| _version_ | 1866901699320348672 |
|---|---|
| author | Kramarenko-Byrd, Pavel |
| author_facet | Kramarenko-Byrd, Pavel |
| contents | <p>Research Note 11 in the "<em>Geometry of the Critical Line</em>" programme.</p> <p>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.</p> <p>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 identity, with all boundary terms vanishing for the regular branch, yields 0 = ∫(A′−2B)|ψ|² > 0, a contradiction.</p> <p>The positivity condition A′−2B > 0 holds for any cross-coupling g with g′ > 0. The spectral gap grows linearly with |m|. No perturbation theory is used.</p> <p>The result is applied to the Symmetric Complex Transcendental (SCT) 5-manifold family, where it provides the spectral exclusion theorem used in the conditional Riemann Hypothesis reduction of Paper 40 (DOI: 10.5281/zenodo.19230354).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19234136 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Real Spectral Exclusion for Singular Chiral Sturm–Liouville Operators Kramarenko-Byrd, Pavel Sturm-Liouville operators non-self-adjoint operators spectral exclusion Friedrichs extension form domain Frobenius analysis Wronskian identity chiral coupling singular boundary spectral gap operator theory Riemann hypothesis SCT manifold Symmetric Complex Transcendental Spectral Theory Operator Theory Differential equations Mathematical physics <p>Research Note 11 in the "<em>Geometry of the Critical Line</em>" programme.</p> <p>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.</p> <p>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 identity, with all boundary terms vanishing for the regular branch, yields 0 = ∫(A′−2B)|ψ|² > 0, a contradiction.</p> <p>The positivity condition A′−2B > 0 holds for any cross-coupling g with g′ > 0. The spectral gap grows linearly with |m|. No perturbation theory is used.</p> <p>The result is applied to the Symmetric Complex Transcendental (SCT) 5-manifold family, where it provides the spectral exclusion theorem used in the conditional Riemann Hypothesis reduction of Paper 40 (DOI: 10.5281/zenodo.19230354).</p> |
| title | Real Spectral Exclusion for Singular Chiral Sturm–Liouville Operators |
| topic | Sturm-Liouville operators non-self-adjoint operators spectral exclusion Friedrichs extension form domain Frobenius analysis Wronskian identity chiral coupling singular boundary spectral gap operator theory Riemann hypothesis SCT manifold Symmetric Complex Transcendental Spectral Theory Operator Theory Differential equations Mathematical physics |
| url | https://doi.org/10.5281/zenodo.19234136 |