# Raising and lowering operators for quantum billiards

## TL;DR

This paper introduces operators for quantum billiards that generate all eigenstates from a single state, analogous to raising and lowering operators in angular momentum, providing a new algebraic framework for integrable billiards.

## Contribution

The paper develops a set of operators that systematically generate the eigenstates of planar integrable billiards from an initial state, establishing an algebraic structure similar to angular momentum theory.

## Key findings

- Operators can generate the entire eigenstate spectrum from a single state.
- The classification of eigenstates is based on a quantum number modulated by a parameter k.
- The approach parallels raising and lowering operators in angular momentum algebra.

## Abstract

For planar integrable billiards, the eigenstates can be classified with respect to a quantity determined by the quantum numbers. Given the quantum numbers as $m, n$, the index which represents a class is $c = m\, \mbox{mod}\, k\,n$ for a natural number, $k$. We show here that the entire tower of states can be generated from an initially given state by application of the operators introduced here. Thus, these operators play the same role for billiards as raising and lowering operators in angular momentum algebra.