Real Spectral Exclusion for Singular Chiral Sturm–Liouville Operators

Research Note 11 in the "Geometry of the Critical Line" programme. We prove that a class of singular non-self-adjoint Sturm–Liouville operators with chiral (first-order) coupling admits no real eigenvalues in the Friedrichs form domain. The operator family is H^(m) = −d²/dx² + V_m(x) + imA(x)d/dx + imB(x) on a bounded interval with confining singularities at both endpoints, where m is a nonzero integer parameter. The proof proceeds in three steps: (1) a Frobenius analysis classifies local solutions into a regular branch (|ψ| = O(ε^{3/2})) and a singular branch (|ψ| = O(ε^{−1/2})); (2) the Friedrichs form-domain condition (finite kinetic and potential energy) excludes the singular branch; (3) a Wronskian…