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arXiv:2508.04840 · Modern Physics Letters A

Exact solutions of the Schrödinger–Dunkl equation for a free particle in a finite and infinite cylindrical well

R. D. Mota, D. Ojeda-Guillén, M. Salazar–Ramírez

Published Aug 18, 2025• 1 citation
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Abstract

In this paper, we study the Schrödinger 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.

DOI
10.1142/s0217732325501482

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The use of the Dunkl derivative, defined by a combination of the difference-differential and reflection operators, allows the classification of the solutions according to even and odd solutions. Recently, we considered the Dunkl formalism to investigate the Bose-Einstein condensation of an ideal Bose gas confined in a gravitational field. In this work, we address another essential problem and examine an ideal Bose gas trapped by a three-dimensional harmonic oscillator within the Dunkl formalism. To this end, we derive an analytic expression for the critical temperature of the N particle system, discuss its value at large-N limit andfinally derive and compare the ground state population with the usual case result. In addition, we explore two thermal quantities, namely the Dunkl-internal energy and the Dunkl-heat capacity functions.
Recent studies have shown that the use of Dunkl derivatives instead of ordinary derivatives leads to deriving parity-dependent dynamic solutions. According to this motivation in this manuscript, we formulate the Dunkl-Schrödinger equation within the position-dependent mass formalism and derive an algebraic solution out of it. Our systematic approach lets us observe some new findings in addition to the earlier ones. For example, we find that the solution of the Dunkl-Schröodinger equation with position-dependent mass cannot be considered independent of the choice of parameters. Similarly, through the sl(2) algebra, the energy spectrum and the corresponding wave functions are derived in terms of possible Dunkl, (μμμ), and mass, (ααα), parameters.