The two-dimensional Hamiltonian system (∗) y′(x) = zJH(x)y(x), x ∈ (a,b), where the Hamiltonian H takes non-negative 2× 2-matrices as values, and J := ( 0 −1 1 0 ) , has attracted a lot of interest over the past decades. Special emphasis has been put on operator models and direct and inverse spectral theorems. Weyl theory plays a prominent role in the spectral theory of the equation, relating t...

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 (cano...

Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl–Titchmarsh–Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one...

We give a comprehensive treatment of Sturm–Liouville operators whose coefficients are measures including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl–Titchmarsh–Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm– Liouville operators,...

We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type A = P 06|α|,|β|6m(−1) Daα,β(x)D β , aα,β(·) ∈ C ∞(Ω), on smooth (bounded or unbounded) domains Ω in R with compact boundary ∂Ω. Using the concept of boundary triples and operator-valued Weyl–Titchmarsh functions, we prove various trace ideal properties of powers of resolvent differences of ...

We discuss direct and inverse spectral theory for a Sturm–Liouville type problem with a quadratic dependence on the eigenvalue parameter, −f ′′ + 1 4 f = z ωf + zυf, which arises as the isospectral problem for the conservative Camassa–Holm flow. In order to be able to treat rather irregular coefficients (that is, when ω is a real-valued Borel measure on R and υ is a non-negative Borel measure o...