A Singular Sturm-Liouville Framework for Heun-Type Orthogonality and Eigenfunction Expansions

Heun functions arise as local solutions of the general second-order Fuchsian differential equation with four regular singular points, but such local solutions do not, in general, form an orthogonal basis. In this paper, we develop a singular Sturm–Liouville framework in which Heun-type solutions can be realized as orthogonal eigenfunctions suitable for spectral expansions. The accessory parameter is interpreted as a spectral parameter, and the general Heun equation is transformed into a formally self-adjoint weighted Sturm–Liouville problem. The singular endpoints are analyzed through Weyl’s limit-point/limit-circle classification, which determines the admissible boundary data for self-adjoint realizations. Within an explicit parameter regime, the resulting operator has compact resolvent, leading to a discrete spectrum and a complete orthogonal system in the associated weighted Hilbert space. The corresponding normalization and eigenfunction expansion formulas are then established within this spectral setting. The proposed framework provides a parameter-specific mathematical foundation for differential equations reducible to Heun form and is intended to support future analytical and spectral applications in wave propagation, quantum models, plasma-equilibrium theory, and tokamak-related configurations.