Finite-dimensional pseudo-bosons: A non-Hermitian version of the truncated harmonic 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 H N , and produces two biorthogonal bases of H N which are eigenstates of the Hamiltonians h=[Formula presented](q 2 +p 2 ), and of its adjoint h † . 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 † and p † . Some examples are discussed.