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 HN, and produces two biorthogonal bases of HN which are eigenstates of the Hamiltonians h=12(q2+p2), 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.