Physical applications of the heun equations with polynomial reduction cases

The scientific community started to encounter the Heun-type solutions more often by the end of the 20th century. Thus, the hypergeometric-type equations that were generaly studied in the mathematical formulation of the limiting cases of the physical phenomena changed by the analysis of the full problem via Heun equations. The mathematical theory of the hypergeometric equation which has a two-term recursion relation and three regular singularities is fully understood. However, the theory of the general Heun equation which has has four regular singularities and a three-term recursion relation is not well-studied. The physical applications of the Heun equations are diverse. Various quantum mechanical problems can be solved in terms of the general and confluent forms of the Heun functions. General relativity applications also play an important role in understanding the various exact solutions of the Einstein equations. Analysis of the Heun-type equations can reduce to finding polynomial solutions if the parameters satisfy some certain conditions. These conditions can put constraints on the physical parameters or yield the energy condition for quantum mechanical systems. Three problems from quantum mechanics and a problem from black hole physics are studied using the polynomial reduction method. Energy spectrum is obtained for the quantum mechanical problems. Some parameter constraints are found for the black hole system. Maple and SageMath codes are given for studying the theorem that yields the energy spectrum easily.