The Characteristic Equation of q-Deformed Bosonic Oscillator

The main goal of this paper is to deduce the characteristic equation of [N]. From the fundamental relation of q-deformed bosonic oscillator, we deduce a strict algebraic equation. After introducing a moderate boundary condition, we obtain the characteristic equation of number operator [N].Using this equation, we extract all the eigenvalues of operator [N] and write out the corresponding eigenstates. Then we obtain the completeness of the Hilbert space.