Algebraization of the Heun Class of Equations: Solutions and Physical Applications

Kar, Priyasri (2022) Algebraization of the Heun Class of Equations: Solutions and Physical Applications. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

This thesis explores new algebraic techniques intending to widen the spectrum of global solutions to the celebrated Heun class of equations, which appear diversely in different branches of physics. Starting with a review of the known su(1, 1) structure of the Heun class that leads to polynomial solutions, the equations have been cast in terms of two new bi-parametric sets of su(1, 1) generators. As a central requirement, the complete su(1, 1) representation theory has been worked out in detail. The use of the new generators has resulted in classes of quasipolynomial type global solutions, in addition to the polynomial solutions that are obtained exploiting the previously known su(1, 1) structure of the Heun class. These solutions are called additional solutions in this thesis. Conditions have been identified for which an equation of the Heun class is quasi-exactly or exactly solvable. Rare situations, where a Heun class equation does not possess any su(1, 1) structure, have also been identified. It may be interesting to note that, even though rare, situations may occur when an equation of the Heun class can not be directly (without any transformation) algebraized using the su(1, 1) generators that were already in the literature, however, it is possible with the use of the generators introduced here. The findings of these new mathematical constructs have then been applied to the physics problem of a massless Dirac particle in the C-metric. The radial and polar parts of the corresponding Dirac equation in this metric, which represents the exterior gravitational field of a uniformly accelerating black hole, turn out to be the General Heun Equations up to some variable transformations. Making use of the two newly found su(1, 1) structures of the Heun class, classes of additional quasi-polynomial type solutions to the radial and polar parts have been obtained.

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