Describe Dubrovin-cycle dynamics for PT-symmetric periodic Schrödinger operators

Determine and characterize the cycles on the hyperelliptic spectral curve Γ defined by w^2 = R(E) = (E - E_1)⋯(E - E_{2g+1}) along which the variables γ_k(x) evolve when they satisfy the Dubrovin equations γ'_k(x) = -2i√{R(γ_k)}/∏_{j≠k}(γ_k - γ_j) for PT-symmetric periodic Schrödinger operators L = -d^2/dx^2 + u(x) with u(x) = ar{u}(-x). Provide a concrete description of these cycles sufficient to enable application of the Dubrovin approach to PT-symmetric finite-gap potentials.

Background

For periodic Schrödinger operators with finite-gap potentials, Bloch functions are parameterized by points on a hyperelliptic spectral curve Γ: w^2 = R(E). The Dubrovin variables γ_k(x), governed by Dubrovin’s equations, play a central role in the algebro-geometric integration and the inverse spectral problem.

In the real-valued case, γk(x) evolve above spectral gaps [E{2k}, E_{2k+1}]. For complex-valued and PT-symmetric potentials, the γ_k(x) do not lie above these lacunae and instead evolve along more complicated cycles on Γ. The paper notes that a description of these cycles is unknown even in the PT-symmetric setting, hindering direct application of the Dubrovin approach to PT-potentials.