Non-minimally coupled complex scalar field in classical and quantum cosmology

We investigate a cosmological model with a complex scalar self-interacting inflaton field non-minimally coupled to gravity. The different geometries of the Euclidean classically forbidden regions are represented. The instanton solutions of the corresponding Euclidean equations of motion are found by numerical calculations. These solutions give rather non-trivial examples of real tunnelling geometries. Possible interpretation of obtained results and their connection with inflationary cosmology is discussed.