Spectral shift functions and Dirichlet-to-Neumann maps.

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh-Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy.

We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy and describe the associated isospectral torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we rely on Weyl-Titchmarsh theory for singular (canonical) Hamiltonian systems. We also briefly discuss real-valued algebro-geometric solutions with a cusp behaviour. While we focus primarily on the case of stationary algebro-geometric CH solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.