Spectral Theory

Our next task is to classify the orbits (i.e. solutions) of the Schrödinger equation $$ i\hbar \frac{{\partial \psi }} {{\partial t}} = H\psi $$ with given initial condition $$ \psi |_{t = 0} = \psi _0 $$ according to their behaviour in space-time. Naturally, we want to distinguish between states which are localized for all time, and those whose essential support moves off to infinity. Such a classification is made with the help of a very important notion — the spectrum of an operator. We begin by describing the general theory, and then we proceed to applications.