Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 33C / a / ii

An even potential $V(x)=V(-x)$ preserves parity. Its Hamiltonian operator commutes with the parity operator, so an even incoming solution produces only an even outgoing solution, and similarly for an odd one. ThusAn asymmetric potential generically couples the parity channels; an accidental vanishing at one energy does not imply the symmetry for all energies.