Quantum deformations of the discrete nonlinear Schrödinger equation

A quantum system with the Hamiltonian and commutation relations depending on a deformation parameter {ε} is introduced. When {ε}=0 the system reduces to the quantum Ablowitz-Ladik (QAL) equation, for {ε}={γ}/3 it represents a quantum discrete nonlinear Schr\"odinger (QDNLS) system, and for {ε}={γ} the system reduces to the usual QDNLS equation. We show that the energy levels of this system can be continuously deformed into the corresponding ones of the QAL and QDNLS equations. The physical significance of this system is also discussed.