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.