So you know ?

Sometimes I met students majoring in theoretical physics who has no clear idea about the following important fact of quantum mechanics at the introductory level based on non-relativistic Schrodinger equation:
1. The single-valuedness and finiteness everywhere of physical wave functions are derived solely from the Born interpretation;
2. The matching conditions (or say interface conditions) used for example in textbook problems such as a particle tunneling through a square potential barrier are derived solely from Schrodinger equation and vary from case to case;
3. The sign of energy, E, determines whether a state is localized or extended : localized for E<0 while extended for E>0, because the notion of 'localized' and 'extended' actually refers to what happens on the boundary at infinity and henceforth, in the infinity, negative energy leads to imaginary wave number while positive to real. One just needs check the asymptotic behaviors of Schrodinger equation.
4. The boundary conditions for localized states: wave functions vanishing in the infinity; while that for extended: wave functions finite everywhere. Therefore, 'localized' or 'extended' are simply states of distinct boundary conditions.
5. Localized states form discrete spectrum (whose values can not be experimentally prepared in arbitrary fasion); while extended ones (whose values can be experimentally prepared in arbitrary fasion) form a continua. A simple illustration: to get localized states one just imagines a particle in a very large box, while to get extended ones one just makes use of free particle states (plane waves). Scattering states are typical extended states.