On the Deformed Oscillator and the Deformed Derivative Associated with the Tsallis q -exponential

The Tsallis q -exponential function e_q(x) = (1+(1-q)x)^1/1-q is found to be associated with the deformed oscillator defined by the relations [N,a^† ] = a^† , [ N , a ] = − a , and [a,a^† ] = ϕ _T(N+1)-ϕ _T(N) , with ϕ T ( N ) = N /(1 + ( q − 1)( N − 1)). In a Bargmann-like representation of this deformed oscillator the annihilation operator a corresponds to a deformed derivative with the Tsallis q -exponential functions as its eigenfunctions, and the Tsallis q -exponential functions become the coherent states of the deformed oscillator. When q = 2 these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when q = 1 this deformed oscillator is a canonical boson oscillator, when 1 < q < 2 its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when q →2 it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state.