Title : Heun equation and some applications in physics
Abstract:
Most physical problems in quantum and relativistic mechanics that are characterized by four singular points lead to a second-order differential equation that cannot generally be solved in terms of classical special functions. The Heun equation constitutes the natural mathematical framework for treating such problems, since Heun functions provide an accurate description of quantum states in these complex systems.
In this work, we have presented the general Heun equation, its solutions, and several of its fundamental properties. We have also briefly discussed other members of the Heun family, including the confluent and double-confluent Heun equations. Subsequently, we solved the Klein–Gordon equation for a particle subjected to a position-dependent potential and a position-dependent mass distribution.
The underlying physical problem consists in describing a quantum particle evolving in a medium with spatially varying mass and potential profiles, a situation that goes beyond the scope of classical special functions. In this context, the Heun equation, particularly its confluent forms, emerges as the natural mathematical language for the analysis of such heterogeneous physical systems. Consequently, the corresponding wave functions are no longer expressed in terms of simple exponential or trigonometric functions, but rather in terms of special Heun functions, such as the confluent Heun function (HeunC) and the biconfluent Heun function (HeunB).
