Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well

Abstract: In this paper, we study the Schr\"odinger equation with Dunkl derivative for a free particle confined in a cylindrical potential well. We consider both the finite and infinite height cases. The Dunkl formalism introduces reflection operators that modify the structure of the Hamiltonian and affect the parity of the solutions. By working in cylindrical coordinates, we obtain exact analytical expressions for the radial and axial wavefunctions in terms of Bessel functions. The energy spectrum and the solutions are classified according to the eigenvalues of the reflection operators in the three coordinates. We analyze in detail the conditions under which the wavefunctions acquire definite parity and discuss the resulting constraints on the Dunkl parameters.