# Asymptotics of Spectral Data for Potentials on a Horizontal Strip

## Abstract

As a final result, we study the asymptotic behavior of the spectral data (*Σ*, *D*) corresponding to a simply periodic solution \(u: X \to \mathbb {C}\) of the sinh-Gordon equation defined on an entire horizontal strip \(X \subset \mathbb {C}\) with positive height. Because such a solution is real analytic on the interior of *X*, we expect a far better asymptotic for such spectral data than for the spectral data of Cauchy data potentials (*u*, *u*_{y}) with only the weak requirements *u* ∈ *W*^{1, 2}([0, 1]), *u*_{y} ∈ *L*^{2}([0, 1]) we have been using throughout most of the paper. More specifically, we expect both the distance of branch points ϰ_{k,1} − ϰ_{k,2} of the spectral curve *Σ* and the distance of the corresponding spectral divisor points to the branch points to fall off exponentially for *k* →±*∞*.