Non-Real Eigenvalues for {{\mathcal{PT}}}-Symmetric Double Wells

We study small, \({{\mathcal{PT}}}\)-symmetric perturbations of self-adjoint double-well Schrodinger operators in dimension \({n\ge 1}\). We prove that the eigenvalues stay real for a very small perturbation, then bifurcate to the complex plane as the perturbation gets stronger.