Non-real eigenvalues for \(\mathcal{PT}\)-symmetric double wells (Q505649)

The authors study \(\mathcal{PT}\)-symmetric perturbations of self-adjoint double-well Schrödinger operator theory in any space dimension \(n\geq 1\) on a smooth compact Riemann manifold. The main result in the paper states that the two eigenvalues are real and simple for small perturbations \(0<\varepsilon< a_0\), there is a double real eigenvalue for \(\varepsilon=a_0\) and are complex for \(a_0<\varepsilon \leq a_1\). Some extensions of the tunneling theory of Helffer-Sjöstrand for non self-adjoint Schrödinger operators are necessary for the proofs.