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

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Abstract

The Tsallis q-exponential function \(e_{q}(x) = (1+(1-q)x)^{\frac {1}{1-q}}\) is found to be associated with the deformed oscillator defined by the relations \(\left [N,a^{\dagger }\right ] = a^{\dagger }\), [N,a] = −a, and \(\left [a,a^{\dagger }\right ] = \phi _{T}(N+1)-\phi _{T}(N)\), with ϕT(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.

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  1. Retired Faculty (Physics), The Institute of Mathematical Sciences, Cental Institutes of Technology (CIT) Campus, Tharamani, Chennai, 600113, India

    Ramaswamy Jagannathan

  2. Department of Mathematics and Sciences, College of Arts and Applied Sciences (CAAS), Dhofar University, Post Box No. 2509, 211, Salalah, Oman

    Sameen Ahmed Khan

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Jagannathan, R., Khan, S.A. On the Deformed Oscillator and the Deformed Derivative Associated with the Tsallis q-exponential. Int J Theor Phys 59, 2647–2669 (2020). https://doi.org/10.1007/s10773-020-04534-w

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