Article

**in Spectral Theory and Analysis, J. Janas (ed.) et al., 107-124, Oper. Theory Adv. Appl. 214, Birkhäuser, Basel, 2011**[DOI: 10.1007/978-3-7643-9994-8_7]

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

### Alice Mikikits-Leitner and Gerald Teschl

We investigate trace formulas for one-dimensional Schrödinger operators which are
trace class perturbations of quasi-periodic finite-gap operators using Krein's
spectral shift theory. In particular, we establish the conserved quantities for the
solutions of the Korteweg-de Vries hierarchy in this class and relate them to
the reflection coefficients via Abelian integrals on the underlying hyperelliptic
Riemann surface.

** MSC2000:** Primary 34L25, 35Q53; Secondary 81U40, 37K15

**Keywords:** *Scattering, KdV hierarchy, Trace formulas*

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