Abstract: In this introductory talk we will introduce the necessary mathematical structure to talk about electronic spectra of quantum systems. We will begin in the spin free single particle setting and give a overview of the rich mathematical structure present in this basic case. In this setting it is possible to give a precise mathematical meaning to a bound versus scattered state of the Hydrogen atom. We will then proceed to discuss multipartite systems mathematically and describe the formalism of density matrices and second quantization, the two primary methods of understanding many body quantum systems.