A sequentially computable function that is not effectively continuous at any point

P. Hertling [Lecture Notes in Computer Science, vol. 2380, Springer, Berlin, 2002, pp. 962–972; Ann. Pure Appl. Logic 132 (2005) 227–246] showed that there exists a sequentially computable function mapping all computable real numbers to computable real numbers that is not effectively continuous. Here, that result is strengthened: a sequentially computable function on the computable real numbers is constructed that is not effectively continuous at any point.     

The spectrum is continuous on the set of quasi-n-hyponormal operators

In this paper it is shown that the spectrum σ, a set-valued function, is continuous when the function is restricted to the set of all ‘quasi-n-hyponormal’ operators acting on an infinite-dimensional separable Hilbert space, where a quasi-n-hyponormal operator is defined to be unitarily equivalent to an n×n upper triangular operator matrix whose diagonal entries are hyponormal operators.     

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^?1/4?ε; in the smoothness direction, a sufficient condition in Hölder classes isq∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^?1/4 and belong toC^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.     

Absolutely continuous spectrum for sparse potentials

In this paper we show that bounded perturbation of the discrete laplacian with a potential that is sparsely supported (a notion made precise in the paper) produces absolutely continuous spectrum in the interval [?2ν, 2ν] for large dimension ν. We note that the potential need not give rise to a compact operator, let alone have decay at infinity.     

Jacobi matrices with absolutely continuous spectrum

Let be a Jacobi matrix defined in as , where is a unilateral weighted shift with nonzero weights such that Define the seqences: If and , then has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution is also given.

     

Floquet operators without singular continuous spectrum

Let U be a unitary operator defined on a infinite-dimensional separable complex Hilbert space H. Assume there exists a self-adjoint operator A on H such that

U^∗AU−A?cI+K     

Substitution-based structures with absolutely continuous spectrum

By generalising Rudin’s construction of an aperiodic sequence, we derive new substitution-based structures which have a purely absolutely continuous diffraction measure and a mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.     

Completeness is determined by any non-algebraic trajectory

It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.     

Absolutely continuous spectrum of perturbed Stark operators

We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate or if it is continuously differentiable with derivative from the Hölder space with any

     

Absolutely continuous spectrum and spectral transition for some continuous random operators

In this paper we consider two classes of random Hamiltonians on $L^2({\mathbb R}^d)$ : one that imitates the lattice case and the other a Schr?dinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.     

Bifurcation of hexagonal patterns in a chemotaxis-diffusion-growth system

In this paper, we study a chemotaxis-diffusion-growth system in a rectangular domain by applying the center manifold theory. It is observed that the trivial solutions are destabilized due to the chemotaxis term. As a result, we obtain the normal form on the center manifold, and it is proved that the locally asymptotically stable hexagonal patterns exist.     

Absence of the absolutely continuous spectrum of a first-order non-selfadjoint Dirac-like system for slowly decaying perturbations

We prove that the absolutely continuous subspace of the completely non-selfadjoint part of a first-order dissipative Dirac-like system is trivial when the imaginary part of the potential is non-integrable.     

Dynamical systems disjoint from any minimal system

Furstenberg showed that if two topological systems and are disjoint, then one of them, say , is minimal. When is nontrivial, we prove that must have dense recurrent points, and there are countably many maximal transitive subsystems of such that their union is dense and each of them is disjoint from . Showing that a weakly mixing system with dense periodic points is in , the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in . We show that a weakly mixing system with dense regular minimal points is in , and each system in has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in and having no periodic points are constructed. Moreover, we show that there is a distal system in .

Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.

     

The pascal automorphism has a continuous spectrum

In this paper we describe the Pascal automorphism and present a sketch of the proof that its spectrum is continuous on the orthogonal complement of the constants.     

Anisotropic Jacobi matrices with absolutely continuous spectrum

We describe a class of anisotropic Jacobi matrices with absolutely continuous spectrum and discrete singular spectrum. The discrete Schrödinger operator perturbed by a potential having different behaviors near +∞ and -∞, belongs to this class. We use the method of the conjugate operator and techniques similar to those for the 3-body problem.