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 study the singular behavior of kth angular derivatives of analytic functions in the unit disk in the complex plane C and positive harmonic functions in the unit ball in R. Faà di Bruno’s formula is a crucial tool in our proofs.

Let G = Z4. We construct examples of G-equivariant entire rational maps from non-singular real algebraic G-varieties to Grassmannians with appropriate actions of G. These examples of strongly algebraicZ4 vector bundles facilitate a key step in the verification of Conjecture1.1 in the general cyclic group action case.

For a certain class of functional dierential equations with perturbations conditions are given such that there exist solutions which converge to solutions of the equations without perturbation. Let ∆ x(t) = x(t) − x(t − σ), σ > 0 being a constant, and consider the neutral functional diierential equation (A) d n dt n ∆ m x(t) + f ? t, x ? g(t) is the m-th iterate of ∆ , i.e. ∆ m x(t) = m X i=0 (...

We prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum. © 2005 Elsevier Inc. All rights reserved.

We prove that any two unitary operators with simple singular spectrum which contains the whole circle are unitarily equivalent up to a rank one operator.

We consider a singular Sturm-Liouville expression with the indefinite weight sgnx. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgnx)(−d2/dx2 + q) with non-real sp...

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is consider...

This paper is dedicated to approximate controllability for Grushin equation on the rectangle (x, y) ∈ (−1, 1) × (0, 1) with an inverse square potential. This model corresponds to the heat equation for the LaplaceBeltrami operator associated to the Grushin metric on R, studied by Boscain and Laurent. The operator is both degenerate and singular on the line {x = 0}. The approximate controllabilit...