This study is broadly divided into two parts.rnThe first deals with a set of useful results in non-relativisticrnquantum mechanics that prove to be useful in a variety of contexts.rnAlthough elementary, they have remained obscure for a long time. Thernadvent of quarkonium spectroscopy has basically revealed their importance.rnThe second part considers the quantum discontinuity phenomenon. Itsrnorigin is carefully traced. Two original partially exactly solvable modelsrnare proposed that exhibit such a phenomenon