Spectral analysis of the complex cubic oscillator

Using the 'exact semiclassical analysis', we study the spectrum of a one-parameter family of complex cubic oscillators. The PT-invariance property of the complex Hamiltonians and the reality property of the spectrum are discussed. Analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated. The global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by Bender and Wu for the quartic oscillator.