Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions

Te Schrödingere quationψ"(x)+κ²ψ(x) = Owhereκ² = k²-V(x) is rewritten as a more popular form of a second order diferential equation by taking a similarity transformation ψ(z) = Ø(z)u(z) with z = z(x). Te Schrodinger invariant I_s{x) can be calculated directly by the Schwarzian derivative {z, x} and the invariant I{z) of the differential equation u_zz + f(z)u_z + g(z)u = 0. We fnd an important relation for a moving particle as ∇² = -I_s(x) and thus explain the reason why the Schrödinger invariant l_s(x) keeps constant. As an illustration, we take the typical Heun's differential equation as an object to construct a class of soluble potentials and generalize the previous results by taking diferent transformation p = z'(x) as before. We get a more general solution z{x) through integrating (z')² = α₁z² + β₁z + γ₁ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail. Te results are also compared with those obtained by Bose, Lemieux, Batic, Ishkhanyan, and their coworkers. It should be recognized that a subtle and diferent choice of the transformation z(x) also related to ρ will lead to difcult connections to the results obtained from other diferent approaches.