Chiral Spectral Confinement and Parity-Resolved Evans-Zero Asymptotics for Singular Sturm–Liouville Problems

Abstract

We study singular non-self-adjoint Sturm–Liouville differential expressions on a bounded interval, -ψ'' + Vψ + im(Aψ' + Bψ), m a nonzero integer, with real V, A, B. Finite-energy endpoint selection annihilates the boundary flux and yields Im λ = -(m/2)⟨A' - 2B⟩_ψ for every two-sided finite-energy spectral value. Consequently, sign-definite chirality confines all corresponding Evans zeros to the sector-selected half λ-plane; for the affine profile g(δ) = kδ this sharpens to Im λ < -2mk when m > 0, with the conjugate statement for m < 0. The affine equation reduces exactly to the standard angular spheroidal differential form with complex order and complex characteristic parameters. Reflection factors the Evans condition into disjoint even/Neumann and odd/Dirichlet half-interval channels. A half-interval Whittaker–Volterra transfer and two-scale Rouché analysis then produce two interleaved captured sequences, for each fixed nonzero winding and every odd sign-positive C³ profile with simple endpoint degeneration: writing κ = g'(δ₀) and L = 2δ₀, their union obeys Λₙ = (π/L)(n + 1/2 + ν) + O(log n / n) with λ = Λ², where ν = ν(m,κ) is the explicit complex endpoint index, the negative-winding law following by conjugation. Squaring yields a captured-branch complex Weyl-type law with asymptotically parabolic geometry in the λ-plane, and the captured depth obeys an explicit law decreasing to π/(κL) per unit winding, recovering π/2 in the affine case.