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We show that the infinite-dimensional representation of the recently introduced logistic algebra can be interpreted as a nontrivial generalization of the Heisenberg or oscillator algebra. This allows us to construct a quantum Hamiltonian having the energy spectrum given by the logistic map. We analyze the Hamiltonian of a solid whose collective modes of vibration are described by this generalized oscillator and compute the thermodynamic properties of the model in the two-cycle and r=3.6785 chaotic region of the logistic map.

Thermodynamic properties of a solid exhibiting the energy spectrum given by the logistic map