Estimates on periodic and Dirichlet eigenvalues for the Zakharov–Shabat system

Consider the 2×2 first order system due to Zakharov–Shabat, $LY:={\rm i}\left(\matrix{1&0\cr 0&-1}\right)Y'+\left(\matrix{0&\psi_{1}\cr \psi _{2}&0}\right)Y=\lambda Y$ with ψ₁,ψ₂ being complex valued functions of period one in the weighted Sobolev space $H^{w}\equiv H^{w}_{\mathbb{C}}.$ Denote by spec(ψ₁,ψ₂) the set of periodic eigenvalues of L(ψ₁,ψ₂) with respect to the interval [0,2] and by spec_Dir(ψ₁,ψ₂) the set of Dirichlet eigenvalues of L(ψ₁,ψ₂) when considered on the interval [0,1]. It is well known that spec(ψ₁,ψ₂) and spec_Dir(ψ₁,ψ₂) are discrete.

Theorem. Assume that w is a weight such that, for some δ>0, w_−δ(k)=(1+|k|)^−δw(k) is a weight as well. Then for any bounded subset ${\mathbb{B}}$ of 1‐periodic elements in H^w×H^w there exist N≥1 and M≥1 so that for any |k|≥N, and $(\psi_{1},\psi_{2})\in{\mathbb{B}} $ , the set $\mathit{spec}(\psi_{1},\psi_{2})\cap \{\lambda \in{\mathbb{C}} \mid |\lambda -k\pi | < \pi/2\}$ contains exactly one isolated pair of eigenvalues {λ⁺_k,λ⁻_k} and $\mathit{spec}_{\rm Dir}(\psi_{1},\psi_{2})\cap \{\lambda \in{\mathbb{C}} \mid |\lambda -k\pi |<{\pi}/{2}\}$ contains a single Dirichlet eigenvalue μ_k. These eigenvalues satisfy the following estimates

(i) Σ_|k|≥Nw(2k)²|λ⁺_k−λ⁻_k|²≤M;

(ii) $\sum _{|k|\geq N}w(2k)^{2}|\frac{(\lambda ^{+}_{k}+\lambda ^{-}_{k})} {2}-\mu _{k}|^{2}\leq M.$

Furthermore spec(ψ₁,ψ₂)\{λ^±_k,|k|≥N} and $\mathit{spec}_{\rm Dir}(\psi_{1},\psi_{2})\backslash \{\mu _{k}\mid |k|\geq N\}$ are contained in $\{\lambda \in{\mathbb{C}} \mid |\lambda | < N\pi -\pi /2\}$ and its cardinality is 4N−2, respectively 2N−1.

When $\psi_{2}=\overline{\psi } _{1}$ (respectively $\psi_{2}=-\overline{\psi } _{1}),L(\psi_{1},\psi_{2})$ is one of the operators in the Lax pair for the defocusing (resp. focusing) nonlinear Schrödinger equation.