Do wave functions of scattering states of a symmetric potential exhibit definite parity?

Question

My question is quite simple. Suppose we are given a potential such as a potential barrier (potential $V = V_0, -a \leq x \leq a$ and 0 otherwise). Will scattering states which are solutions to this potential exhibit definite parity ($\psi(x) = \pm\psi(-x)$)?

Answer 1

Yeah, Of course! Since $$V(x)=V(-x)\rightarrow \psi(x)=\pm \psi(-x)$$