The Self-Adjoint Fractional Heun Operator and Its Spectral Properties

This paper introduces a rigorously defined fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives. By deriving a compatible fractional Pearson-type equation, a new weight function and Hilbert space setting are established, ensuring the operator’s self-adjointness under natural fractional boundary conditions. Within this framework, we prove the existence of a real, discrete spectrum and demonstrate that the corresponding eigenfunctions form a complete orthogonal system in Lωα2(a,b). The central theoretical result shows that the fractional eigenpairs (λn(α),un(α)) converge continuously to their classical Heun counterparts (λn(1),un(1)) as α→1−. This provides a rigorous analytic bridge between fractional and classical spectral theories. A numerical study based on the fractional Legendre case confirms the predicted self-adjointness and spectral convergence, illustrating the smooth deformation of the classical eigenfunctions into their fractional counterparts. The results establish the fractional Heun operator as a mathematically consistent generalization capable of generating new families of orthogonal fractional functions.