The algebraic symmetry of the hydrogen atom

The hydrogen atom system is a fundamental example of a quantum mechanical system. Symmetry plays the main role in our current understanding of the system. In this talk I will describe a new type of algebraic symmetry for the system. I will show that the collection of all regular solutions of the Schrödinger equation is an algebraic family of representations of different algebras. Such a family is known as an algebraic family of Harish-Chandra modules. The algebraic family has a canonical filtration from which the physically relevant solutions and the spectrum of the Schrödinger operator can be recovered.

If time permits I will relate the spectral theory of the Schrödinger operator to the algebraic family. No prior knowledge about quantum mechanics or representation theory will be assumed.