Factorization of the Heun’s differential operator

摘要

The differential equation with four regular singularities located at z=0,1,aand∞, called the Heun’s equation (HE), isy″(z)+γz+δz−1+εz−ay′(z)+αβz−qz(z−1)(z−a)y(z)=0with α+β+1=γ+δ+ε, and defines the Heun’s operator H byH[y(z)]={P3(z)D2+P2(z)D+P1(z)}[y(z)]with D≡d/dz and Pi(z) polynomials of degree i.H can be factorized in the formH=[L(z)D+M(z)][L(z)D+M(z)]Polynomials L, L, M and M are given explicitly in the cases where this factorization is possible. It is shown that the value of the parameters α, β and q allowing the factorization coincides with those obtained from the F-homotopic transformation: y(z)=zρ(z−1)σ(z−a)τỹ(z) forcing ỹ(z) to be solution of a HE as y(z).