q-Schrödinger equations for V = u² + 1/u² and Morse potentials in terms of the q-canonical transformation

The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2, 1) of the Schrödinger equations for the Morse and the V = u² + 1/u² potentials were known to be related by a canonical transformation, q-deformed analog of this transformation connecting two different realizations of the sl_q(2) algebra is presented. By the virtue of the q-canonical transformation, a q-deformed Schrödinger equation for the Morse potential is obtained from the q-deformed V = u²+1/u² Schrödinger equation. Wave functions and eigenvalues of the q-Schrödinger equations yielding a new definition of the q-Laguerre polynomials are studied.