MSC: 34L05 34L40 47E05 47B25 47B36 81Q10 Keywords: Schrödinger operator Local point interaction Self-adjointness Lower semiboundedness Discreteness Spectral properties of 1-D Schrödinger operators H X,αOur paper is devoted to the case d * = 0. We consider H X,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of H X,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators H X,α to be self-adjoint, lower semibounded, and discrete in the case d * = 0. The operators with δ -type interactions are investigated too. The obtained results demonstrate that in the case d * = 0, as distinguished from the case d * > 0, the spectral properties of the operators with δ-and δ -type interactions are substantially different.

Abstract. We apply the method of nonlinear steepest descent to compute the longtime asymptotics of the Camassa-Holm equation for decaying initial data, completing previous results by A. Boutet de Monvel and D. Shepelsky.

We develop Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials such as perturbed spherical Schrödinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schrödinger operators.2000 Mathematics Subject Classification. Primary 34B20, 34L05; Secondary 34B24, 47A10.

Abstract. Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form, υ is a non-negative Borel measure on [0, L) and z is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings.

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x) ∈ L 1 (0, 1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular mfunction belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.