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Finite-dimensional pseudo-bosons: a non-Hermitian version of the truncated harmonic oscillator (1806.10063v1)
Published 26 Jun 2018 in math-ph, math.MP, and quant-ph
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 $h=\frac{1}{2}(q2+p2)$, 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.