# Basic Behavior of the Spectral Data

## Abstract

We would like to show that the spectral data \((\varSigma ,\mathcal {D})\) corresponding to a potential behave like the spectral data \((\varSigma _0,\mathcal {D}_0)\) of the vacuum (as described in Chap. 4) “asymptotically” for *λ* near *∞*, and for *λ* near 0. In particular, we would like to show that the classical spectral divisor *D* corresponding to \(\mathcal {D}\), like the classical spectral divisor *D*_{0} of the vacuum, is composed of a \(\mathbb {Z}\)-sequence of points \((\lambda _k,\mu _k)_{k\in \mathbb {Z}}\), and that for |*k*| large, (*λ*_{k}, *μ*_{k}) ∈ *D* is near (*λ*_{k,0}, *μ*_{k,0}) ∈ *D*_{0}. Similarly, we will show that the set of zeros of *Δ*^{2} − 4 with multiplicities (corresponding to the branch points resp. singularities of the spectral curve *Σ*) is enumerated by two \(\mathbb {Z}\)-sequences (ϰ_{k,1}) and (ϰ_{k,2}) such that for |*k*| large, ϰ_{k,1} and ϰ_{k,2} are near *λ*_{k,0}.

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