# ABC of ladder operators for rationally extended quantum harmonic oscillator systems

## TL;DR

This paper develops a comprehensive framework for constructing ladder operators in rationally extended quantum harmonic oscillators, revealing their algebraic structure and connection to Darboux-Crum-Krein-Adler transformations.

## Contribution

It introduces a trinity of primary ladder operators and shows how higher-order operators are derived, linking spectral features to algebraic identities in REQHO systems.

## Key findings

- Identified a set of basic ladder operators reflecting spectral properties.
- Demonstrated how higher-order operators are composed from basic ones.
- Connected ladder operator structure to Darboux-Crum-Krein-Adler transformation schemes.

## Abstract

The problem of construction of ladder operators for rationally extended quantum harmonic oscillator (REQHO) systems of a general form is investigated in the light of existence of different schemes of the Darboux-Crum-Krein-Adler transformations by which such systems can be generated from the quantum harmonic oscillator. Any REQHO system is characterized by the number of separated states in its spectrum, the number of `valence bands' in which the separated states are organized, and by the total number of the missing energy levels and their position. All these peculiarities of a REQHO system are shown to be detected and reflected by a trinity $(\mathcal{A}^\pm$, $\mathcal{B}^\pm$, $\mathcal{C}^\pm$) of the basic (primary) lowering and raising ladder operators related between themselves by certain algebraic identities with coefficients polynomially-dependent on the Hamiltonian. We show that all the secondary, higher-order ladder operators are obtainable by a composition of the basic ladder operators of the trinity which form the set of the spectrum-generating operators. Each trinity, in turn, can be constructed from the intertwining operators of the two complementary minimal schemes of the Darboux-Crum-Krein-Adler transformations.