Global ordering of Dirichlet eigenvalues

Determine whether the Dirichlet eigenvalues μ_j associated with the periodic non-self-adjoint Zakharov–Shabat operator L defined by L = iσ3(∂x − Q) with Q(x) as in equation (2.4) are globally lexicographically nondecreasing, i.e., establish whether μ_j ≼ μ_{j+1} holds for all j ∈ Z. The known results guarantee μ_j ≼ μ_{j+1} only for sufficiently large |j|, leaving the general case unresolved.

Background

In the development of spectral data for the periodic non-self-adjoint Zakharov–Shabat problem, the authors introduce a lexicographic ordering on complex numbers and organize the main spectrum and the Dirichlet spectrum accordingly. While asymptotic results and counting lemmas ensure that, beyond a large index threshold J, the Dirichlet eigenvalues exhibit a nondecreasing ordering, the authors explicitly note that they cannot assert the same ordering property for all indices.

This unresolved issue concerns the global monotonicity (in the lexicographic sense) of the Dirichlet eigenvalues μ_j across all j ∈ Z, which impacts the structure and potential simplifications in the inverse spectral formulation and the Riemann–Hilbert problem for periodic focusing NLS under the presented framework.