Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator

TL;DR

This paper constructs ladder operators for a rationally extended quantum harmonic oscillator, revealing a unique spectrum structure with isolated ground states and generalized Jordan states, advancing understanding of algebraic and spectral properties.

Contribution

It introduces a new ladder operator framework for the REQHO, uncovering its spectrum's algebraic structure and the role of generalized Jordan states.

Findings

The spectrum includes a trivial and an infinite-dimensional irreducible representation.

The ground state is isolated but accessible via ladder operators through non-physical states.

Six generalized Jordan states are involved in the discrete chains of physical states.

Abstract

Ladder operators for the simplest version of a rationally extended quantum harmonic oscillator (REQHO) are constructed by applying a Darboux transformation to the quantum harmonic oscillator system. It is shown that the physical spectrum of the REQHO carries a direct sum of a trivial and an infinite-dimensional irreducible representation of the polynomially deformed bosonized osp(1|2) superalgebra. In correspondence with this the ground state of the system is isolated from other physical states but can be reached by ladder operators via non-physical energy eigenstates, which belong to either an infinite chain of similar eigenstates or to the chains with generalized Jordan states. We show that the discrete chains of the states generated by ladder operators and associated with physical energy levels include six basic generalized Jordan states, in comparison with the two basic Jordan states entering in analogous discrete chains for the quantum harmonic oscillator.