Marchenko–Ostrovki Mapping for Periodic Zakharov–Shabat Systems

Abstract

Consider the Zakharov–Shabat (or Dirac) operator T in the Hilbert space L2(R)⊕L2(R), with real periodic vector potential q=(q1, q2)∈H=L2(T)⊕L2(T). The spectrum of T is absolutely continuous and consists of intervals separated by gaps gn=(z−n, z+n), n∈Z. From the Dirichlet eigenvalue mn, n∈Z, of the Zakharov–Shabat equation with Dirichlet boundary conditions at 0, 1, the square of the height of vertical slits on the quasimomentum domain, and the points on these slits, we construct the Marchenko–Ostrovki (vertical slits) mapping for the periodic Zakharov–Shabat systems h:H→ℓ2⊕ℓ2. Using nonlinear functional analysis in Hilbert spaces, we show that this mapping is a real analytic isomorphism. In the second part of our paper we prove a new identity for the effective masses.