In this thesis we focus on two elementary unstable quantum systems, the inverted harmonic oscillator and the inverse square potential, using the methods of effective field theory (EFT) and the renormalization group (RG). We demonstrate that the phenomenon of fall to the centre associated with the inverse square potential is an example of a PT symmetry breaking transition. We also demonstrate a mapping between the inverted harmonic oscillator and the inverse square potential including a one-to-one mapping between the quantum states and boundary conditions using an EFT framework in a renormalization group invariant way. We apply these methods to the phenomenon of Schwinger pair production and study finite size effects using the RG scheme for the quantum inverted harmonic oscillator. / Thesis / Doctor of Philosophy (PhD)