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We investigate trace formulas for one-dimensional Schroedinger operatorswhich are trace class perturbations of quasi-periodic finite-gap operatorsusing Krein's spectral shift theory. In particular, we establish the conservedquantities for the solutions of the Korteweg-de Vries hierarchy in this classand relate them to the reflection coefficients via Abelian integrals on theunderlying hyperelliptic Riemann surface.

Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds