A Frobenius‐Normalized Characteristic Determinant Method for Singular Fourth‐Order Radial Boundary Spectral Problems

ABSTRACT

At a singular endpoint, a determinant written in special‐function coordinates has spectral meaning only after the operator domain has been fixed. We address this issue for a specified fourth‐order radial expression motivated by higher‐order hyperbolic problems. In the natural weighted Hilbert space, resonant Levelt–Frobenius analysis and the Green form identify a two‐dimensional maximal/minimal boundary quotient with graph‐continuous traces. A fixed real projective condition at the center, together with clamped data at the regular endpoint, defines a self‐adjoint realization with compact resolvent. Its complete four‐row boundary determinant reduces exactly to a two‐column determinant on the center‐admissible plane. For the regular‐center realization, the recursive Frobenius determinant is times the characteristic determinant. An exact‐rational certificate excludes all earlier positive zeros and proves that the first positive zero is simple and lies in ; hence, the first positive eigenvalue lies in . The construction provides a reproducible route from a singular differential expression to a certified eigenvalue while keeping the endpoint realization, coordinate normalization, and computer‐assisted inequalities logically separate.