QUANTUM DEFORMATIONS OF THE DISCRETE NONLINEAR SCHRODINGER-EQUATION

A quantum system with the Hamiltonian and commutation relations depending on a deformation parameter epsilon is introduced. When epsilon = 0 the system reduces to the quantum Ablowitz-Ladik (QAL) equation, for epsilon = gamma/3 it represents a quantum discrete nonlinear Schrodinger (QDNLS) system, and for epsilon = gamma 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.