# Spectral intertwining relations in exactly solvable quantum-mechanical systems

## TL;DR

This paper introduces spectral intertwining relations that unify ladder and intertwining operators in exactly solvable quantum systems, enabling new connections between eigenfunctions of different energies and Hamiltonians.

## Contribution

It proposes the spectral intertwining relation as a novel framework that extends beyond shape invariance, providing new operators for hydrogen and Rosen--Morse Hamiltonians.

## Key findings

- Spectral intertwining relations connect eigenfunctions of different energies.
- New spectral intertwining operators are found for hydrogen and Rosen--Morse systems.
- The framework extends solvable models beyond traditional shape invariance.

## Abstract

In exactly solvable quantum-mechanical systems, ladder and intertwining operators play a central role because, if they are found, the energy spectra can be obtained algebraically. In this paper, we propose the spectral intertwining relation as a unified relation of ladder and intertwining operators in a way that can depend on the energy eigenvalues. It is shown that the spectral intertwining relations can connect eigenfunctions of different energy eigenvalues belonging to two different Hamiltonians, which cannot be obtained by previously known structures such as shape invariance. As an application, we find new spectral intertwining operators for the Hamiltonians of the hydrogen atom and the Rosen--Morse potential.