We obtain retarded Green's functions for massless scalar fields in the background of near-extreme, nearhorizon rotating charged black holes of five-dimensional minimal gauged supergravity. The radial part of the (separable) massless Klein-Gordon equation in such general black hole backgrounds is Heun's equation, due to the singularity structure associated with the three black hole horizons. On the other hand, we find the scaling limit for the near-extreme, near-horizon background where the radial equation reduces to a hypergeometric equation whose SL(2,R) 2 symmetry signifies the underlying two-dimensional conformal invariance, with the two sectors governed by the respective Frolov-Thorne temperatures. We obtain retarded Green's functions for massless scalar fields in the background of near-extreme, near-horizon rotating charged black holes of five-dimensional minimal gauged supergravity. The radial part of the (separable) massless Klein-Gordon equation in such general black hole backgrounds is Heun's equation, due to the singularity structure associated with the three black hole horizons. On the other hand, we find the scaling limit for the near-extreme, near-horizon background where the radial equation reduces to a hypergeometric equation whose SLð2; RÞ 2 symmetry signifies the underlying two-dimensional conformal invariance, with the two sectors governed by the respective Frolov-Thorne temperatures.

We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. As a new example we find that while the Dirac equation written in the background of Nutku helicoid metric yields Mathieu functions as its solutions in four spacetime dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations.

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We study the solutions of the Dirac equation in the background of the Nutku helicoid metric. This metric has curvature singularities, which necessitates imposing a boundary to exclude this point. We use the AtiyahPatodi-Singer non local spectral boundary conditions for both the four and the five dimensional manifolds.

We present an exact solution of the one-dimensional Bosonic oscillator for spin 1 and spin 0 particles with the Snyder-de Sitter model, where the energy eigenvalues and eigenfunctions are determined for both cases. The wave functions can be given in terms of Gegenbauer polynomials. We also comment on the thermodynamic properties of the system.

The Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an Appendix which gives a new example wherein one encounters a form of the Heun equation.

The problem of a particle of mass m in the field of the inverse square potential α/r 2 is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.