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
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.