What question did this study set out to answer?

This research aims to explore the connection matrix and its spectral implications related to the open chiral SCT operator.

March 28, 2026Open Access

The Connection Matrix of the Open Chiral SCT Operator and Its Spectral Consequences

Key Points

This research aims to explore the connection matrix and its spectral implications related to the open chiral SCT operator.
Constructed the full 2×2 connection matrix mapping data from left-endpoints to right-endpoints.
Proved the involution theorem M² = I and derived properties such as tr(M) = 0 and det(M) = −1.
Analyzed chiral symmetry and confined spectral conditions within the connection matrix.
Conducted numerical reconnaissance to investigate the real axis properties of the connection matrix.
Confirmed M₁₁ ≠ 0 and M₂₁ ≠ 0 on the real axis.
Identified the eigenspace transport as significantly r₁-dominated.
Established the real axis as spectrally inert in the tested regime.

Abstract

Paper 41 in the Geometry of the Critical Line programme. Constructs the full 2×2 connection matrix M(λ,m) mapping left-endpoint Frobenius data to right-endpoint data for the open chiral SCT operator. Proves the involution theorem M² = I from the δ → −δ reflection symmetry, derives tr(M) = 0 and det(M) = −1, proves the chiral symmetry M(−m, λ̄) = M̄(m, λ), and identifies the confined spectral condition as M₂₁ = 0. Numerical reconnaissance shows M₁₁ ≠ 0 and M₂₁ ≠ 0 on the real axis, with eigenspace transport overwhelmingly r₁-dominated (|ρ±| ≈ 1.47 × 10⁻³). The real axis is spectrally inert in the tested regime.

The Connection Matrix of the Open Chiral SCT Operator and Its Spectral Consequences

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Abstract

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