Φασματικές μέθοδοι για την επίλυση της ημικλασικής εξίσωσης του Schrodinger

We consider the time-dependent linear Schrödinger equation in a semiclassical scaling. This equation forms a canonical example of a dispersive model whose solution exhibits high-frequency oscillations. Because of this, the design of efficient numerical methods which produce an accurate approximation of the solutions, or of the associated physical observables, is a mathematical and computational challenge. In this thesis we study two time-splitting spectral methods for the semiclassical Schrödinger equation and investigate their effectiveness in approximating the observables