Bounded version of bosonic creation and annihilation operators and their related quasicoherent states

Coherent states are usually defined as eigenstates of an unbounded operator, the so called annihilation operator. We propose here possible constructions of quasicoherent states, which turn out to be quasieigenstate of a bounded operator related to an annihilationlike operator. We use this bounded operator to construct a sort of modified harmonic oscillator and we analyze the dynamics of this oscillator from an algebraic point of view