting the KK mode expansion of $\Psi(x^{\mu},y)$ in the Dirac field lagrangian given in eqn.(\[dirac lagrangian\]), we obtain the following equations of motion for $\zeta_{L,R}(y)$ as follows : $$\begin{aligned}
    &e&^{-A(y)}e^{-\frac{2\kappa}{\sqrt{3}}\big(<\Phi>+\xi(y)\big)}
   \bigg[\pm e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\bigg(\partial_{y}-2A'(y)\nonumber\\ 
   &+&\frac{\kappa}{\sqrt{3}}\xi'(y)\bigg) + m_5\bigg]\zeta^n_{R,L}(y) = -m_n\zeta^n_{L,R}(y)
   \label{eom for wave function}
  \end{aligned}$$

where $m_{n}$ is the mass of nth KK mode. The 4D fermions obey the canonical equation of motion $i\gamma^{\mu}\partial_{\mu}\chi^n_{L,R} = m_{n}\chi^n_{L,R}$. Moreover eqn.(\[eom for wave function\]) is obtained provided the following normalization conditions hold : $$\int_{0}^{\pi} dy e^{-3A(y)}e^{-\frac{2\kappa}{\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\zeta_{L,R}^{m}\zeta_{L,R}^{n} = \delta_{m,n}
   \label{norm1}$$ $$\int_{0}^{\pi} dy e^{-3A(y)}e^{-\frac{2\kappa}{\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\zeta_{L}^{m}\zeta_{R}^{n} = 0
   \label{norm2}$$

In the next two subsections, we discuss the localization scenario for massless and massive KK modes respectively.

Massless KK mode
----------------

For massless mode, equation of motion of $\zeta_{L,R}$ takes the following form $$\begin{aligned}
   \bigg[\pm e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\bigg(\partial_{y}-2A'(y) 
   - \frac{\kappa}{\sqrt{3}}\xi'(y)\bigg) + m_5\bigg]\zeta^n_{R,L}(y) = 0
   \label{eom for wave function1}
  \end{aligned}$$

where $A(y)$ and $\xi(y)$ are given in eqn. (\[sol\_warp\_final\]) and eqn.(\[sol\_scalar\_final\]) 