Bridging supersymmetric quantum mechanics, Heun's equation and the asymptotic iteration method

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The proposed research program aims to bridge three areas of mathematical physics: Supersymmetric quantum mechanics, Heun's differential equations, and the asymptotic iteration method. By bridging, I mean finding, establishing, and solidifying connections and links among these three disciplines. The importance of this research program comes from the applications of its findings in a number of different areas of mathematical physics. Indeed, the recent developments in nanotechnology have renewed interest in the modeling of spatially confined quantum systems under the influence of a wide variety of confining potentials. The findings of the present program are expected to contribute substantially in this direction. Further, the recent applications of Heun's different equations in general relativity, quantum-, plasma-, atomic-, molecular-, and nano-physics indicates the need for a fresh and comprehensive approach to the study of this class of equations. In order to achieve this bridging, the plan is to implement supersymmetric quantum mechanics techniques in the study of Heun's differential equations and to analyze their solutions further by using the asymptotic iteration method. The purpose is to span these three topics in as unified and concise way as possible in order to gain a better understanding of the theory of supersymmetric quantum mechanics.