Finite-gap Potential, Heun’s Differential Equation and Wkb Analysis

with the condition γ+ δ+ ǫ = α+β+1 [12]. It has four singularities {0, 1, t,∞} and they are all regular. Heun’s equation is known to be a standard form of the secondorder Fuchsian differential equation with four singularities. The parameter q is not determined by the local monodromy, and is called an accessory parameter. Heun’s differential equation frequently appears in Physics, i.e. black hole (general relativity, Kerr’s solution), crystalline materials [13], fluid dynamics [3], quantum mechanics (Inozemtsev model [11]) and so on. A standard form of the second-order Fuchsian differential equation with three singularities is given by the hypergeometric differential equation