Some classes of finite-dimensional ladder operators

TL;DR

This paper introduces and analyzes special classes of ladder operators in finite-dimensional Hilbert spaces, including truncated quons, pseudo-versions, and operators on a closed chain, exploring their eigenstates and resolution of identity.

Contribution

It presents new classes of ladder operators in finite dimensions, including discrete coherent states and biorthogonal eigenstates, expanding the understanding of operator structures in quantum systems.

Findings

Existence of discrete coherent states as eigenvectors of the annihilation operator.

Resolution of the identity involving these states and biorthogonal vectors.

Introduction of new classes of ladder operators in finite-dimensional Hilbert spaces.

Abstract

We introduce and study some special classes of ladder operators in finite-dimensional Hilbert spaces. In particular we consider a truncated version of quons, their {\em psudo-}version, and a third family of operators acting on a closed chain. In this latter situation, we discuss the existence of what could be considered {\em discrete coherent states}, as suitable eigenvectors of the annihilation operator of the chain. We see that, under reasonable assumptions, a resolution of the identity can be recovered, involving these states, together with a biorthogonal family of vectors, which turn out to be eigenstates of the raising operator of the chain.