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Non-real eigenvalues for \(\mathcal{PT}\)-symmetric double wells. (English) Zbl 1372.35201

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.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35Q40 PDEs in connection with quantum mechanics
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
47B25 Linear symmetric and selfadjoint operators (unbounded)
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