# Coupled Supersymmetry and Ladder Structures Beyond the Harmonic Oscillator

## TL;DR

This paper introduces coupled supersymmetry, a generalization of SUSY quantum mechanics, providing a method to determine all eigenvalues of certain Hamiltonians beyond the harmonic oscillator, and develops related coherent states and uncertainty principles.

## Contribution

It presents coupled supersymmetry, extending algebraic methods to Hamiltonians not factorizable into canonical pairs, and constructs explicit differential operator realizations with orthonormal eigenfunctions.

## Key findings

- All eigenvalues can be obtained for a class of Hamiltonians using coupled SUSY.
- Developed coherent states and generalized uncertainty principles.
- Realized coupled SUSY through infinite families of differential operators.

## Abstract

The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general -- except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalize the Heisenberg uncertainty principle. Coupled SUSY is explicitly realized by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalized harmonic oscillators.