Finite-dimensional pseudo-bosons: a non-Hermitian version of the truncated harmonic oscillator

TL;DR

This paper introduces a non-Hermitian, finite-dimensional deformation of the truncated harmonic oscillator, leading to new eigenstates and ladder operators within a modified commutation framework.

Contribution

It develops a novel finite-dimensional pseudo-bosonic model with a deformed commutation relation, expanding the understanding of non-Hermitian quantum systems.

Findings

Constructed two biorthogonal bases of eigenstates.

Defined ladder operators connecting these eigenstates.

Provided explicit examples of the deformed oscillator.

Abstract

We propose a deformed version of the commutation rule introduced in 1967 by Buchdahl to describe a particular model of the truncated harmonic oscillator. The rule we consider is defined on a $N$-dimensional Hilbert space $\Hil_N$, and produces two biorhogonal bases of $\Hil_N$ which are eigenstates of the Hamiltonians $h=\frac{1}{2}(q^2+p^2)$, and of its adjoint $h^\dagger$. Here $q$ and $p$ are non-Hermitian operators obeying $[q,p]=i(\1-Nk)$, where $k$ is a suitable orthogonal projection operator. These eigenstates are connected by ladder operators constructed out of $q$, $p$, $q^\dagger$ and $p^\dagger$. Some examples are discussed.