A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation by Sebastian Klein

Author:Sebastian Klein
Language: eng
Format: epub, pdf
ISBN: 9783030012762
Publisher: Springer International Publishing

(12.8)

holds.

To show that b is holomorphic in λ, we note that by (12.4), has at λ ∗ a zero of order m (with respect to λ), so we can represent d in the form with a holomorphic function germ ζ in λ. We now calculate:

(12.9)

Because is obviously holomorphic, it remains to show that is holomorphic. For this purpose, we continue to calculate:

(12.10)

Δ 2 − 4 has (with respect to λ) a zero of order , whereas (λ − λ ∗)m has by (12.8) a zero of order at most . Therefore is holomorphic in λ. Moreover, we have

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