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 specDir(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 specDir(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 Hw×Hw 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 μk. 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.
