Complex scalar field in quantum cosmology: From instanton to inflation

We investigate the cosmological model with the complex scaler self-interact ing inflaton field non-minimally coupled to gravity. The different geometri es of the Euclidean classically forbidden regions are represented. The inst anton solutions of the corresponding Euclidean equations of motion are foun d by numerical calculations supplemented by the qualitative analysis of Lor entzian and Euclidean trajectories. The applications of these solutions to the no-boundary and tunneling proposals for the wave function of the Univer se are studied. Possible interpretation of obtained results and their conne ction with inflationary cosmology is discussed.