One must then have \hat{\mathcal P}^2\, \psi{\left(r\right)} = e^{i\phi}\psi{\left(r\right)}, since an overall phase is unobservable. The operator \hat{\mathcal P}^2, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases e^{i\phi}. If \hat{\mathcal P}^2 is an element e^{iQ} of a continuous U(1) symmetry group of phase rotations, then e^{-iQ}is part of this U(1) and so is also a symmetry. In particular, we can define \hat{\mathcal P}' \equiv \hat{\mathcal P}\, e^{-{iQ}/{2}}, which is also a symmetry, and so we can choose to call \hat{\mathcal P}' our parity operator, instead of \hat{\mathcal P}. Note that {\hat{\mathcal P}'}^2 = 1 and so \hat{\mathcal P}' has eigenvalues \pm 1. Wave functions with eigenvalue +1 under a parity transformation are even functions, while eigenvalue −1 corresponds to odd functions. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than \pm 1.
