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Basic Behavior of the Spectral Data

  • Sebastian Klein
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2229)

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 D0 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, (λk, μk) ∈ D is near (λk,0, μk,0) ∈ D0. Similarly, we will show that the set of zeros of Δ2 − 4 with multiplicities (corresponding to the branch points resp. singularities of the spectral curve Σ) is enumerated by two \(\mathbb {Z}\)-sequences (ϰk,1) and (ϰk,2) such that for |k| large, ϰk,1 and ϰk,2 are near λk,0.

References

  1. [Ki-S-S]
    M. Kilian, M. Schmidt, N. Schmitt, Flows of constant mean curvature tori in the 3-sphere: the equivariant case. J. Reine Angew. Math. 707, 45–86 (2015)MathSciNetzbMATHGoogle Scholar
  2. [Pö-T]
    J. Pöschel, E. Trubowitz, Inverse Spectral Theory (Academic Press, London, 1987)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sebastian Klein
    • 1
  1. 1.School of Business Informatics & MathematicsUniversity of MannheimMannheimGermany

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