# Families of Symmetries and the Hydrogen Atom

## TL;DR

This paper introduces a novel algebraic family of symmetries for the hydrogen atom, linking algebraic, spectral, and scattering theories through advanced representation techniques.

## Contribution

It constructs and characterizes an algebraic family of Harish-Chandra modules from hydrogen atom solutions, connecting symmetries with spectral and scattering analysis.

## Key findings

- Algebraic family of symmetries parametrized by energy
- Physical state subspaces derived via Jantzen filtration
- Connection established between algebraic methods and spectral theory

## Abstract

We study a new type of symmetry for the hydrogen atom involving algebraic families of groups parametrized by the energy value in the time-independent Schr\"odinger equation. We construct an algebraic family of Harish-Chandra modules from the solutions of the Schr\"odinger equation, and we characterize this family. We show that the subspaces of physical states may be obtained from our algebraic family using a Jantzen filtration, and we relate our algebraic methods with spectral theory and scattering theory using the limiting absorption principle