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

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 ${\text{Hil}}_N$, and produces two biorhogonal bases of ${\text{Hil}}_N$ which are eigenstates of the Hamiltonians $h=\frac{1}{2}(q^2+p^2)$, and of its adjoint $h^{†}$. Here $q$ and $p$ are non-Hermitian operators obeying $[q,p]=i(\text{1}-Nk)$, 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.